Biggs-Smith Graph

Author

Eric W. Weisstein
June 19, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/Biggs-SmithGraph.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Biggs-SmithGraph.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Embeddings

Names

GraphData["BiggsSmithGraph","StandardName"]

GraphData["BiggsSmithGraph","StandardNames"]

Primary

GraphData["BiggsSmithGraph"]//StyleGraphs[#,ImageSize->Small]&

Construction from Read and Wilson

g=Graph[Range[102],UndirectedEdge@@@{{1,2},{1,3},{1,4},{2,7},{2,9},{3,8},{3,13},{4,6},{4,24},{5,7},{5,36},{5,55},{6,33},{6,58},{7,25},{8,11},{8,27},{9,12},{9,19},{10,12},{10,18},{10,80},{11,16},{11,81},{12,29},{13,14},{13,26},{14,18},{14,34},{15,22},{15,23},{15,24},{16,21},{16,89},{17,20},{17,27},{17,30},{18,91},{19,21},{19,41},{20,25},{20,38},{21,39},{22,34},{22,45},{23,29},{23,49},{24,31},{25,32},{26,28},{26,53},{27,35},{28,36},{28,44},{29,37},{30,44},{30,59},{31,70},{31,71},{32,68},{32,71},{33,42},{33,95},{34,40},{35,67},{35,70},{36,96},{37,64},{37,68},{38,39},{38,66},{39,43},{40,61},{40,67},{41,42},{41,63},{42,47},{43,57},{43,64},{44,46},{45,51},{45,72},{46,54},{46,61},{47,48},{47,49},{48,50},{48,57},{49,79},{50,51},{50,54},{51,52},{52,53},{52,58},{53,76},{54,56},{55,62},{55,100},{56,62},{56,66},{57,60},{58,100},{59,60},{59,82},{60,74},{61,65},{62,63},{63,85},{64,69},{65,79},{65,83},{66,84},{67,73},{68,75},{69,72},{69,88},{70,77},{71,78},{72,86},{73,84},{73,92},{74,76},{74,81},{75,82},{75,94},{76,90},{77,87},{77,97},{78,86},{78,98},{79,87},{80,83},{80,96},{81,95},{82,87},{83,85},{84,86},{85,93},{88,90},{88,91},{89,91},{89,92},{90,99},{92,93},{93,101},{94,96},{94,99},{95,97},{97,101},{98,100},{98,102},{99,102},{101,102}},VertexCoordinates->(10^(-4)){{3285,14863},{3724,10119},{4588,19445},{4612,14740},{4641,11767},{4735,16068},{5006,10484},{5781,18851},{5847,5855},{6107,7723},{6375,20045},{6910,6658},{7458,23247},{8356,22262},{8827,13010},{8934,4481},{9086,15715},{9341,23160},{9366,2645},{9561,10394},{10068,3779},{10305,18143},{11190,8221},{11249,14125},{11417,12309},{11507,25755},{11747,15879},{11988,24512},{12229,10677},{12319,19966},{12576,14002},{12699,12674},{12741,2480},{12846,17334},{12941,15285},{13231,24993},{13293,11480},{13494,6782},{13577,9448},{13744,16349},{13807,924},{14052,2235},{14278,10582},{14396,18294},{14855,20939},{14877,17051},{15276,8789},{15521,10100},{16162,6274},{16188,17296},{16188,18629},{16188,25297},{16188,26631},{16854,10100},{17014,1990},{17099,8789},{17498,17051},{17521,25297},{17571,20930},{17980,18294},{18097,10582},{18324,2235},{18569,924},{18632,16349},{18799,9448},{18834,6764},{19082,11480},{19435,15285},{19530,17334},{19677,12674},{19800,14002},{20100,19940},{20146,10677},{20387,24512},{20628,15879},{20869,25755},{20959,12309},{21127,14125},{21148,8186},{21175,3077},{21630,24030},{22101,18102},{22308,3779},{22792,10349},{23010,2645},{23303,15666},{23544,12960},{24020,22262},{24662,5594},{24918,23247},{25005,21363},{25465,6658},{26529,5855},{26595,18851},{27005,9202},{27189,17657},{27370,10484},{27764,14740},{27788,19445},{27887,13412},{28652,10119},{29091,14863}}]

All

GraphData["BiggsSmithGraph","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["BiggsSmithGraph","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["BiggsSmithGraph","EdgeList"]]

Default embedding

GraphPlot@GraphData["BiggsSmithGraph","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["BiggsSmithGraph","Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData["BiggsSmithGraph","Graph","Degenerate"]

Bilateral

GraphData["BiggsSmithGraph","Graph","Bilateral"]//StyleGraphs

Circular

GraphData["BiggsSmithGraph","Graph","Circular"]//StyleGraphs

Planar

GraphData["BiggsSmithGraph","Planar"]

GraphData["BiggsSmithGraph","Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData["BiggsSmithGraph",{"RCN","CN"}]

GraphData["BiggsSmithGraph","Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

(cn=CrossingNumber[GraphData["BiggsSmithGraph"],TimeConstraint->10])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BiggsSmithGraph"],TimeConstraint->60])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BiggsSmithGraph"],TimeConstraint->600])//AbsoluteTiming

(cn=CrossingNumber[GraphData["BiggsSmithGraph"],TimeConstraint->3600])//AbsoluteTiming

(g=MinimalCrossingGraph[GraphData["BiggsSmithGraph"],"CrossingNumber"->84])//PolyhedralGraphQ

faces=ReverseSort@SymmetricallyDistinctFaces[g]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[g,"OuterFace"->#]&/@faces

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["BiggsSmithGraph"],"CrossingNumber"->cn,RandomSeed->n]],{n,50}],Max[OuterFace[#]]<=GraphData["BiggsSmithGraph","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["BiggsSmithGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,50}]//Union

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["BiggsSmithGraph","Circulant"]

GraphData["BiggsSmithGraph","Graph","Circulant"]//StyleGraphs

Line graph

GraphData["BiggsSmithGraph","Line"]

LineGraphQ[GraphData["BiggsSmithGraph"]]//Timing

RootGraph

Select[GraphData[All],GraphData[#,"LineGraph","Name"]=== "BiggsSmithGraph"&]

RootGraph[GraphData["BiggsSmithGraph"]]

Halin

GraphData["BiggsSmithGraph","Halin"]

GraphData["BiggsSmithGraph","Graph","Halin"]//StyleGraphs

Rigid

GraphData["BiggsSmithGraph","Rigid"]

GraphData["BiggsSmithGraph","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["BiggsSmithGraph","Flexible"]

GraphData["BiggsSmithGraph","Graph","Flexible"]//StyleGraphs

LCF

GraphData["BiggsSmithGraph","LCF"]

GraphData["BiggsSmithGraph","HamiltonianCycleCount"]

GraphData["BiggsSmithGraph","LCFSignature"]

GraphData["BiggsSmithGraph","LCFNotations"]

GraphData["BiggsSmithGraph","Graph","LCF"]//StyleGraphs

LCF compute all

GraphData["BiggsSmithGraph","HamiltonianCycleCount"]

Length[FindHamiltonianCycle[GraphData["BiggsSmithGraph"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["BiggsSmithGraph"],All])[[All,2]]]//Timing

lcf

LCFGraph@@@lcf//StyleGraphs

(gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFGraph@#,#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFGraph@@@Cases[lcf,{_,1}]]}])//Timing

Length[gs]

UnitDistance

GraphData["BiggsSmithGraph","UnitDistance"]

UnitDistanceGraphQ[GraphData["BiggsSmithGraph"],Debug->True]//Timing

GraphData["BiggsSmithGraph","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["BiggsSmithGraph","VertexList"],GraphData["BiggsSmithGraph","EdgeList"]],MaxIterations->500]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["BiggsSmithGraph","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["BiggsSmithGraph"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["BiggsSmithGraph"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"BiggsSmithGraph",l=FindSubgraphUnitDistanceEmbedding[GraphData["BiggsSmithGraph"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["BiggsSmithGraph","Matchstick"]

GraphData["BiggsSmithGraph","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["BiggsSmithGraph","Edges"]]

3D All

GraphData["BiggsSmithGraph","Graph",{"3D","All"}]

3D UnitDistance

GraphData["BiggsSmithGraph","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["BiggsSmithGraph","Polyhedral"]

GraphMatrices

GraphDataMatrices["BiggsSmithGraph"]

Construction

HGraph expansion (17; 3, 5, 6, 7)

Gerbracht

Exact Gerbracht construction involving complicated trig expressions

E. Gerbracht
1/2010

F102 = G (17; 1, 2, 4, 8)

EdA1=Table[{i+1,Mod[i+1,17]+1},{i,0,16}];
EdA2=Table[{17+i+1,17+Mod[i+4,17]+1},{i,0,16}];
EdB1=Table[{2*17+i+1,2*17+Mod[i+2,17]+1},{i,0,16}];
EdB2=Table[{3*17+i+1,3*17+Mod[i+8,17]+1},{i,0,16}];
EdH=Union[
Table[{i,4*17+i},{i,1,17}],
Table[{17+i,4*17+i},{i,1,17}],
Table[{2*17+i,5*17+i},{i,1,17}],
Table[{3*17+i,5*17+i},{i,1,17}],
Table[{4*17+i,5*17+i},{i,1,17}]
];
edges=UndirectedEdge@@@Union[EdA1,EdA2,EdB1,EdB2,EdH];

RecognizeGraph[Graph[edges]]

rA1=Sin[1/2*(Pi-2*k/n*Pi)]/Sin[k*2*Pi/n]/.{n->17,k->1};
rA2=Sin[1/2*(Pi-2*k/n*Pi)]/Sin[k*2*Pi/n]/.{n->17,k->4};
rB1=Sin[1/2*(Pi-2*k/n*Pi)]/Sin[k*2*Pi/n]/.{n->17,k->2};
rB2=Sin[1/2*(Pi-2*k/n*Pi)]/Sin[k*2*Pi/n]/.{n->17,k->8};
xA101= 0;
yA101=rA1;
A1=Table[
{FullSimplify[Re[(xA101+I*yA101)*Exp[I*(i-1)*(2*Pi)/17]]],FullSimplify[Im[(xA101+I*yA101)*Exp[I*(i-1)*(2*Pi)/17]]]},{i,17}];
xA201= 0;
yA201=rA2;
A2=Table[
{FullSimplify[Re[(xA201+I*yA201)*Exp[I*(i-1)*(2*Pi)/17]]],FullSimplify[Im[(xA201+I*yA201)*Exp[I*(i-1)*(2*Pi)/17]]]},{i,1,17}];

P01=(A1[[1]][[1]]-xH101)^2+(A1[[1]][[2]]-yH101)^2-1;
P02=(A2[[1]][[1]]-xH101)^2+(A2[[1]][[2]]-yH101)^2-1;
S1=Solve[{P01,P02}==0,{xH101,yH101}];

xH1201=FullSimplify[xH101/.S1[[1]]];
yH1201=FullSimplify[yH101/.S1[[1]]];
H12=Table[{Re[(xH1201+I*yH1201)*Exp[I*(i-1)*(2*Pi)/17]],Im[(xH1201+I*yH1201)*Exp[I*(i-1)*(2*Pi)/17]]},{i,17}];

X={0,rB1};
Y={0,rB2};
P03=(X[[1]]-x)^2+(X[[2]]-y)^2-1;
 P04=(Y[[1]]-x)^2+(Y[[2]]-y)^2-1;
Rad=Solve[{P03,P04}==0,{x,y}];
rH34=FullSimplify[Sqrt[x^2+y^2]/.Rad[[1]]];

P05=(H12[[1]][[1]]-xH34)^2+(H12[[1]][[2]]-yH34)^2-1;
P06=xH34^2+yH34^2-rH34^2;
S2=Solve[{P05,P06}==0,{xH34,yH34}];

xH3401=xH34/.S2[[1]];
yH3401=yH34/.S2[[1]];
H34=Table[{Re[(xH3401+I*yH3401)*Exp[I*(i-1)*(2*Pi)/17]],Im[(xH3401+I*yH3401)*Exp[I*(i-1)*(2*Pi)/17]]},{i,17}];

P07=(x1-xB1)^2+(y1-yB1)^2-1;
P08=xB1^2+yB1^2-rB1^2;
S3=Solve[{P07,P08}==0,{xB1,yB1}];
T3=S3/.{x1->H34[[1]][[1]],y1->H34[[1]][[2]]};
P09=(x2-xB2)^2+(y2-yB2)^2-1;
P10=xB2^2+yB2^2-rB2^2;
S4=Solve[{P09,P10}==0,{xB2,yB2}];
T4=S4/.{x2->H34[[1]][[1]],y2->H34[[1]][[2]]};

xB101=xB1/.T3[[2]];
yB101=yB1/.T3[[2]];
B1=Table[{Re[(xB101+I*yB101)*Exp[I*(i-1)*(2*Pi)/17]],Im[(xB101+I*yB101)*Exp[I*(i-1)*(2*Pi)/17]]},{i,17}];
xB201=xB2/.T4[[2]];
yB201=yB2/.T4[[2]];
B2=Table[{Re[(xB201+I*yB201)*Exp[I*(i-1)*(2*Pi)/17]],Im[(xB201+I*yB201)*Exp[I*(i-1)*(2*Pi)/17]]},{i,17}];

VRulesA1Final=Table[1+i->A1[[1+Mod[i,17]]],{i,0,16}];
VRulesA2Final=Table[17+1+i->A2[[1+Mod[i,17]]],{i,0,16}];
VRulesB1Final=Table[2*17+1+i->B1[[1+Mod[i,17]]],{i,0,16}];
VRulesB2Final=Table[3*17+1+i->B2[[1+Mod[i,17]]],{i,0,16}];
VRulesH12Final=Table[4*17+1+i->H12[[1+Mod[i,17]]],{i,0,16}];
VRulesH34Final=Table[5*17+1+i->H34[[1+Mod[i,17]]],{i,0,16}];
VRulesFinal=Union[VRulesA1Final,VRulesA2Final,VRulesB1Final,VRulesB2Final,VRulesH12Final,VRulesH34Final];
vertices=VRulesFinal[[All,-1]];

g=Graph[Range[102],edges,VertexCoordinates->vertices]

RecognizeGraph[g]

g=;

LabelGraph[g]

vertices=GraphEmbedding[g];

groups=Sort/@SplitBy[SortBy[MapIndexed[{N@#,#2[[1]]}&,Norm/@vertices],First],First][[All,All,2]]

groups={{52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68},{18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34},{86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102},{35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51},{69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85},{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}};

Length/@groups

r1

Norm@vertices[[groups[[1,1]]]]//FullSimplify

ToAlgebraicRoot[Norm@vertices[[groups[[1,1]]]],"MaxDegree"->64,WorkingPrecision->10000]//Timing

a1

groups[[1]]

ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->10000]&/@vertices[[52]]//Timing

a1=Table[RotationMatrix[2Pi i/17].{(NumericalApproximation[Root, (RowDefault["0.0349", FEExpressionsNumericalApproximationElider]), Root[311467152058624-350856571663616832 ((#1)^2)+89576973305408208288 ((#1)^4)-10482815632947350685072 ((#1)^6)+682674889559324544675141 ((#1)^8)-27161824389172490215747150 ((#1)^10)+705995317065357726580808753 ((#1)^12)-12712553335512762742251398760 ((#1)^14)+«27»+2096050127062617746636983576297472 ((#1)^52)-381312849292287068019046882476032 ((#1)^54)+55632326936090229893680727064576 ((#1)^56)-4770190402147905978782153965568 ((#1)^58)+347478960522270032809099788288 ((#1)^60)+277324292120453182893588480 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,21], ]),(NumericalApproximation[Root, (RowDefault["0.501", FEExpressionsNumericalApproximationElider]), Root[196249-9426537043 ((#1)^2)+30938926861842 ((#1)^4)-20255932097951418 ((#1)^6)+5795379296245649655 ((#1)^8)-904866473684364144194 ((#1)^10)+85833856324016158211867 ((#1)^12)-5256985419092185984030134 ((#1)^14)+«27»+9448613966150765411646593363345408 ((#1)^52)-1710769050616949164508719014215680 ((#1)^54)+208151420700982977239003443494912 ((#1)^56)-16247191405194081103409278091264 ((#1)^58)+779223581257438669119680413696 ((#1)^60)-19209329300876723801762562048 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,35], ])},{i,0,16}]

ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->10000]&/@vertices[[53]]//Timing

r2

Norm@vertices[[groups[[2,1]]]]//FullSimplify

ToAlgebraicRoot[Norm@vertices[[groups[[2,1]]]],"MaxDegree"->64,WorkingPrecision->10000]//Timing

a2

a2=vertices[[groups[[2]]]]//RootReduce

r3

Norm@vertices[[groups[[3,1]]]]//FullSimplify

ToAlgebraicRoot[Norm@vertices[[groups[[3,1]]]],"MaxDegree"->64,WorkingPrecision->10000]//Timing

a3

groups[[3]]

ToAlgebraicRoot[#,"MaxDegree"->32]&/@vertices[[86]]//Timing

a3=Table[RotationMatrix[2Pi i/17].{(NumericalApproximation[Root, (RowDefault["-0.830", FEExpressionsNumericalApproximationElider]), Root[17-133144 ((#1)^2)+63387152 ((#1)^4)-6610311552 ((#1)^6)-5686018560 ((#1)^8)+3668381816832 ((#1)^10)-42988678246400 ((#1)^12)+113308278161408 ((#1)^14)+102308322344960 ((#1)^16)-687543533699072 ((#1)^18)+414173453352960 ((#1)^20)+1065746062901248 ((#1)^22)-1583734344122368 ((#1)^24)+722966207791104 ((#1)^26)-147927263608832 ((#1)^28)+9345848836096 ((#1)^30)+18691697672192 ((#1)^32)&,2], ]),(NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[126590191433-5250520254472 ((#1)^2)+63554091162512 ((#1)^4)-341288873203328 ((#1)^6)+805446383384064 ((#1)^8)+5340883339264 ((#1)^10)-4382751503216640 ((#1)^12)+8857592733138944 ((#1)^14)-1511335944912896 ((#1)^16)-21460294451068928 ((#1)^18)+42023170367029248 ((#1)^20)-41394139077541888 ((#1)^22)+24807573748187136 ((#1)^24)-9413291365367808 ((#1)^26)+2235264189595648 ((#1)^28)-308413011591168 ((#1)^30)+18691697672192 ((#1)^32)&,20], ])},{i,0,16}]

groups[[3]][[{1,2}]]

ArcTan@@vertices[[86]]//N

ArcTan@@vertices[[87]]//N

%-%%

2Pi/17.

RotationMatrix[2Pi/17].{(NumericalApproximation[Root, (RowDefault["-0.830", FEExpressionsNumericalApproximationElider]), Root[17-133144 ((#1)^2)+63387152 ((#1)^4)-6610311552 ((#1)^6)-5686018560 ((#1)^8)+3668381816832 ((#1)^10)-42988678246400 ((#1)^12)+113308278161408 ((#1)^14)+102308322344960 ((#1)^16)-687543533699072 ((#1)^18)+414173453352960 ((#1)^20)+1065746062901248 ((#1)^22)-1583734344122368 ((#1)^24)+722966207791104 ((#1)^26)-147927263608832 ((#1)^28)+9345848836096 ((#1)^30)+18691697672192 ((#1)^32)&,2], ]),(NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[126590191433-5250520254472 ((#1)^2)+63554091162512 ((#1)^4)-341288873203328 ((#1)^6)+805446383384064 ((#1)^8)+5340883339264 ((#1)^10)-4382751503216640 ((#1)^12)+8857592733138944 ((#1)^14)-1511335944912896 ((#1)^16)-21460294451068928 ((#1)^18)+42023170367029248 ((#1)^20)-41394139077541888 ((#1)^22)+24807573748187136 ((#1)^24)-9413291365367808 ((#1)^26)+2235264189595648 ((#1)^28)-308413011591168 ((#1)^30)+18691697672192 ((#1)^32)&,20], ])}

N[%]

vertices[[87]]//N

ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->10^4]&/@vertices[[87]]//Timing

r4

Norm@vertices[[groups[[4,1]]]]//FullSimplify

ToAlgebraicRoot[Norm@vertices[[35]],"MaxDegree"->64,WorkingPrecision->10000]//Timing

a4

groups[[4]]

ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->10^4]&/@vertices[[35]]//Timing

a4=Table[RotationMatrix[2Pi i/17].{(NumericalApproximation[Root, (RowDefault["0.0961", FEExpressionsNumericalApproximationElider]), Root[311467152058624-110692599791766016 ((#1)^2)+16316091107690050080 ((#1)^4)-1327308895910128211188 ((#1)^6)+67113388771674261312893 ((#1)^8)-2263117551110450234045642 ((#1)^10)+53612944367126224091900309 ((#1)^12)-930633594773161725434331128 ((#1)^14)+«27»+425678549078693922538174998904832 ((#1)^52)-93407811611685509538202349731840 ((#1)^54)+16166957705649824205105642078208 ((#1)^56)-1837184476971333381622071296000 ((#1)^58)+171103792114685369079698554880 ((#1)^60)-8104032091964354122334863360 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,21], ]),(NumericalApproximation[Root, (RowDefault["1.38", FEExpressionsNumericalApproximationElider]), Root[196249-8104056179 ((#1)^2)+20688043268526 ((#1)^4)-6637395698577430 ((#1)^6)+983339566171971011 ((#1)^8)-85808324573568960062 ((#1)^10)+4898958505842888297619 ((#1)^12)-194202106221975848522938 ((#1)^14)+«27»+713017873467037865882183010353152 ((#1)^52)-158944678601834303430770617745408 ((#1)^54)+25876411732149018218750474715136 ((#1)^56)-2833842595674528874464223952896 ((#1)^58)+230192354622483260581559140352 ((#1)^60)-10827972916791916496534110208 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,40], ])},{i,0,16}]

ArcTan@@vertices[[35]]//N

ArcTan@@vertices[[36]]//N

%-%%

2Pi/17.

RotationMatrix[2Pi/17].{(NumericalApproximation[Root, (RowDefault["0.0961", FEExpressionsNumericalApproximationElider]), Root[311467152058624-110692599791766016 ((#1)^2)+16316091107690050080 ((#1)^4)-1327308895910128211188 ((#1)^6)+67113388771674261312893 ((#1)^8)-2263117551110450234045642 ((#1)^10)+53612944367126224091900309 ((#1)^12)-930633594773161725434331128 ((#1)^14)+«27»+425678549078693922538174998904832 ((#1)^52)-93407811611685509538202349731840 ((#1)^54)+16166957705649824205105642078208 ((#1)^56)-1837184476971333381622071296000 ((#1)^58)+171103792114685369079698554880 ((#1)^60)-8104032091964354122334863360 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,21], ]),(NumericalApproximation[Root, (RowDefault["1.38", FEExpressionsNumericalApproximationElider]), Root[196249-8104056179 ((#1)^2)+20688043268526 ((#1)^4)-6637395698577430 ((#1)^6)+983339566171971011 ((#1)^8)-85808324573568960062 ((#1)^10)+4898958505842888297619 ((#1)^12)-194202106221975848522938 ((#1)^14)+«27»+713017873467037865882183010353152 ((#1)^52)-158944678601834303430770617745408 ((#1)^54)+25876411732149018218750474715136 ((#1)^56)-2833842595674528874464223952896 ((#1)^58)+230192354622483260581559140352 ((#1)^60)-10827972916791916496534110208 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,40], ])}

N[%]

vertices[[36]]//N

N[vertices[[36,2]]-(Cos[((2 π)/17)] (NumericalApproximation[Root, (RowDefault["1.38", FEExpressionsNumericalApproximationElider]), Root[196249-8104056179 ((#1)^2)+20688043268526 ((#1)^4)-6637395698577430 ((#1)^6)+983339566171971011 ((#1)^8)-85808324573568960062 ((#1)^10)+4898958505842888297619 ((#1)^12)-194202106221975848522938 ((#1)^14)+«27»+713017873467037865882183010353152 ((#1)^52)-158944678601834303430770617745408 ((#1)^54)+25876411732149018218750474715136 ((#1)^56)-2833842595674528874464223952896 ((#1)^58)+230192354622483260581559140352 ((#1)^60)-10827972916791916496534110208 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,40], ])+(NumericalApproximation[Root, (RowDefault["0.0961", FEExpressionsNumericalApproximationElider]), Root[311467152058624-110692599791766016 ((#1)^2)+16316091107690050080 ((#1)^4)-1327308895910128211188 ((#1)^6)+67113388771674261312893 ((#1)^8)-2263117551110450234045642 ((#1)^10)+53612944367126224091900309 ((#1)^12)-930633594773161725434331128 ((#1)^14)+«27»+425678549078693922538174998904832 ((#1)^52)-93407811611685509538202349731840 ((#1)^54)+16166957705649824205105642078208 ((#1)^56)-1837184476971333381622071296000 ((#1)^58)+171103792114685369079698554880 ((#1)^60)-8104032091964354122334863360 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,21], ]) Sin[((2 π)/17)]),10^5]

RootReduce[Cos[((2 π)/17)]]

ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->10^4]&/@vertices[[36]]//Timing

From Adam S., Oct 21, 2021:

The maximum possible degree is the product of degrees of all numbers involved, so the worst case scenario would be

8 64^2 16

However, here RootReduce of both products gives Root objects of degree 64, so now the degree is bounded by 64^2 = 4096. Unfortunately, to get the sum of the two roots, we need to compute the resultant of two bivariate polynomials of degree 64, and the resultant will have degree 4096 (if it ever finishes -- this is where the code gets stuck). Then we need to factor the resultant, and a degree 4096 polynomial might be out of range for Factor, unless there are some low-degree  factors.

The Resultant computation has finished after a couple of hours, and I was able to factor the resultant using the FLINT library in 58 seconds (in a development branch). The resultant has 11 factors of degrees 128 and 512, the number is a root of a factor of degree 128.    Adam

N[(NumericalApproximation[Root, (RowDefault["1.32", FEExpressionsNumericalApproximationElider]), Root[81817239611583093798849137366236785889-495316582610793036683956034545951290131144 ((#1)^2)+837416378046061033012979654947161993708261264 ((#1)^4)-335636585862625181492885364494921034914404085888 ((#1)^6)+«87»+60164292582694182846626340058109050602137126060737345489456471614685184 ((#1)^124)-2043646553231436946629674490314473149514696736270610061992382051647488 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,79], ])-(Cos[((2 π)/17)] (NumericalApproximation[Root, (RowDefault["1.38", FEExpressionsNumericalApproximationElider]), Root[196249-8104056179 ((#1)^2)+20688043268526 ((#1)^4)-6637395698577430 ((#1)^6)+983339566171971011 ((#1)^8)-85808324573568960062 ((#1)^10)+4898958505842888297619 ((#1)^12)-194202106221975848522938 ((#1)^14)+«27»+713017873467037865882183010353152 ((#1)^52)-158944678601834303430770617745408 ((#1)^54)+25876411732149018218750474715136 ((#1)^56)-2833842595674528874464223952896 ((#1)^58)+230192354622483260581559140352 ((#1)^60)-10827972916791916496534110208 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,40], ])+(NumericalApproximation[Root, (RowDefault["0.0961", FEExpressionsNumericalApproximationElider]), Root[311467152058624-110692599791766016 ((#1)^2)+16316091107690050080 ((#1)^4)-1327308895910128211188 ((#1)^6)+67113388771674261312893 ((#1)^8)-2263117551110450234045642 ((#1)^10)+53612944367126224091900309 ((#1)^12)-930633594773161725434331128 ((#1)^14)+«27»+425678549078693922538174998904832 ((#1)^52)-93407811611685509538202349731840 ((#1)^54)+16166957705649824205105642078208 ((#1)^56)-1837184476971333381622071296000 ((#1)^58)+171103792114685369079698554880 ((#1)^60)-8104032091964354122334863360 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,21], ]) Sin[((2 π)/17)]),10^6]//Timing

Oct 22, 2021

I am not sure if RootApproximant can be used here.
"DegreeCost" needs to be a nonnegative integer and is zero by default, so you cannot force it to use
higher degrees. You could try "IntegerRelationFinder" -> "PSLQ" (I have no reason to think it would be better, but it is the only alternative to the default "LLL").

r5

Norm@vertices[[groups[[5,1]]]]//FullSimplify

ToAlgebraicRoot[Norm@vertices[[groups[[5,1]]]],"MaxDegree"->64,WorkingPrecision->10000]//Timing

a5

groups[[5]]

a5=Map[ToAlgebraicRoot[#,"MaxDegree"->64,Debug->True]&,vertices[[groups[[5]]]],{2}]

vertices[[groups[[5,1]]]]//N

ToAlgebraicRoot[#,"MaxDegree"->64]&/@vertices[[groups[[5,1]]]]

r6

Norm@vertices[[groups[[6,1]]]]//FullSimplify

ToAlgebraicRoot[Norm@vertices[[groups[[6,1]]]],"MaxDegree"->64,WorkingPrecision->10000]//Timing

a6

groups[[6]]

a6=Map[RootReduce,vertices[[groups[[6]]]],{2}]

vertices[[groups[[6,1]]]]

Reassemble (leaving sin and cos of pi/17)

(aa=Sort[Join@@(Transpose/@{{groups[[1]],a1},{groups[[2]],a2},{groups[[3]],a3},{groups[[4]],a4},{groups[[5]],a5},{groups[[6]],a6}})][[All,2]]);

N[aa-vertices,10000]//Abs//Max

e=EdgeList[g]

e={1\[UndirectedEdge]2,1\[UndirectedEdge]69,2\[UndirectedEdge]3,2\[UndirectedEdge]70,3\[UndirectedEdge]4,3\[UndirectedEdge]71,4\[UndirectedEdge]5,4\[UndirectedEdge]72,5\[UndirectedEdge]6,5\[UndirectedEdge]73,6\[UndirectedEdge]7,6\[UndirectedEdge]74,7\[UndirectedEdge]8,7\[UndirectedEdge]75,8\[UndirectedEdge]9,8\[UndirectedEdge]76,9\[UndirectedEdge]10,9\[UndirectedEdge]77,10\[UndirectedEdge]11,10\[UndirectedEdge]78,11\[UndirectedEdge]12,11\[UndirectedEdge]79,12\[UndirectedEdge]13,12\[UndirectedEdge]80,13\[UndirectedEdge]14,13\[UndirectedEdge]81,14\[UndirectedEdge]15,14\[UndirectedEdge]82,15\[UndirectedEdge]16,15\[UndirectedEdge]83,16\[UndirectedEdge]17,16\[UndirectedEdge]84,17\[UndirectedEdge]1,17\[UndirectedEdge]85,18\[UndirectedEdge]22,18\[UndirectedEdge]69,19\[UndirectedEdge]23,19\[UndirectedEdge]70,20\[UndirectedEdge]24,20\[UndirectedEdge]71,21\[UndirectedEdge]25,21\[UndirectedEdge]72,22\[UndirectedEdge]26,22\[UndirectedEdge]73,23\[UndirectedEdge]27,23\[UndirectedEdge]74,24\[UndirectedEdge]28,24\[UndirectedEdge]75,25\[UndirectedEdge]29,25\[UndirectedEdge]76,26\[UndirectedEdge]30,26\[UndirectedEdge]77,27\[UndirectedEdge]31,27\[UndirectedEdge]78,28\[UndirectedEdge]32,28\[UndirectedEdge]79,29\[UndirectedEdge]33,29\[UndirectedEdge]80,30\[UndirectedEdge]34,30\[UndirectedEdge]81,31\[UndirectedEdge]18,31\[UndirectedEdge]82,32\[UndirectedEdge]19,32\[UndirectedEdge]83,33\[UndirectedEdge]20,33\[UndirectedEdge]84,34\[UndirectedEdge]21,34\[UndirectedEdge]85,35\[UndirectedEdge]37,35\[UndirectedEdge]86,36\[UndirectedEdge]38,36\[UndirectedEdge]87,37\[UndirectedEdge]39,37\[UndirectedEdge]88,38\[UndirectedEdge]40,38\[UndirectedEdge]89,39\[UndirectedEdge]41,39\[UndirectedEdge]90,40\[UndirectedEdge]42,40\[UndirectedEdge]91,41\[UndirectedEdge]43,41\[UndirectedEdge]92,42\[UndirectedEdge]44,42\[UndirectedEdge]93,43\[UndirectedEdge]45,43\[UndirectedEdge]94,44\[UndirectedEdge]46,44\[UndirectedEdge]95,45\[UndirectedEdge]47,45\[UndirectedEdge]96,46\[UndirectedEdge]48,46\[UndirectedEdge]97,47\[UndirectedEdge]49,47\[UndirectedEdge]98,48\[UndirectedEdge]50,48\[UndirectedEdge]99,49\[UndirectedEdge]51,49\[UndirectedEdge]100,50\[UndirectedEdge]35,50\[UndirectedEdge]101,51\[UndirectedEdge]36,51\[UndirectedEdge]102,52\[UndirectedEdge]60,52\[UndirectedEdge]86,53\[UndirectedEdge]61,53\[UndirectedEdge]87,54\[UndirectedEdge]62,54\[UndirectedEdge]88,55\[UndirectedEdge]63,55\[UndirectedEdge]89,56\[UndirectedEdge]64,56\[UndirectedEdge]90,57\[UndirectedEdge]65,57\[UndirectedEdge]91,58\[UndirectedEdge]66,58\[UndirectedEdge]92,59\[UndirectedEdge]67,59\[UndirectedEdge]93,60\[UndirectedEdge]68,60\[UndirectedEdge]94,61\[UndirectedEdge]52,61\[UndirectedEdge]95,62\[UndirectedEdge]53,62\[UndirectedEdge]96,63\[UndirectedEdge]54,63\[UndirectedEdge]97,64\[UndirectedEdge]55,64\[UndirectedEdge]98,65\[UndirectedEdge]56,65\[UndirectedEdge]99,66\[UndirectedEdge]57,66\[UndirectedEdge]100,67\[UndirectedEdge]58,67\[UndirectedEdge]101,68\[UndirectedEdge]59,68\[UndirectedEdge]102,69\[UndirectedEdge]86,70\[UndirectedEdge]87,71\[UndirectedEdge]88,72\[UndirectedEdge]89,73\[UndirectedEdge]90,74\[UndirectedEdge]91,75\[UndirectedEdge]92,76\[UndirectedEdge]93,77\[UndirectedEdge]94,78\[UndirectedEdge]95,79\[UndirectedEdge]96,80\[UndirectedEdge]97,81\[UndirectedEdge]98,82\[UndirectedEdge]99,83\[UndirectedEdge]100,84\[UndirectedEdge]101,85\[UndirectedEdge]102};

g=Graph[Range[Length[aa]],e,VertexCoordinates->aa]

RecognizeGraph[g]

UnitDistanceGraphEmbeddingQ[g]

evertices=GraphEmbedding[g]

Put[evertices,"~/biggs.m"]

ToAlgebraicRoot

g=GraphData["BiggsSmithGraph","Graph","UnitDistance"][[1]]

(evertices=Monitor[Table[ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->5000]&/@vertices[[i]],{i,Length[vertices]}],i])//Timing

exact=Get["~/biggs2.m"]

Position[exact,_?(Head[#]=!=Root&&!FreeQ[#,Root]&),{2},Heads->False]

exact[[36]]

N[%]

g=Graph[Range[Length[exact]],e,VertexCoordinates->exact]

RecognizeGraph[g]

Un-root-reduced left to go

v=GraphData["BiggsSmithGraph","Embeddings","UnitDistance"][[1]]

Position[v,_?(Head[#]=!=Root&&!FreeQ[#,Root]&),{2},Heads->False]

v[[2,2]]

From Adam S., Oct 21, 2021:

This is the one Adam got:

Position[v,(NumericalApproximation[Root, (RowDefault["-0.409", FEExpressionsNumericalApproximationElider]), Root[39663582454289605109+9210522532420525554308 #1+544478951329459392141964 ((#1)^2)+5760322564386318146384768 ((#1)^3)-74850008915753400679204160 ((#1)^4)-1057086700106709860184544640 ((#1)^5)+«83»+696578235021268799177510276023975936 ((#1)^61)+29157105185965497355037630716182528 ((#1)^62)-50456267130323272018646680104599552 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,14], ])]

v[[69,2]]

N[%]

The maximum possible degree is the product of degrees of all numbers involved, so the worst case scenario would be:

8 64^2 16

Here RootReduce of both products gives Root objects of degree 64:

RootReduce[Cos[((2 π)/17)] (NumericalApproximation[Root, (RowDefault["1.38", FEExpressionsNumericalApproximationElider]), Root[196249-8104056179 ((#1)^2)+20688043268526 ((#1)^4)-6637395698577430 ((#1)^6)+983339566171971011 ((#1)^8)-85808324573568960062 ((#1)^10)+«30»+713017873467037865882183010353152 ((#1)^52)-158944678601834303430770617745408 ((#1)^54)+25876411732149018218750474715136 ((#1)^56)-2833842595674528874464223952896 ((#1)^58)+230192354622483260581559140352 ((#1)^60)-10827972916791916496534110208 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,40], ])]+RootReduce[(NumericalApproximation[Root, (RowDefault["0.0961", FEExpressionsNumericalApproximationElider]), Root[311467152058624-110692599791766016 ((#1)^2)+16316091107690050080 ((#1)^4)-1327308895910128211188 ((#1)^6)+67113388771674261312893 ((#1)^8)-2263117551110450234045642 ((#1)^10)+«33»+16166957705649824205105642078208 ((#1)^56)-1837184476971333381622071296000 ((#1)^58)+171103792114685369079698554880 ((#1)^60)-8104032091964354122334863360 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,21], ]) Sin[((2 π)/17)]]

N[%]

so now the degree is bounded by 64^2 = 4096.

Unfortunately, to get the sum of the two roots, we need to compute the resultant of two bivariate polynomials of degree 64, and the resultant will have degree 4096 (if it ever finishes -- this is where the code gets stuck). Then we need to factor the resultant, and a degree 4096 polynomial might be out of range for Factor, unless there are some low-degree  factors.

The Resultant computation has finished after a couple of hours, and I was able to factor the resultant using the FLINT library in 58 seconds (in a development branch). The resultant has 11 factors of degrees 128 and 512, the number is a root of a factor of degree 128.    Adam

N[(NumericalApproximation[Root, (RowDefault["1.32", FEExpressionsNumericalApproximationElider]), Root[81817239611583093798849137366236785889-495316582610793036683956034545951290131144 ((#1)^2)+837416378046061033012979654947161993708261264 ((#1)^4)-335636585862625181492885364494921034914404085888 ((#1)^6)+«87»+60164292582694182846626340058109050602137126060737345489456471614685184 ((#1)^124)-2043646553231436946629674490314473149514696736270610061992382051647488 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,79], ])-(Cos[((2 π)/17)] (NumericalApproximation[Root, (RowDefault["1.38", FEExpressionsNumericalApproximationElider]), Root[196249-8104056179 ((#1)^2)+20688043268526 ((#1)^4)-6637395698577430 ((#1)^6)+983339566171971011 ((#1)^8)-85808324573568960062 ((#1)^10)+4898958505842888297619 ((#1)^12)-194202106221975848522938 ((#1)^14)+«27»+713017873467037865882183010353152 ((#1)^52)-158944678601834303430770617745408 ((#1)^54)+25876411732149018218750474715136 ((#1)^56)-2833842595674528874464223952896 ((#1)^58)+230192354622483260581559140352 ((#1)^60)-10827972916791916496534110208 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,40], ])+(NumericalApproximation[Root, (RowDefault["0.0961", FEExpressionsNumericalApproximationElider]), Root[311467152058624-110692599791766016 ((#1)^2)+16316091107690050080 ((#1)^4)-1327308895910128211188 ((#1)^6)+67113388771674261312893 ((#1)^8)-2263117551110450234045642 ((#1)^10)+53612944367126224091900309 ((#1)^12)-930633594773161725434331128 ((#1)^14)+«27»+425678549078693922538174998904832 ((#1)^52)-93407811611685509538202349731840 ((#1)^54)+16166957705649824205105642078208 ((#1)^56)-1837184476971333381622071296000 ((#1)^58)+171103792114685369079698554880 ((#1)^60)-8104032091964354122334863360 ((#1)^62)+394416771015755637893103616 ((#1)^64)&,21], ]) Sin[((2 π)/17)]),10^6]//Timing

Oct 22, 2021

I am not sure if RootApproximant can be used here.
"DegreeCost" needs to be a nonnegative integer and is zero by default, so you cannot force it to use
higher degrees. You could try "IntegerRelationFinder" -> "PSLQ" (I have no reason to think it would be better, but it is the only alternative to the default "LLL").

Algebraic root objects

From ToAlgebraicRoot of Gerbracht's construction

Do[
Print[{i,ToAlgebraicRoot[#,"MaxDegree"->128,WorkingPrecision->20000]&/@vertices[[i]]//Timing}],
{i,Length[vertices]}]

With[{v={{0,(NumericalApproximation[Root, (RowDefault["2.72", FEExpressionsNumericalApproximationElider]), Root[1-17 ((#1)^2)+119 ((#1)^4)-442 ((#1)^6)+935 ((#1)^8)-1122 ((#1)^10)+714 ((#1)^12)-204 ((#1)^14)+17 ((#1)^16)&,16], ])},{(NumericalApproximation[Root, (RowDefault["-0.293", FEExpressionsNumericalApproximationElider]), Root[-144868772621434576663-11482942785492549605104 #1-179208203095248135724196 ((#1)^2)+5750509355061827732712592 ((#1)^3)+«88»+74753854785294519814511372368084992 ((#1)^62)+62863545932861781531428650622124032 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,17], ]),(NumericalApproximation[Root, (RowDefault["-0.408", FEExpressionsNumericalApproximationElider]), Root[7748595065607438394046768430283449814729-9768061347234198759220793591807617217452856 ((#1)^2)+«95»+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,20], ])},{(NumericalApproximation[Root, (RowDefault["-0.664", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-12 ((#1)^2)-168 ((#1)^3)+128 ((#1)^4)+608 ((#1)^5)-320 ((#1)^6)-640 ((#1)^7)+256 ((#1)^8)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.331", FEExpressionsNumericalApproximationElider]), Root[1-408 ((#1)^2)+12240 ((#1)^4)-144704 ((#1)^6)+844288 ((#1)^8)-2628608 ((#1)^10)+4386816 ((#1)^12)-3620864 ((#1)^14)+1114112 ((#1)^16)&,11], ])},{(NumericalApproximation[Root, (RowDefault["0.213", FEExpressionsNumericalApproximationElider]), Root[39663582454289605109-9008772710184850428504 #1+564824709087037880124776 ((#1)^2)-9579923671378821035366528 ((#1)^3)-19903388992390343821911008 ((#1)^4)+«83»+11224125421356887431208228278669148160 ((#1)^59)+10349295745597761951963530424941019136 ((#1)^60)+369064847380510080882293698019196928 ((#1)^61)-262413946673689476195338676445642752 ((#1)^62)+9098671121861573642706778379517952 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,25], ]),(NumericalApproximation[Root, (RowDefault["0.455", FEExpressionsNumericalApproximationElider]), Root[81817239611583093798849137366236785889-372286602726802146675681330779269065657520 ((#1)^2)+«90»+61981860308361705181253355337043112395326826947395829768938104505237504 ((#1)^124)-647918743190549309828691577612255129759095349597679186041776298262528 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,63], ])},{(NumericalApproximation[Root, (RowDefault["-0.859", FEExpressionsNumericalApproximationElider]), Root[-1462806817859922002999-1510197299941086266168 #1+514763933903125422924772 ((#1)^2)-552309772793542469627312 ((#1)^3)-69389701065082920401268864 ((#1)^4)+160928073409441902816081024 ((#1)^5)+«81»+1219144384836935002438253415539671040 ((#1)^60)+185333727112918985847180684605521920 ((#1)^61)-131103579346823583851729488468508672 ((#1)^62)-2481455760507701902556394103504896 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,5], ]),(NumericalApproximation[Root, (RowDefault["1.09", FEExpressionsNumericalApproximationElider]), Root[27803728988860330106995809732390811242764197849-61170118847672982341108678976286478250680458662152 ((#1)^2)+«90»+67705199554903144055640327057867734596189838957133935166019381357445120 ((#1)^124)-2462364896243703237669969364125138555441377152272489324777563777662976 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,78], ])},{(NumericalApproximation[Root, (RowDefault["0.363", FEExpressionsNumericalApproximationElider]), Root[-1462806817859922002999-1809911316752684772520 #1+766146648582622284045796 ((#1)^2)-2752809484285326790192896 ((#1)^3)-115743640311267384609310192 ((#1)^4)+642718660483424719398099264 ((#1)^5)+«79»+68907365891528317952760092534578872320 ((#1)^59)+6824011419051650634705547835400519680 ((#1)^60)+2461591190174887643055200469457567744 ((#1)^61)+697599250672727697356166292347813888 ((#1)^62)+102566838100985011972330956278202368 ((#1)^63)+6617215361353871740150384276013056 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((#1)^9)+365568 ((#1)^10)-5709824 ((#1)^11)+4595712 ((#1)^12)-1392640 ((#1)^13)+2785280 ((#1)^14)-3342336 ((#1)^15)+1114112 ((#1)^16)&,9], ]),(NumericalApproximation[Root, (RowDefault["-1.68", FEExpressionsNumericalApproximationElider]), Root[132871729-10435745052 ((#1)^2)+266344601568 ((#1)^4)-3345259712768 ((#1)^6)+24486087138048 ((#1)^8)-114261458771968 ((#1)^10)+358457217343488 ((#1)^12)-782294167486464 ((#1)^14)+1219043002089472 ((#1)^16)-1395424624902144 ((#1)^18)+1223158697820160 ((#1)^20)-866229388574720 ((#1)^22)+509579783831552 ((#1)^24)-237039110848512 ((#1)^26)+77267535396864 ((#1)^28)-14894946582528 ((#1)^30)+1241245548544 ((#1)^32)&,2], ])},{(NumericalApproximation[Root, (RowDefault["2.17", FEExpressionsNumericalApproximationElider]), Root[-1+10 #1+20 ((#1)^2)-152 ((#1)^3)-128 ((#1)^4)+672 ((#1)^5)+192 ((#1)^6)-768 ((#1)^7)+256 ((#1)^8)&,8], ]),(NumericalApproximation[Root, (RowDefault["-1.64", FEExpressionsNumericalApproximationElider]), Root[1-340 ((#1)^2)+12784 ((#1)^4)-177344 ((#1)^6)+1175040 ((#1)^8)-3986432 ((#1)^10)+6754304 ((#1)^12)-5013504 ((#1)^14)+1114112 ((#1)^16)&,1], ])},{(NumericalApproximation[Root, (RowDefault["1.49", FEExpressionsNumericalApproximationElider]), Root[-67+34 #1+16048 ((#1)^2)+73576 ((#1)^3)-300560 ((#1)^4)-1493280 ((#1)^5)+1069504 ((#1)^6)+8160000 ((#1)^7)+639744 ((#1)^8)-18295808 ((#1)^9)-8077312 ((#1)^10)+18487296 ((#1)^11)+12185600 ((#1)^12)-7798784 ((#1)^13)-6406144 ((#1)^14)+1114112 ((#1)^15)+1114112 ((#1)^16)&,13], ]),(NumericalApproximation[Root, (RowDefault["0.901", FEExpressionsNumericalApproximationElider]), Root[1-4523836 ((#1)^2)+402774336 ((#1)^4)-11707879872 ((#1)^6)+168725703936 ((#1)^8)-1412044157952 ((#1)^10)+7437354299392 ((#1)^12)-25661280976896 ((#1)^14)+58591977865216 ((#1)^16)-86406569394176 ((#1)^18)+74749285761024 ((#1)^20)-24057617580032 ((#1)^22)-16082857361408 ((#1)^24)+15244046893056 ((#1)^26)+543044927488 ((#1)^28)-4344359419904 ((#1)^30)+1241245548544 ((#1)^32)&,26], ])},{(NumericalApproximation[Root, (RowDefault["0.502", FEExpressionsNumericalApproximationElider]), Root[-858974878598450863+110280441280964664180 #1-4289215437833520392060 ((#1)^2)+65562112631971982875872 ((#1)^3)-40257109426758996875840 ((#1)^4)-11436536235532993224442368 ((#1)^5)+«78»+5736809074199367013832192377430736896 ((#1)^60)+923631437107723523261420336494673920 ((#1)^61)+169772931614735271833233296581459968 ((#1)^62)+50456267130323272018646680104599552 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,35], ]),(NumericalApproximation[Root, (RowDefault["0.0115", FEExpressionsNumericalApproximationElider]), Root[532185576379061829319942843851588889-13535035937454590321533476758403255255768 ((#1)^2)+«91»+177713288380062242209880371529016921205027693159626031895330834865979392 ((#1)^124)-3904616966619287129030985040584097175855498585167851229926523056160768 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,42], ])},{(NumericalApproximation[Root, (RowDefault["1.30", FEExpressionsNumericalApproximationElider]), Root[-30691701112735380481259+3133883520201966909355748 #1-47250602797121017040623544 ((#1)^2)-213574121344264486128886832 ((#1)^3)+11209306232862453853160549216 ((#1)^4)-104419466552260399656304909824 ((#1)^5)+«76»+1030976814243611997971397122335965184 ((#1)^59)-1109832704418163756453440072944975872 ((#1)^60)+734019783657679280488478368304463872 ((#1)^61)-115697874833671601206691875075915776 ((#1)^62)-28950317205923188863157931207557120 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,38], ]),(NumericalApproximation[Root, (RowDefault["-0.470", FEExpressionsNumericalApproximationElider]), Root[52803260235101467213051670229889-377553440534525936738019168386781024 ((#1)^2)+«89»+16028580504165556699415883387520175390706646117608703075298104238407680 ((#1)^124)-1834287381725303801109527053409140446551356528269670430599791188639744 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,16], ])},{(NumericalApproximation[Root, (RowDefault["1.03", FEExpressionsNumericalApproximationElider]), Root[-9179+70346 #1-75616 ((#1)^2)-550664 ((#1)^3)+1196528 ((#1)^4)+1456288 ((#1)^5)-4971072 ((#1)^6)-1307776 ((#1)^7)+9822464 ((#1)^8)+26112 ((#1)^9)-11088896 ((#1)^10)-417792 ((#1)^11)+8495104 ((#1)^12)+1253376 ((#1)^13)-4456448 ((#1)^14)-557056 ((#1)^15)+1114112 ((#1)^16)&,14], ]),(NumericalApproximation[Root, (RowDefault["-1.40", FEExpressionsNumericalApproximationElider]), Root[73441-23987068 ((#1)^2)+2162925504 ((#1)^4)-57886925760 ((#1)^6)+759305151232 ((#1)^8)-5809257085952 ((#1)^10)+28075013468160 ((#1)^12)-89010554126336 ((#1)^14)+187621892096000 ((#1)^16)-262196227735552 ((#1)^18)+241337105383424 ((#1)^20)-152326526468096 ((#1)^22)+83512552062976 ((#1)^24)-53470530895872 ((#1)^26)+29945048858624 ((#1)^28)-9619653001216 ((#1)^30)+1241245548544 ((#1)^32)&,2], ])},{(NumericalApproximation[Root, (RowDefault["1.04", FEExpressionsNumericalApproximationElider]), Root[3687340463257779679873-141794892826387031859428 #1+14056680120221857252436 ((#1)^2)+79192016157923786437026896 ((#1)^3)-1514532900781230336890869184 ((#1)^4)+8349672342654325250419392704 ((#1)^5)+«81»+5854149742214331759161385576239726592 ((#1)^59)-4268179838034669057546359935191744512 ((#1)^60)+711389424091799144908393753329270784 ((#1)^61)+125727091865723563062857301244248064 ((#1)^62)-62863545932861781531428650622124032 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,37], ]),(NumericalApproximation[Root, (RowDefault["-0.909", FEExpressionsNumericalApproximationElider]), Root[143607302859651314004874206238357369-1225575476921223333324805896579055364952 ((#1)^2)+«90»+31872988224083377398473254722190827098475566280180631846591329218854912 ((#1)^124)-1915704837311022246589584389983436497703766609158924731696909857587200 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,5], ])},{(NumericalApproximation[Root, (RowDefault["-0.738", FEExpressionsNumericalApproximationElider]), Root[-5808389-91953334 #1+320521708 ((#1)^2)+7097361224 ((#1)^3)+267215456 ((#1)^4)-237555148192 ((#1)^5)-293821010816 ((#1)^6)+5176465311616 ((#1)^7)+3134699285760 ((#1)^8)-60580892558848 ((#1)^9)+2504097218560 ((#1)^10)+«16»+2980973826277376 ((#1)^22)-3747081445441536 ((#1)^23)-5938258743656448 ((#1)^24)-4222391072849920 ((#1)^25)-1024433720393728 ((#1)^26)+2503574017802240 ((#1)^27)+3928907233361920 ((#1)^28)+2734536957886464 ((#1)^29)+1046735069642752 ((#1)^30)+214954523230208 ((#1)^31)+18691697672192 ((#1)^32)&,8], ]),(NumericalApproximation[Root, (RowDefault["-1.07", FEExpressionsNumericalApproximationElider]), Root[202605487434199249-16743859392822664612 ((#1)^2)+633348252736802809008 ((#1)^4)-19070224756702075838016 ((#1)^6)+492009619256127481707008 ((#1)^8)-10046438092674499624473600 ((#1)^10)+«30»+7745932655222379480131139796992 ((#1)^52)-2489722203415747105508334501888 ((#1)^54)+500423208667988458255417344000 ((#1)^56)-76720288697920040376775737344 ((#1)^58)+18669970337354799745251409920 ((#1)^60)-4105209851956377020008497152 ((#1)^62)+349379561868627831490084864 ((#1)^64)&,4], ])},{(NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[1537716403-29195473710 #1+176701778484 ((#1)^2)-175614933232 ((#1)^3)-1996098535216 ((#1)^4)+6165731609568 ((#1)^5)+3681008013248 ((#1)^6)-35521815267072 ((#1)^7)+23879108654336 ((#1)^8)+56918209560576 ((#1)^9)-66096099592192 ((#1)^10)+«18»+1485341391323136 ((#1)^23)+2052877816168448 ((#1)^24)+509004224659456 ((#1)^25)-1583372979666944 ((#1)^26)-432847877832704 ((#1)^27)+696995882729472 ((#1)^28)+36507222016000 ((#1)^29)-72430328479744 ((#1)^30)-42056319762432 ((#1)^31)+18691697672192 ((#1)^32)&,18], ]),(NumericalApproximation[Root, (RowDefault["-0.834", FEExpressionsNumericalApproximationElider]), Root[2066494587616369-677848721476004148 ((#1)^2)+87796507001063536432 ((#1)^4)-5916636431258898063488 ((#1)^6)+231903391050775175961856 ((#1)^8)-5669299775449126522219520 ((#1)^10)+«32»+27105194105543798040948595752960 ((#1)^52)-9224040814936162371830015852544 ((#1)^54)+2389168906885429023208399437824 ((#1)^56)-464485541590014643528766849024 ((#1)^58)+67039932961370063822093549568 ((#1)^60)-6703720343354296516716003328 ((#1)^62)+349379561868627831490084864 ((#1)^64)&,8], ])},{(NumericalApproximation[Root, (RowDefault["0.234", FEExpressionsNumericalApproximationElider]), Root[-144868772621434576663+11482942785492549605104 #1-179208203095248135724196 ((#1)^2)-5750509355061827732712592 ((#1)^3)+161848464213597435676008192 ((#1)^4)+«80»+67112239638722424694819307701470756864 ((#1)^59)-9385862468925176285620546363493187584 ((#1)^60)+1202536725189787101684165048753192960 ((#1)^61)+74753854785294519814511372368084992 ((#1)^62)-62863545932861781531428650622124032 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,22], ]),(NumericalApproximation[Root, (RowDefault["-0.444", FEExpressionsNumericalApproximationElider]), Root[7748595065607438394046768430283449814729-9768061347234198759220793591807617217452856 ((#1)^2)+«95»+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,18], ])},{(NumericalApproximation[Root, (RowDefault["0.528", FEExpressionsNumericalApproximationElider]), Root[-52388423+1895439878 #1+23495682964 ((#1)^2)-40524874088 ((#1)^3)-571434715936 ((#1)^4)+2026762273632 ((#1)^5)+3559181226240 ((#1)^6)-42734648915584 ((#1)^7)+61165579431168 ((#1)^8)+«20»+6647299870556160 ((#1)^22)-724176222552064 ((#1)^23)-5374707595476992 ((#1)^24)-2403089030053888 ((#1)^25)+2598602316709888 ((#1)^26)+2046010750664704 ((#1)^27)-566592085688320 ((#1)^28)-702398951587840 ((#1)^29)-15187004358656 ((#1)^30)+93458488360960 ((#1)^31)+18691697672192 ((#1)^32)&,11], 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((#1)^62)+31431772966430890765714325311062016 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.701", FEExpressionsNumericalApproximationElider]), Root[17092657914137050571915724079827688129-625380755556663994693392950426829814199963128 ((#1)^2)+«91»+34810099415640390831169992240193055115002471174340055816513580886392832 ((#1)^124)-2125064008817155392109731826888769200667106817159864363089500720594944 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,67], ])},{(NumericalApproximation[Root, (RowDefault["1.28", FEExpressionsNumericalApproximationElider]), Root[6265230016424677+172832583686147664 #1-19764035647792183268 ((#1)^2)-214605173059567907568 ((#1)^3)+15139434674245915542912 ((#1)^4)+«85»+788314552667537903887641970537922560 ((#1)^61)-82405010046859934014060254187225088 ((#1)^62)-31431772966430890765714325311062016 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,39], 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((#1)^30)-4672924418048 ((#1)^31)+18691697672192 ((#1)^32)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.636", FEExpressionsNumericalApproximationElider]), Root[14027095807753681-10457105872280950216 ((#1)^2)+8056046239883800112 ((#1)^4)+4108601487301744486400 ((#1)^6)+5696720724531153767680 ((#1)^8)-1863824311468983888962560 ((#1)^10)+«31»+2984373452419826070147408330752 ((#1)^52)-964162324355586069073500307456 ((#1)^54)-60507511813602357110697361408 ((#1)^56)+89872262639561118675023953920 ((#1)^58)-9072950497275928999008141312 ((#1)^60)-2249130929529291665217421312 ((#1)^62)+349379561868627831490084864 ((#1)^64)&,31], ])},{(NumericalApproximation[Root, (RowDefault["0.159", FEExpressionsNumericalApproximationElider]), Root[359372512395629+53548347019264292 #1-528498998303277340 ((#1)^2)-48123224913919537184 ((#1)^3)+362738118103429935440 ((#1)^4)+«84»+708442695376196248899108035331358720 ((#1)^61)+115801268823692755452631724830228480 ((#1)^62)-60382090172354079628872256518619136 ((#1)^63)+6617215361353871740150384276013056 ((#1)^64)&,22], ]),(NumericalApproximation[Root, (RowDefault["-1.37", FEExpressionsNumericalApproximationElider]), Root[572433919713682631341918057882623983381798401-363028603321796659311563024237094807605443097208 ((#1)^2)+«89»+65432545027202528992010417763969415594812362710252820162049258217799680 ((#1)^124)-2090170813566133201189707254071213750173216782493041091190735576760320 ((#1)^126)+43787539138537651350619071771049977090371808209346851010215082459136 ((#1)^128)&,2], ])}}},
g=Graph[Range[102],UnitEdges[v],VertexCoordinates->v]]

RecognizeGraph[g]

Construction from scratch

g0=GraphData["BiggsSmithGraph","Graph",{"Labeled","UnitDistance"}][[1]]

v0=GraphEmbedding[g0];

a1=Norm@v0[[76]]

cc=CirclePoints[1,17];
EuclideanDistance@@cc[[{1,9}]]//FullSimplify

1./%

(a1=(1/(2 Cos[(π/34)])))//N

RootReduce[a1]

a2=Norm@v0[[80]]

a3=Norm@v0[[101]]

a4=Norm@v0[[99]]

a5=Norm@v0[[33]]

a6=Norm@v0[[35]]

N[Csc[Pi/17]/2]

a6=Csc[Pi/17]/2;

q2=VectorAngle@@v0[[{76,80}]]-Pi/2

q3=VectorAngle@@v0[[{76,75}]]-Pi/2

q4=VectorAngle@@v0[[{76,92}]]-Pi/2

q5=VectorAngle@@v0[[{76,88}]]-Pi/2

q6=VectorAngle@@v0[[{76,59}]]-Pi/2

GraphData["BiggsSmithGraph","V"]

17 6

v=Join[CirclePoints[{a1,-Pi/2},17],CirclePoints[{a2,q2},17],CirclePoints[{a3,q3},17],CirclePoints[{a4,q4},17],CirclePoints[{a5,q5},17],CirclePoints[{a6,q6},17]];
ee=Select[Subsets[Range[Length[v]],{2}],Abs[EuclideanDistance@@v[[#]]-1]<10^-2&];
g=Graph[Range[102],ee,VertexCoordinates->v]

RecognizeGraph[g]

ee=Select[Subsets[Range[Length[v]],{2}],Abs[EuclideanDistance@@v[[#]]-1]<10^-2&]

ee={{1,9},{1,10},{1,37},{2,10},{2,11},{2,38},{3,11},{3,12},{3,39},{4,12},{4,13},{4,40},{5,13},{5,14},{5,41},{6,14},{6,15},{6,42},{7,15},{7,16},{7,43},{8,16},{8,17},{8,44},{9,17},{9,45},{10,46},{11,47},{12,48},{13,49},{14,50},{15,51},{16,35},{17,36},{18,22},{18,31},{18,69},{19,23},{19,32},{19,70},{20,24},{20,33},{20,71},{21,25},{21,34},{21,72},{22,26},{22,73},{23,27},{23,74},{24,28},{24,75},{25,29},{25,76},{26,30},{26,77},{27,31},{27,78},{28,32},{28,79},{29,33},{29,80},{30,34},{30,81},{31,82},{32,83},{33,84},{34,85},{35,67},{35,70},{36,68},{36,71},{37,52},{37,72},{38,53},{38,73},{39,54},{39,74},{40,55},{40,75},{41,56},{41,76},{42,57},{42,77},{43,58},{43,78},{44,59},{44,79},{45,60},{45,80},{46,61},{46,81},{47,62},{47,82},{48,63},{48,83},{49,64},{49,84},{50,65},{50,85},{51,66},{51,69},{52,54},{52,67},{53,55},{53,68},{54,56},{55,57},{56,58},{57,59},{58,60},{59,61},{60,62},{61,63},{62,64},{63,65},{64,66},{65,67},{66,68},{69,87},{70,88},{71,89},{72,90},{73,91},{74,92},{75,93},{76,94},{77,95},{78,96},{79,97},{80,98},{81,99},{82,100},{83,101},{84,102},{85,86},{86,87},{86,102},{87,88},{88,89},{89,90},{90,91},{91,92},{92,93},{93,94},{94,95},{95,96},{96,97},{97,98},{98,99},{99,100},{100,101},{101,102}};

Clear[a2,q2,a3,q3,a4,q4,a5,q5,q6]

v=Join[CirclePoints[{(1/2) Sec[(π/34)],-Pi/2},17],CirclePoints[{a2,q2},17],CirclePoints[{a3,q3},17],CirclePoints[{a4,q4},17],CirclePoints[{a5,q5},17],CirclePoints[{Csc[Pi/17]/2,q6},17]];

eqn=DeleteCases[UnitEdgeEquations[v,ee]//Simplify//TrigExpand//Simplify//Union,True]

Length[eqn]

eqn/.a2->(1/2) Csc[((4 π)/17)]

soln=FindRoot[eqn,{{a2,0.7421734135331807`},{a3,1.3019276027352977`},{a4,1.3841167241622125`},{a5,1.7376798732548868`},{q2,-1.3946851930390745`},{q3,-1.5493753473850211`},{q4,-1.5707951912354967`},{q5,-1.3112567116360805`},{q6,-1.7469067766340123`}},WorkingPrecision->10000];//Timing

N[soln,10]

esoln1=Take[soln,4]/.x_?InexactNumberQ:>RootApproximant[N[x,1000],64]

Solve[4 (x^2) ((Sin[((4 π)/17)])^2)==1,x,PositiveReals]

N[(1/2) Csc[((4 π)/17)]]

Solve[4 (x^2) ((Sin[((2 π)/17)])^2)==1==1,x,PositiveReals]

N[(1/2) Csc[((2 π)/17)]]

(a3 and a5 are not the correct roots)

Take[soln,{2}]/.x_?InexactNumberQ:>RootApproximant[N[x,5000],128]//Timing

Ran for 6+ hours:

Take[soln,{2}]/.x_?InexactNumberQ:>RootApproximant[N[x,7000],256]//Timing

Take[soln,{4}]/.x_?InexactNumberQ:>RootApproximant[N[x,5000],128]//Timing

eqn/.{a2->(1/2) Csc[((4 π)/17)],a4->(1/2) Csc[((2 π)/17)]}

gb=Join[Subtract@@@TrigExpand[eqn]/.{a2->(NumericalApproximation[Root, (RowDefault["0.742", FEExpressionsNumericalApproximationElider]), Root[1-17 ((#1)^2)+119 ((#1)^4)-442 ((#1)^6)+935 ((#1)^8)-1122 ((#1)^10)+714 ((#1)^12)-204 ((#1)^14)+17 ((#1)^16)&,13], ]),a4->(NumericalApproximation[Root, (RowDefault["1.38", FEExpressionsNumericalApproximationElider]), Root[1-17 ((#1)^2)+119 ((#1)^4)-442 ((#1)^6)+935 ((#1)^8)-1122 ((#1)^10)+714 ((#1)^12)-204 ((#1)^14)+17 ((#1)^16)&,15], ])}/.{
Cos[q1]->q1c,Sin[q1]->q1s,
Cos[q2]->q2c,Sin[q2]->q2s,
Cos[q3]->q3c,Sin[q3]->q3s,
Cos[q4]->q4c,Sin[q4]->q3s,
Cos[q5]->q5c,Sin[q5]->q5s,
Cos[q6]->q6c,Sin[q6]->q6s
},{q1c^2+q1s^2-1,q2c^2+q2s^2-1,q3c^2+q3s^2-1,q4c^2+q4s^2-1,q5c^2+q5s^2-1,q6c^2+q6s^2-1}]//RootReduce//FullSimplify

gb//Expand//Column

gb[[{7,8}]]//Expand//Column

{((gb[[7]])/((NumericalApproximation[Root, (RowDefault["0.738", FEExpressionsNumericalApproximationElider]), Root[65536-65536 #1-40960 ((#1)^2)+10240 ((#1)^3)+3840 ((#1)^4)-384 ((#1)^5)-112 ((#1)^6)+4 ((#1)^7)+((#1)^8)&,5], ])-8)),gb[[8]]}//Expand//RootReduce//FullSimplify//Expand

Equal@@%

gb2=DeleteCases[Delete[gb,7],0]

vars=Variables[gb2]

Length/@{gb2,vars}

GroebnerBasis[gb,a2,Complement[vars,{a2}],MonomialOrder->EliminationOrder]

Take[soln,{4}]/.x_?InexactNumberQ:>RootApproximant[N[x,7000],256]//Timing

(eqn2=eqn/.{a2->(NumericalApproximation[Root, (RowDefault["0.742", FEExpressionsNumericalApproximationElider]), Root[1-17 ((#1)^2)+119 ((#1)^4)-442 ((#1)^6)+935 ((#1)^8)-1122 ((#1)^10)+714 ((#1)^12)-204 ((#1)^14)+17 ((#1)^16)&,13], ]),a4->(NumericalApproximation[Root, (RowDefault["1.38", FEExpressionsNumericalApproximationElider]), Root[1-17 ((#1)^2)+119 ((#1)^4)-442 ((#1)^6)+935 ((#1)^8)-1122 ((#1)^10)+714 ((#1)^12)-204 ((#1)^14)+17 ((#1)^16)&,15], ])})//Column

eqn3=FullSimplify[eqn2]

eqn4=RootReduce[TrigExpand[Subtract@@@DeleteCases[RootReduce[eqn3],True]]]

Length[eqn4]

vars=Variables[eqn4/.{Cos->Identity,Sin->Identity}]

Length[vars]

Once extra variable, so one degree of freedom

soln=FindRoot[Thread[eqn4==0]/.q2->-1/60,{{a3,1.3019276027352977`},{a5,1.7376798732548868`},{q3,-1.5493753473850211`},{q4,-1.5707951912354967`},{q5,-1.3112567116360805`},{q6,-1.7469067766340123`}},WorkingPrecision->100]

distanceregular.org

ToEntity[g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/biggssmith.am"]]

VertexCount[g]

GraphDiameter[g]//Timing

RegularParameters[g]//Timing

IntersectionArray[g]//Timing

Tally[Sort[Eigenvalues[AdjacencyMatrix[g]]]]//SpectrumForm

GroupOrder[GraphAutomorphismGroup[g]]

HamiltonianGraphQ[g]

GraphData

GraphClassString

GraphClassString["BiggsSmithGraph"]