Graph Density

Author

Eric W. Weisstein
January 20, 2025

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

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

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

GraphDataCountTable

GraphDataCountTable["Density"]

Parametrized families

<<MathWorld`Graphs`

Alkane

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

FindSequenceFunction[seq,n]

AlternatingGroupGraph

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

Andrasfai

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

FindSequenceFunction[seq,n]

Antelope n×n

seq=Table[GraphData[{"Antelope",{n,n}},"Density"],{n,20}]

Antiprism

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

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

FindSequenceFunction[seq,n-2]//Simplify

Apollonian

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

FindSequenceFunction[seq,n]//Head

(seq=Table[GraphDensity[ApollonianGraph[n]],{n,6}])//Timing

FindSequenceFunction[seq,n]//Head//Timing

Table[(((3/2) (1+(3^n)))/(Binomial[(1/2) (5+(3^n)),2])),{n,6}]

(((3/2) (1+(3^n)))/(Binomial[(1/2) (5+(3^n)),2]))//Simplify

Barbell

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

FindSequenceFunction[seq,n-2]//Factor

Bishop n×n

seq=Table[GraphData[{"Bishop",{n,n}},"Density"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Factor

BlackBishop n×n

seq=Table[GraphData[{"BlackBishop",{n,n}},"Density"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Head//Timing

Table[(((1/12) (((-1))^n) (-1+n) (-3+(((-1))^n) (3-2 n+4 (n^2))))/(Binomial[(1+2 (n^2)-(((-1))^n) )/4,2])),{n,20}]//Quiet

FullSimplify[(((1/12) (((-1))^n) (-1+n) (-3+(((-1))^n) (3-2 n+4 (n^2))))/(Binomial[(1+2 (n^2)-(((-1))^n) )/4,2])),n∈Integers]

Book

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

FindSequenceFunction[seq,n]//Factor

Bruhat

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

FindSequenceFunction[Rest@seq,n-1]//Head//Timing

Table[(((n-1)n!/2)/(Binomial[n!,2])),{n,6}]//Quiet

(((n-1)n!/2)/(Binomial[n!,2]))//Simplify

Camel n×n

seq=Table[GraphData[{"Camel",{n,n}},"Density"],{n,20}]

Table[((Piecewise[{{0,n<3}},4 (-3+n) (-1+n)])/(Binomial[(n^2),2])),{n,20}]//Quiet

Simplify[((Piecewise[{{0,n<3}},4 (-3+n) (-1+n)])/(Binomial[(n^2),2]))]

Centipede

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

FindSequenceFunction[seq,n]

CocktailParty

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

FindSequenceFunction[seq,n]//Factor

Complete

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

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

CompleteBipartite n×n

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

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

FindSequenceFunction[seq,n]

CompleteBipartite m×n

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

((Entity["Graph",{"CompleteBipartite",{m,n}}]["EdgeCount"])/(Entity["Graph",{"Complete",m+n}]["EdgeCount"]))

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"CompleteBipartite",{m,n}},"Density"],
((2 m n)/((-1+m+n) (m+n)))
},{n,10},{m,n}],1],{_,x_,x_}]

CompleteTripartite n×n×n

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

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

FindSequenceFunction[seq,n]

CompleteTripartite k×m×n

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

((Entity["Graph",{"CompleteTripartite",{k,m,n}}]["EdgeCount"])/(Entity["Graph",{"Complete",k+m+n}]["EdgeCount"]))

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"Density"],
((2 (k m+k n+m n))/((-1+k+m+n) (k+m+n)))
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CrossedPrism

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

FindSequenceFunction[seq,n-1]//Factor

Crown

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

FindSequenceFunction[seq,n]//Factor

CubeConnectedCycle

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

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

FindSequenceFunction[seq,n-2]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

Table[((3 (2^(-1+n)) n)/(Binomial[(2^n) n,2])),{n,3,8}]

((3 (2^(-1+n)) n)/(Binomial[(2^n) n,2]))

Cycle

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

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

FindSequenceFunction[seq,n-2]

CycleComplement

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

FindSequenceFunction[seq,n-2]

DiagonalIntersection

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

del[m_,n_]:=Boole[Divisible[n,m]]

A007569[n_]:=If[n<4,n,n+Binomial[n,4]+del[2,n] (-5 n^3+45 n^2-70 n+24)/24-del[4,n] (3 n/2)+del[6,n] (-45 n^2+262 n)/6+del[12,n]*42 n+del[18,n]*60 n+del[24,n]*35 n-del[30,n]*38 n-del[42,n]*82 n-del[60,n]*330 n-del[84,n]*144 n-del[90,n]*96 n-del[120,n]*144 n-del[210,n]*96 n]

A007678[n_]:=If[n<3,0,(n^4-6 n^3+23 n^2-42 n+24)/24+del[2,n] (-5 n^3+42 n^2-40 n-48)/48-del[4,n] (3 n/4)+del[6,n] (-53 n^2+310 n)/12+del[12,n] (49 n/2)+del[18,n]*32 n+del[24,n]*19 n-del[30,n]*36 n-del[42,n]*50 n-del[60,n]*190 n-del[84,n]*78 n-del[90,n]*48 n-del[120,n]*78 n-del[210,n]*48 n]

Table[((A007569[n]+A007678[n]-1)/(Binomial[A007569[n],2])),{n,3,14}]

Dipyramid

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

FindSequenceFunction[seq,n-2]//Factor

DorogovtsevGoltsevMendes

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"Density"],{n,0,7}]

FindSequenceFunction[seq,n+1]//FullSimplify

DoubleCone

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

FindSequenceFunction[seq,n-2]//Factor

Empty

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

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

FibonacciCube

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

FindSequenceFunction[seq,n]//Head//Timing

Table[(((1/10) ((4+5 n) Fibonacci[n]+n LucasL[n]))/(Binomial[Fibonacci[n+2],2])),{n,15}]

(((1/10) ((4+5 n) Fibonacci[n]+n LucasL[n]))/(Binomial[Fibonacci[n+2],2]))//FullSimplify

Fiveleaper n×n

seq=Table[GraphData[{"Fiveleaper",{n,n}},"Density"],{n,20}]

Table[((Piecewise[{{0,n<5}}, 2*(24 - 19*n + 3*n^2)])/(Binomial[(n^2),2])),{n,20}]//Quiet

FullSimplify[((Piecewise[{{0,n<5}}, 2*(24 - 19*n + 3*n^2)])/(Binomial[(n^2),2]))]//Factor

Flower

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

FindSequenceFunction[seq,n-4]//Factor

FoldedCube

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

FindSequenceFunction[seq,n-1]//Factor

Table[((Piecewise[{{1,n==2}},(2^(n-2))n])/(Binomial[(2^(-1+n)),2])),{n,2,10}]

FullSimplify[((Piecewise[{{1,n==2}},(2^(n-2))n])/(Binomial[(2^(-1+n)),2]))]

Gear

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

FindSequenceFunction[seq,n-2]//Factor

Giraffe n×n

seq=Table[GraphData[{"Giraffe",{n,n}},"Density"],{n,20}]

FindSequenceFunction[Drop[seq,4],n-4]//Factor

Goldberg

seq=Table[GraphData[{"Goldberg",n},"Density"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

Grid n×n

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

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

FindSequenceFunction[Rest@seq,n-1]//Factor

Grid m×n

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

Sort@DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Grid",{m,n}},"Density"],
Piecewise[{{Indeterminate,m==n==1}},((2 m n-m-n)/(Binomial[m n,2]))]
},{n,10},{m,n}],1],{_,x_,x_}]

((2 m n-m-n)/(Binomial[m n,2]))

Grid n×n×n

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

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

FindSequenceFunction[Rest@seq,n-1]//Factor

Grid k×m×n

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

Sort@DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"Grid",{k,m,n}},"Density"],
Piecewise[{{Indeterminate,k==m==n==1}},((3 k m n-k m-k n-m n)/(Binomial[k m n,2]))]
},{n,5},{m,n},{k,m}],2],{_,x_,x_}]

((3 k m n-k m-k n-m n)/(Binomial[k m n,2]))

HalvedCube

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

FindSequenceFunction[Rest@seq,n-1]//Head//Timing

Table[(((2^(-3+n)) (-1+n) n)/(Binomial[(2^(-1+n)),2])),{n,10}]//Quiet

(((2^(-3+n)) (-1+n) n)/(Binomial[(2^(-1+n)),2]))//FullSimplify

Hanoi

seq=Table[GraphData[{"Hanoi",n},"Density"],{n,6}]

FindSequenceFunction[seq,n]

Helm

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

FindSequenceFunction[seq,n-2]//Factor

Hypercube

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

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

FindSequenceFunction[Rest@seq,n+1]

KC

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"KC",{n,r}},"Density"],
((r(n+1)n/2)/(Binomial[r n,2]))
},{n,10},{r,3,10}],1],{_,x_,x_}]

((r(n+1)n/2)/(Binomial[r n,2]))//Simplify

Keller

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

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

FindSequenceFunction[seq,n]//Head//Timing

((Entity["Graph",{"Keller",n}]["EdgeCount"])/(Binomial[Entity["Graph",{"Keller",n}]["VertexCount"],2]))

King n×n

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

Table[((2 (-1+n) (-1+2 n))/(Binomial[(n^2),2])),{n,20}]//Quiet

((2 (-1+n) (-1+2 n))/(Binomial[(n^2),2]))//Simplify

King m×n

Sort@DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"King",{m,n}},"Density"],
Piecewise[{{Indeterminate,m==n==1}},((2 (2-3 m-3 n+4 m n))/(m n (-1+m n)))]
},{n,10},{m,n}],1],{_,x_,x_}]

Knight n×n

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

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

Table[((4 (-2+n) (-1+n))/(Binomial[(n^2),2])),{n,20}]//Quiet

((4 (-2+n) (-1+n))/(Binomial[(n^2),2]))//FullSimplify

Knight m×n

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

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"Knight",{m,n}},"Density"],
Piecewise[{{Indeterminate,m==n==1},{0,Min[m,n]==1}},((4 (4-3 n+m (-3+2 n)))/(m n (-1+m n)))]
},{n,10},{m,n}],1],{_,x_,x_}]

FullSimplify[((Piecewise[{{0,Min[m,n]==1}},8+4 m n-6 (m+n)])/(Binomial[m n,2])),m>1&&n>1]//Factor

KP

DeleteCases[Flatten[Table[{
{n,r},
GraphData[{"KP",{n,r}},"Density"],
Piecewise[{{Indeterminate,r==n==1}},((r Binomial[n,2]+(r-1)n)/(Binomial[r n,2]))]
},{r,10},{n,10}],1],{_,x_,x_}]

Ladder

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

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

FindSequenceFunction[%,n]

LadderRung

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

FindSequenceFunction[%,n]

LindgrenSousselier

seq=Table[GraphData[{"LindgrenSousselier",n},"Density"],{n,11}]

FindSequenceFunction[%,n]

LucasCube

seq=Table[GraphData[{"LucasCube",n},"Density"],{n,15}]

FindSequenceFunction[DeleteMissing@seq,n-1]//Head//Timing

Table[((n Fibonacci[n-1])/(Binomial[LucasL[n],2])),{n,15}]//Quiet

((n Fibonacci[n-1])/(Binomial[LucasL[n],2]))//FullSimplify

MengerSponge

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

Table[(((2^(1+2 n)) (-(2^n)+(5^n)))/(Binomial[(20^n),2])),{n,3}]

(((2^(1+2 n)) (-(2^n)+(5^n)))/(Binomial[(20^n),2]))//FullSimplify

MoebiusLadder

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

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

FindSequenceFunction[%,n-2]//Factor

Mycielski

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

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

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Table[Piecewise[{{Indeterminate,n==1}},((8 (18-27 (2^n)+14 (3^n)))/(9 (32-9 (2^(2+n))+9 (4^n))))],{n,11}]

Odd

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

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

FindSequenceFunction[Rest@seq,n-1]//Head//Timing

FullSimplify[((Entity["Graph",{"Odd",n}]["EdgeCount"])/(Binomial[Entity["Graph",{"Odd",n}]["VertexCount"],2])),n>1]

Table[Piecewise[{{Indeterminate,n==1}},(n/(-1+(((2^(-1+2 n)) Gamma[(1/2)+n])/((Sqrt[π]) Gamma[1+n]))))],{n,7}]

FunctionExpand[(n/(-1+(((2^(-1+2 n)) Gamma[(1/2)+n])/((Sqrt[π]) Gamma[1+n]))))]

FullSimplify[(n/(-1+(((2^(-1+2 n)) Gamma[(1/2)+n])/((Sqrt[π]) Gamma[1+n]))))/.Gamma[(1/2)+n]->(((2n-1)!!)/(2^n))(Sqrt[π])]

Table[((2 n Gamma[1+n])/((2^n) (-1+2 n)!!-2 Gamma[1+n])),{n,10}]

Table[Piecewise[{{Indeterminate,n==1}},((2 n n!)/((2^n) (2 n-1)!!-2 n!))],{n,10}]

Pan

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

FindSequenceFunction[seq,n-2]//Factor

Path

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

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

FindSequenceFunction[Rest@seq,n-1]//Factor

PathComplement

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

FindSequenceFunction[Rest@seq,n-1]//FullSimplify

Pell

seq=Table[GraphData[{"Pell",n},"Density"],{n,9}]

FindSequenceFunction[seq,n]//Head//Timing

Table[((n(Fibonacci[n,2]+(-I)^n ChebyshevT[n,I])/2)/(Binomial[Fibonacci[1+n,2],2])),{n,9}]

((n(Fibonacci[n,2]+(-I)^n ChebyshevT[n,I])/2)/(Binomial[Fibonacci[1+n,2],2]))//FullSimplify

Table[((n(Fibonacci[n,2]+(-I)^n ChebyshevT[n,I]))/(2Binomial[Fibonacci[1+n,2],2])),{n,9}]

PlummerToft

seq=Table[GraphData[{"PlummerToft",n},"Density"],{n,20}]

FindSequenceFunction[seq,n]//Factor

Prism

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

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

FindSequenceFunction[seq,n-2]//Factor

Queen n×n

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

FindSequenceFunction[Rest@seq,n-1]//Factor

Rook n×n

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

FindSequenceFunction[Rest@seq,n-1]//Factor

RookComplement n×n

seq=Table[GraphData[{"RookComplement",{n,n}},"Density"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Factor

SierpinskiCarpet

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

FindSequenceFunction[seq,n]//Head//Timing

Table[((8(8^n-3^n)/5)/(Binomial[(8^n),2])),{n,4}]

((8(8^n-3^n)/5)/(Binomial[(8^n),2]))//FullSimplify

SierpinskiGasket

seq=Table[GraphData[{"SierpinskiGasket",n},"Density"],{n,7}]

FindSequenceFunction[seq,n]//Head//Timing

Table[((3^n)/(Binomial[(1/2) (3+(3^n)),2])),{n,7}]

FullSimplify[((3^n)/(Binomial[(1/2) (3+(3^n)),2]))]

SierpinskiTetrahedron

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

FindSequenceFunction[seq,n]//Head//Timing

Table[((6 (4^(n-1)))/(Binomial[2+(2^(-1+2 n)),2])),{n,6}]

((6 (4^(n-1)))/(Binomial[2+(2^(-1+2 n)),2]))//FullSimplify

Spoke n×n

seq=Table[GraphData[{"Spoke",{n,n}},"Density"],{n,5}]

FindSequenceFunction[seq,n]

StackedBook m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"StackedBook",{m,n}},"Density"],
XXX
},{n,10},{m,n}],1],{_,x_,x_}]

StackedBook n×n

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

FindSequenceFunction[seq,n]//Factor

StackedPrism m×n

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"StackedPrism",{m,n}},"Density"],
XXX
},{m,3,8},{n,5}],1],{_,x_,x_}]

StackedPrism n×n

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

FindSequenceFunction[seq,n-2]

Star

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

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

FindSequenceFunction[Rest@seq,n-1]//Factor

Sun

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

FindSequenceFunction[seq,n-2]//Factor

Sunlet

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

FindSequenceFunction[seq,n-2]//Factor

Tetrahedral

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

FindSequenceFunction[seq,n-5]//Factor

TorusGrid n×n

seq=Table[GraphData[{"TorusGrid",{n,n}},"Density"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

Transposition

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

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

FindSequenceFunction[Rest@seq,n-1]//Head//Timing

Table[((Entity["Graph",{"Transposition",n}]["EdgeCount"])/(Binomial[Entity["Graph",{"Transposition",n}]["VertexCount"],2])),{n,6}]//Quiet

((Entity["Graph",{"Transposition",n}]["EdgeCount"])/(Binomial[Entity["Graph",{"Transposition",n}]["VertexCount"],2]))//FullSimplify

Trapezohedron

Entity["Graph",{"Trapezohedron",n}]["Density"]

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

FindSequenceFunction[seq,n]//Factor

Triangular

seq=Table[GraphData[{"Triangular",n},"Density"],{n,2,20}]

FindSequenceFunction[Rest@seq,n-2]//Factor

TriangularGrid

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

FindSequenceFunction[Rest@seq,n]//Factor

TriangularHoneycombAcuteKnight

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

FindSequenceFunction[Rest@seq,n-1]//Factor

TriangularHoneycombBishop

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"Density"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Factor

TriangularHoneycombObtuseKnight

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

FindSequenceFunction[Rest@seq,n-1]//Factor

TriangularHoneycombQueen

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"Density"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Factor

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"Density"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Factor

TriangularSnake

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

FindSequenceFunction[Rest@seq,n-1]//Factor

Web

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

FindSequenceFunction[Rest@seq,n-2]//Factor

Wheel

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

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

FindSequenceFunction[seq,n-3]//Factor

WheelComplement

seq=Table[GraphData[{"WheelComplement",n},"Density"],{n,4,20}]

FindSequenceFunction[seq,n-3]//Factor

WhiteBishop n×n

seq=Table[GraphData[{"WhiteBishop",{n,n}},"Density"],{n,2,20}]

FindSequenceFunction[seq,n-1]//Head//Timing

Table[(((n-1) (4 (n^2)-2n-3+3(((-1))^n))/12)/(Binomial[(1/4) (-1+(((-1))^n)+2 (n^2)),2])),{n,2,20}]

FullSimplify[(((n-1) (4 (n^2)-2n-3+3(((-1))^n))/12)/(Binomial[(1/4) (-1+(((-1))^n)+2 (n^2)),2]))]

FullSimplify[((8 (-1+n) (-3+3 (((-1))^n)-2 n+4 (n^2)))/(3 (-5+(((-1))^n)+2 (n^2)) (-1+(((-1))^n)+2 (n^2)))),Mod[n,2]==#]&/@{0,1}

WhiteCamel n×n

seq=Table[GraphData[{"Bishop",{n,n}},"Density"],{n,{5}}]

FindSequenceFunction[seq,n-1]//Factor

Zebra n×n

seq=Table[GraphData[{"Zebra",{n,n}},"Density"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Factor