Minimum Cyclic Edge Cut

Author

Eric W. Weisstein
November 21, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/MinimumCyclicEdgeCut.nb.

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Table

GraphDataCountTable["MinimumCyclicEdgeCutCount",Join[{"Antiprism"->071355,"Barbell"->000012,"Complete"->351860,"CrossedPrism"->000217,"Dipyramid"->028552,"Flower"->000096,{"Grid",2}->010709,"Hanoi"->008585,"Ladder"->000027,"MoebiusLadder"->000096,"Prism"->000096,"Trapezohedron"->005563,"TriangularGrid"->010701,"TriangularSnake"->005843,"Web"->000096},Thread[{"Alkane","Bishop","Book","Camel","Centipede","Cycle","Empty","Helm","Gear","LadderRung","Pan","Path","Star","Sunlet","TriangularHoneycombAcuteKnight","TriangularHoneycombRook","Wheel"}->000007]]]

Parametrized families

<<MathWorld`Graphs`

data=Partition[#,UpTo[10]]&/@Table[With[{gn={"Complete",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,6,8}];
GraphicsGrid[Flatten[data,1],Dividers->{Thread[{1,-1}->Thick],Thread[(1+Accumulate[Prepend[Length/@data,0]])->Thick]}]

Alkane

seq=Table[GraphData[{"Alkane",n},"MinimumCyclicEdgeCutCount"],{n,10}]

AlternatingGroupGraph

seq=Table[GraphData[{"AlternatingGroupGraph",n},"MinimumCyclicEdgeCutCount"],{n,2,6}]

Andrasfai

seq=Table[GraphData[{"Andrasfai",n},"MinimumCyclicEdgeCutCount"],{n,10}]

Antelope n×n

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumCyclicEdgeCutCount"],{n,15}]

Antiprism

Entity["Graph",{"Antiprism",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Antiprism",n},"MinimumCyclicEdgeCutCount"],{n,3,25}]

Table[Piecewise[{{4,n==3}},n(2n-5)],{n,3,25}]

GraphicsGrid[Flatten[Partition[#,UpTo[5]]&/@Table[With[{gn={"Antiprism",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,5}],1],Dividers->All]

Apollonian

seq=Table[GraphData[{"Apollonian",n},"MinimumCyclicEdgeCutCount"],{n,5}]

GraphicsGrid[Table[With[{gn={"Apollonian",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,2,2}],Dividers->All]

Barbell

seq=Table[GraphData[{"Barbell",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

BlackBishop n×n

seq=Table[GraphData[{"BlackBishop",{n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]

GraphicsGrid[Table[With[{gn={"BlackBishop",{n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,4}],Dividers->All]

Bishop n×n

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]

Book

seq=Table[GraphData[{"Book",n},"MinimumCyclicEdgeCutCount"],{n,20}]

Bruhat

seq=Table[GraphData[{"Bruhat",n},"MinimumCyclicEdgeCutCount"],{n,6}]

FindSequenceFunction[seq,n]

Camel n×n

seq=Table[GraphData[{"Camel",{n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]

CocktailParty

seq=Table[GraphData[{"CocktailParty",n},"MinimumCyclicEdgeCutCount"],{n,20}]

GraphicsGrid[Table[With[{gn={"CocktailParty",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,3}],Dividers->All]

Complete

A351860

Entity["Graph",{"Complete",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Complete",n},"MinimumCyclicEdgeCutCount"],{n,30}]

Table[Piecewise[{{0,n<6},{10,n==6}},Binomial[n,3]],{n,30}]==seq

gf=x FindGeneratingFunction[seq,x]

StringReplace[ToString[gf,InputForm]," "->""]

CoefficientList[Series[((x^5 (10-5 x-24 (x^2)+30 (x^3)-10 (x^4)))/(((-1+x))^4)),{x,0,20}],x]

data=Partition[#,UpTo[10]]&/@Table[With[{gn={"Complete",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,6,8}];

pos=1+Accumulate[Prepend[Length/@data,0]]

GraphicsGrid[Flatten[data,1],Dividers->{Thread[{1,-1}->Thick],Thread[pos->Thick]}]

FullSimplify[(2^(-(n-5)))Binomial[n,3]Binomial[n-3,3]]

5 (2^(7-n)) Binomial[n,6]//FunctionExpand//FullSimplify

Table[5 (2^(7-n)) Binomial[n,6],{n,8}]

CompleteBipartite m×n

Entity["Graph",{"CompleteBipartite",{m,n}}]["MinimumCyclicEdgeCutCount"]

CompleteBipartite n×n

Entity["Graph",{"CompleteBipartite",{n,n}}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"CompleteBipartite",{n,n}},"MinimumCyclicEdgeCutCount"],{n,20}]

data=Partition[#,UpTo[10]]&/@Table[With[{gn={"CompleteBipartite",{n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,4}];
GraphicsGrid[Flatten[data,1],Dividers->{Thread[{1,-1}->Thick],Thread[(1+Accumulate[Prepend[Length/@data,0]])->Thick]}]

CompleteTripartite k×m×n

Entity["Graph",{"CompleteTripartite",{k,m,n}}]["MinimumCyclicEdgeCutCount"]

CompleteTripartite n×n×n

Entity["Graph",{"CompleteTripartite",{n,n,n}}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"CompleteTripartite",{n,n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]

data=Partition[#,UpTo[10]]&/@Table[With[{gn={"CompleteTripartite",{n,n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,2,3}];
GraphicsGrid[Flatten[data,1],Dividers->{Thread[{1,-1}->Thick],Thread[(1+Accumulate[Prepend[Length/@data,0]])->Thick]}]

CrossedPrism

seq=Table[GraphData[{"CrossedPrism",2n},"MinimumCyclicEdgeCutCount"],{n,2,12}]

Table[Piecewise[{{3,n==2}},Binomial[n,2]],{n,2,12}]

GraphicsGrid[Table[With[{gn={"CrossedPrism",2n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,2,4}],Dividers->All]

Crown

seq=Table[GraphData[{"Crown",n},"MinimumCyclicEdgeCutCount"],{n,20}]

GraphicsGrid[Flatten[Partition[#,20]&/@Table[With[{gn={"Crown",n}},HighlightGraph[GraphData[gn,"Graph","Circular"][[1]],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,6}],1],Dividers->All]

Cycle

Entity["Graph",{"Cycle",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Cycle",n},"MinimumCyclicEdgeCutCount"],{n,3,30}]

CycleComplement

seq=Table[GraphData[{"CycleComplement",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

Table[GraphData[{"CycleComplement",n},"CyclicEdgeConnectivity"],{n,3,20}]

GraphicsGrid[Table[With[{gn={"CycleComplement",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,6,8}],Dividers->All]

DiagonalIntersection

seq=Table[GraphData[{"DiagonalIntersection",n},"MinimumCyclicEdgeCutCount"],{n,3,14}]

Dipyramid

A028552

seq=Table[GraphData[{"Dipyramid",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

GraphicsGrid[Flatten[Partition[#,UpTo[10]]&/@Table[With[{gn={"Dipyramid",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,6}],1],Dividers->All]

DoubleCone

seq=Table[GraphData[{"DoubleCone",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

Empty

seq=Table[GraphData[{"Empty",n},"MinimumCyclicEdgeCutCount"],{n,30}]

FibonacciCube

seq=Table[GraphData[{"FibonacciCube",n},"MinimumCyclicEdgeCutCount"],{n,15}]

Flower

Computed up to n==10; inferred beyond

seq=Table[GraphData[{"Flower",n},"MinimumCyclicEdgeCutCount"],{n,5,20}]

Table[Piecewise[{{1,n==5},{3,n==6}},n(n+3)/2],{n,5,20}]

data=Partition[#,UpTo[10]]&/@Table[With[{gn={"Flower",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,5,8}];

GraphicsGrid[Flatten[data,1],Dividers->{Thread[{1,-1}->Thick],Thread[(1+Accumulate[Prepend[Length/@data,0]])->Thick]}]

FoldedCube

seq=Table[GraphData[{"FoldedCube",n},"MinimumCyclicEdgeCutCount"],{n,2,10}]

Gear

seq=Table[GraphData[{"Gear",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

Giraffe n×n

seq=Table[GraphData[{"Giraffe",{n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]

Goldberg

seq=Table[GraphData[{"FoldedCube",n},"MinimumCyclicEdgeCutCount"],{n,2,10}]

Grid m×n

Entity["Graph",{"Grid",{m,n}}]["MinimumCyclicEdgeCutCount"]

DeleteCases[Sort@Flatten[Table[{
{m,n},
GraphData[{"Grid",{m,n}},"MinimumCyclicEdgeCutCount"],
Piecewise[{{0,Min[m,n]==1||m==n==2},{6,m==n==4},{Max[m,n]-3,Min[m,n]==2||Min[m,n]==3},{Max[m,n]+1,Min[m,n]==4}},4]
},{n,10},{m,n}],1],{_,x_,x_}]

Grid n×n

Entity["Graph",{"Grid",{n,n}}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Grid",{n,n}},"MinimumCyclicEdgeCutCount"],{n,20}]

GraphicsGrid[Table[With[{gn={"Grid",{n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,8}],Dividers->None,ImageSize->Large]

c1=Table[GraphData[{"Grid",{n,n}},"MinimumCyclicEdgeCuts","List"],{n,5,10}]

c2=Table[{{3,2n+1,2n+2,2n+3},{2n-5,4n-6,4n-3,4n-2},{(-1+n) (-5+2 n),(-1+n) (-5+2 n)+2,5-5 n+2 (n^2),3-3 n+2 (n^2)},{1-5 n+2 (n^2),1-5 n+2 (n^2)+1,-3-3 n+2 (n^2),-1-2 n+2 (n^2)}},{n,5,10}]

c1===c2

Grid k×m×n

Entity["Graph",{"Grid",{k,m,n}}]["MinimumCyclicEdgeCutCount"]

DeleteCases[Select[Sort@Flatten[Table[{
{k,m,n},
GraphData[{"Grid",{k,m,n}},"MinimumCyclicEdgeCutCount"],
Piecewise[{{0,k m n===Max[k,m,n]}},24]
},{n,10},{m,n},{k,m}],2],FreeQ[GraphData]],{_,x_,x_}]//Quiet

Tally[GraphData[{"Grid",{2,3,4}},"DistanceMatrix"]//Normal//Flatten]

Grid n×n×n

Entity["Graph",{"Grid",{n,n,n}}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Grid",{n,n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]

GraphicsGrid[Flatten[Partition[#,UpTo[8]]&/@Table[With[{gn={"Grid",{n,n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,2,2}],1],Dividers->All,ImageSize->Large]

GraphData[{"Grid",{3,3,3}},"Graph","Labeled"]

cut={{13,22},{14,23},{16,25},{17,26},{19,22},{20,23},{23,24},{26,27}};

HighlightGraph[GraphData[{"Grid",{3,3,3}}],UndirectedEdge@@@cut,GraphHighlightStyle->"Dotted"]

HalvedCube

seq=Table[GraphData[{"HalvedCube",n},"MinimumCyclicEdgeCutCount"],{n,10}]

Hanoi

seq=Table[GraphData[{"Hanoi",n},"MinimumCyclicEdgeCutCount"],{n,5}]

GraphicsGrid[Flatten[Partition[#,UpTo[4]]&/@Table[With[{gn={"Hanoi",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,2,4}],1],Dividers->All,ImageSize->Large]

Helm

seq=Table[GraphData[{"Helm",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

Hypercube

Entity["Graph",{"Hypercube",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Hypercube",n},"MinimumCyclicEdgeCutCount"],{n,0,10}]

GraphicsGrid[Flatten[Partition[#,UpTo[8]]&/@Table[With[{gn={"Hypercube",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,4}],1],Dividers->All,ImageSize->Large]

Keller

Entity["Graph",{"Keller",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Keller",n},"MinimumCyclicEdgeCutCount"],{n,5}]

FindSequenceFunction[seq,n]

King n×n

seq=Table[GraphData[{"King",{n,n}},"MinimumCyclicEdgeCutCount"],{n,20}]

GraphicsGrid[Flatten[Partition[#,UpTo[8]]&/@Table[With[{gn={"King",{n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,3}],1],Dividers->All,ImageSize->Large]

Knight m×n

Entity["Graph",{"Knight",{m,n}}]["MinimumCyclicEdgeCutCount"]

Knight n×n

Entity["Graph",{"Knight",{n,n}}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Knight",{n,n}},"MinimumCyclicEdgeCutCount"],{n,20}]

GraphicsGrid[Flatten[Partition[#,UpTo[8]]&/@Table[With[{gn={"Knight",{n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,4}],1],Dividers->All,ImageSize->Large]

Ladder

Entity["Graph",{"Ladder",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Ladder",n},"MinimumCyclicEdgeCutCount"],{n,20}]

Table[Piecewise[{{0,n==1||n==2}},n-3],{n,20}]

Table[With[{gn={"Ladder",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,6}]

LadderRung

seq=Table[GraphData[{"LadderRung",n},"MinimumCyclicEdgeCutCount"],{n,20}]

MengerSponge

seq=Table[GraphData[{"MengerSponge",n},"MinimumCyclicEdgeCutCount"],{n,3}]

GraphicsGrid[Flatten[Partition[#,UpTo[8]]&/@Table[With[{gn={"MengerSponge",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,1}],1],Dividers->None]

MoebiusLadder

Entity["Graph",{"MoebiusLadder",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"MoebiusLadder",n},"MinimumCyclicEdgeCutCount"],{n,3,25}]

Table[n(n-3)/2,{n,3,25}]

Binomial[n,2]-n//Factor

GraphicsGrid[Table[With[{gn={"MoebiusLadder",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,6}],Dividers->All,ImageSize->Large]

Mycielski

Entity["Graph",{"Mycielski",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Mycielski",n},"MinimumCyclicEdgeCutCount"],{n,11}]

Odd

Entity["Graph",{"Odd",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Odd",n},"MinimumCyclicEdgeCutCount"],{n,7}]

Path

Entity["Graph",{"Path",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Path",n},"MinimumCyclicEdgeCutCount"],{n,20}]

PathComplement

seq=Table[GraphData[{"PathComplement",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

GraphicsGrid[Table[With[{gn={"PathComplement",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,6,9}],Dividers->All]

Prism

Entity["Graph",{"Prism",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Prism",n},"MinimumCyclicEdgeCutCount"],{n,3,25}]

Table[Piecewise[{{1,n==3},{3,n==4}},n(n-3)/2],{n,3,25}]

Binomial[n,2]-n//Factor

GraphicsGrid[Table[With[{gn={"Prism",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,6}],Dividers->All]

Queen n×n

seq=Table[GraphData[{"Queen",{n,n}},"MinimumCyclicEdgeCutCount"],{n,20}]

FullSimplify[FindSequenceFunction[seq,n],n∈Integers]//Factor

Rook n×n

seq=Table[GraphData[{"Rook",{n,n}},"MinimumCyclicEdgeCutCount"],{n,20}]

FullSimplify[FindSequenceFunction[seq,n],n∈Integers]//Factor

RookComplement n×n

seq=Table[GraphData[{"RookComplement",{n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]

FindSequenceFunction[Drop[seq,2],n-2]//Factor

SierpinskiCarpet

seq=Table[GraphData[{"SierpinskiCarpet",n},"MinimumCyclicEdgeCutCount"],{n,4}]

SierpinskiSieve

seq=Table[GraphData[{"SierpinskiSieve",n},"MinimumCyclicEdgeCutCount"],{n,5}]

Table[12(n-2)+3,{n,5}]

GraphicsGrid[Flatten[Partition[#,UpTo[6]]&/@Table[With[{gn={"SierpinskiSieve",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,4}],1],Dividers->All]

SierpinskiTetrahedron

seq=Table[GraphData[{"SierpinskiTetrahedron",n},"MinimumCyclicEdgeCutCount"],{n,6}]

StackedBook

seq=Table[GraphData[{"StackedBook",{n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]

StackedPrism

seq=Table[GraphData[{"StackedPrism",{n,n}},"MinimumCyclicEdgeCutCount"],{n,3,5}]

GraphicsGrid[Table[With[{gn={"StackedPrism",{n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,4}],Dividers->All]

Star

Entity["Graph",{"Star",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Star",n},"MinimumCyclicEdgeCutCount"],{n,25}]

Sun

seq=Table[GraphData[{"Sun",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

GraphicsGrid[Flatten[Partition[#,UpTo[10]]&/@Table[With[{gn={"Sun",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,6}],1],Dividers->All]

Sunlet

seq=Table[GraphData[{"Sunlet",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

TorusGrid n×n

seq=Table[GraphData[{"TorusGrid",{n,n}},"MinimumCyclicEdgeCutCount"],{n,3,10}]

GraphicsGrid[Table[With[{gn={"TorusGrid",{n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,4}]/._Missing->Nothing,Dividers->All,ImageSize->Large]

Transposition

Entity["Graph",{"Transposition",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Transposition",n},"MinimumCyclicEdgeCutCount"],{n,6}]

Trapezohedron

A005563

seq=Table[GraphData[{"Trapezohedron",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

Range[20]^2-1

seq=Table[n (n+2),{n,20}]

gf=FindGeneratingFunction[seq,x]

CoefficientList[Series[((-3+x)/(((-1+x))^3)),{x,0,20}],x]

FindLinearRecurrence[seq]

LinearRecurrence[{3,-3,1},{3,8,15},20]

Triangular

seq=Table[GraphData[{"Triangular",n},"MinimumCyclicEdgeCutCount"],{n,2,10}]

GraphicsGrid[Table[With[{gn={"Triangular",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,5}]/._Missing->Nothing,Dividers->All,ImageSize->Large]

TriangularGrid

seq=Table[GraphData[{"TriangularGrid",n},"MinimumCyclicEdgeCutCount"],{n,0,20}]

GraphicsGrid[Table[With[{gn={"TriangularGrid",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,5}]/._Missing->Nothing,Dividers->All,ImageSize->Large]

TriangularHoneycombAcuteKnight

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimumCyclicEdgeCutCount"],{n,20}]

TriangularHoneycombBishop

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimumCyclicEdgeCutCount"],{n,9}]

GraphicsGrid[Table[With[{gn={"TriangularHoneycombBishop",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,5}]/._Missing->Nothing,Dividers->All,ImageSize->Large]

TriangularHoneycombObtuseKnight

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"MinimumCyclicEdgeCutCount"],{n,20}]

TriangularHoneycombQueen

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"MinimumCyclicEdgeCutCount"],{n,10}]

data=Partition[#,UpTo[10]]&/@Table[With[{gn={"TriangularHoneycombQueen",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,4}];
GraphicsGrid[Flatten[data,1],Dividers->{Thread[{1,-1}->Thick],Thread[(1+Accumulate[Prepend[Length/@data,0]])->Thick]}]

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimumCyclicEdgeCutCount"],{n,10}]

TriangularSnake

seq=Table[GraphData[{"TriangularSnake",n},"MinimumCyclicEdgeCutCount"],{n,1,20,2}]

GraphicsGrid[Table[With[{gn={"TriangularSnake",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dotted"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,7,11,2}],Dividers->All]

Web

seq=Table[GraphData[{"Web",n},"MinimumCyclicEdgeCutCount"],{n,3,20}]

Table[Piecewise[{{1,n==3},{3,n==4}},n(n-3)/2],{n,3,20}]

Binomial[n,2]-n//Factor

GraphicsGrid[Table[With[{gn={"Web",n}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,3,5}],Dividers->All,ImageSize->Large]

Wheel

Entity["Graph",{"Wheel",n}]["MinimumCyclicEdgeCutCount"]

seq=Table[GraphData[{"Wheel",n},"MinimumCyclicEdgeCutCount"],{n,4,20}]

WhiteBishop n×n

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumCyclicEdgeCutCount"],{n,2,10}]

GraphicsGrid[Table[With[{gn={"WhiteBishop",{n,n}}},HighlightGraph[GraphData[gn],#,GraphHighlightStyle->"Dashed"]&/@GraphData[gn,"MinimumCyclicEdgeCuts"]],{n,4,5}]/._Missing->Nothing,Dividers->All,ImageSize->Large]

Zebra n×n

seq=Table[GraphData[{"Zebra",{n,n}},"MinimumCyclicEdgeCutCount"],{n,10}]