Isolated Vertex

Author

Eric W. Weisstein
September 4, 2025

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

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

©2025 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

Graphs with no isolated points

Graphs/@Range[9];

Length/@(iso=Select[#,GraphicalPartitionQ[CombinatoricaGraph[#]]]&/@Graphs/@Range[9])//Timing

Length/@Table[Select[GraphData[n],GraphData[#,"MinimumVertexDegree"]>0&],{n,7}]//Timing

Length/@Table[Select[GraphData[n],GraphData[#,"IsolatedPointCount"]==0&],{n,7}]//Timing

Graphic

GraphicsGrid[Map[StyleGraphs[GraphData[#],VertexSize->{"Scaled",.1}]&,Flatten[Partition[#,UpTo[8]]&/@Table[Select[GraphData[n],GraphData[#,"IsolatedPointCount"]==0&],{n,5}],1],{2}],ImageSize->500,Dividers->{All,{Thick,Thick,Thick,Thick,{Automatic},Thick}}]

GraphDataCountTable

<<MathWorld`Graphs`

table=GraphDataCountTable["IsolatedPointCount"]

TextGrid[OEISUpdates[table],Dividers->All]

Parametrized families

Andrasfai

seq=Table[GraphData[{"Andrasfai",n},"IsolatedPointCount"],{n,20}]

Antelope n×n

Table[GraphData[{"Antelope",{n,n}},"Connected"],{n,10}]

seq=Table[GraphData[{"Antelope",{n,n}},"IsolatedPointCount"],{n,10}]

Antiprism

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

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

Apollonian

seq=Table[GraphData[{"Apollonian",n},"IsolatedPointCount"],{n,6}]

Barbell

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

Bishop n×n

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

BlackBishop m×n

Table[GraphData[{"BlackBishop",{m,n}},"IsolatedPointCount"],{n,10},{m,n}]//Column

BlackBishop n×n

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

Book

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

Bruhat

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

Centipede

seq=Table[GraphData[{"Centipede",n},"IsolatedPointCount"],{n,10}]

CocktailParty

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

Complete

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

seq=Table[GraphData[{"Complete",n},"IsolatedPointCount"],{n,20}]

CompleteBipartite m×n

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

DeleteCases[Flatten[Table[{{m,n},GraphData[{"CompleteBipartite",{m,n}},"IsolatedPointCount"],0},{n,20},{m,n}],1],{_,x_,x_}]

CompleteBipartite n×n

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

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

CompleteTripartite k×m×n

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

DeleteCases[Flatten[Table[{{m,n},GraphData[{"CompleteTripartite",{k,m,n}},"IsolatedPointCount"],0},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

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

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

CrossedPrism

seq=Table[GraphData[{"CrossedPrism",2n},"IsolatedPointCount"],{n,2,10}]

Crown

seq=Table[GraphData[{"Crown",n},"IsolatedPointCount"],{n,3,20}]

CubeConnectedCycle

Entity["Graph",{"CubeConnectedCycle",n}]["IsolatedPointCount"]

seq=Table[GraphData[{"CubeConnectedCycle",n},"IsolatedPointCount"],{n,3,8}]

Cycle

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

seq=Table[GraphData[{"Cycle",n},"IsolatedPointCount"],{n,3,20}]

DiagonalIntersection

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

Dipyramid

seq=Table[GraphData[{"Dipyramid",n},"IsolatedPointCount"],{n,3,10}]

Empty

Entity["Graph",{"Empty",n}]["IsolatedPointCount"]

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

FibonacciCube

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

Fiveleaper n×n

Table[GraphData[{"Fiveleaper",{n,n}},"Connected"],{n,10}]

seq=Table[GraphData[{"Fiveleaper",{n,n}},"IsolatedPointCount"],{n,10}]

Gear

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

GeneralizedPetersen

Sort[{"GeneralizedPetersen"/.GraphData[#,"NotationRules"],GraphData[#,"IsolatedPointCount"]}&/@GraphData["GeneralizedPetersen"]]

DeleteCases[%,{_,0}]

Grid m×n

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

DeleteCases[Flatten[Table[{{m,n},GraphData[{"Grid",{m,n}},"IsolatedPointCount"],Piecewise[{{1,m==n==1}}, 0]},{n,10},{m,n}],1],{_,x_,x_}]

Grid n×n

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

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

Grid k×m×n

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

DeleteCases[Flatten[Table[{{k,m,n},GraphData[{"Grid",{k,m,n}},"IsolatedPointCount"],Piecewise[{{1, k==m==n==1}}, 0]},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

Grid n×n×n

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

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

HalvedCube

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

Hanoi

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

Helm

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

Hypercube

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

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

Keller

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

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

King m×n

Table[GraphData[{"King",{m,n}},"IsolatedPointCount"],{n,10},{m,n}]//Column

King n×n

seq=Table[GraphData[{"King",{n,n}},"IsolatedPointCount"],{n,10}]

Knight m×n

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

DeleteCases[Flatten[Table[{{m,n},GraphData[{"Knight",{m,n}},"IsolatedPointCount"], 0},{n,10},{m,n}],1],{_,x_,x_}]

Knight n×n

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

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

Ladder

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

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

LadderRung

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

MoebiusLadder

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

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

Mycielski

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

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

Odd

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

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

Pan

seq=Table[GraphData[{"Pan",n},"IsolatedPointCount"],{n,3,20}]

Path

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

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

PathComplement

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

Prism

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

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

Queen m×n

DeleteCases[Flatten[Table[{{m,n},GraphData[{"Queen",{m,n}},"IsolatedPointCount"],0},{n,10},{m,n}],1],{_,x_,x_}]

Queen n×n

seq=Table[GraphData[{"Queen",{n,n}},"IsolatedPointCount"],{n,10}]

Rook n×n

seq=Table[GraphData[{"Rook",{n,n}},"IsolatedPointCount"],{n,10}]

RookComplement  n×n

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

StackedBook n×n

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

Star

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

seq=Table[GraphData[{"Star",n},"IsolatedPointCount"],{n,20}]

Sun

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

Sunlet

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

Tetrahedral

seq=Table[GraphData[{"Tetrahedral",n},"IsolatedPointCount"],{n,6,10}]

TorusGrid n×n

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

Transposition

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

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

Triangular

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

TriangularGrid

seq=Table[GraphData[{"TriangularGrid",n},"IsolatedPointCount"],{n,0,10}]

TriangularHoneycombAcuteKnight

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

TriangularHoneycombBishop

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

TriangularHoneycombObtuseKnight

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

TriangularHoneycombQueen

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

TriangularHoneycombRook

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

Web

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

Wheel

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

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

WhiteBishop n×n

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