Molecular Topological Index

Author

Eric W. Weisstein
March 13, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Examples

3-methylpentane [112]

Devellers and Balaban, pp. 139–140

ShowLabeledGraph[g=AddEdge[AddVertex[Path[5],{3,1}],{3,6}]]

MolecularTopologicalIndex[g]

RecognizeGraph[g]

113, 114, 155 (all have MTI = 440)

Devellers and Balaban, p. 140

113

ShowLabeledGraph[g1=AddEdges[AddVertices[Path[6],{{4,1},{4,2},{3,1},{5,1}}],{{4,7},{7,8},{9,7},{7,10}}]]

MolecularTopologicalIndex[g1]

RecognizeGraph[g1]

114

ShowLabeledGraph[g2=AddEdges[AddVertices[Path[7],{{3,1},{5,1},{3,-1}}],{{4,8},{4,9},{3,10}}]]

MolecularTopologicalIndex[g2]

RecognizeGraph[g2]

115

ShowLabeledGraph[g3=AddEdges[AddVertices[Path[6],{{3,1},{5,1},{5,2},{2,1}}],{{2,10},{4,7},{4,8},{8,9}}]]

MolecularTopologicalIndex[g3]

RecognizeGraph[g3]

All

GraphicsRow[GraphPlot[#,Method->None]&/@{g1,g2,g3},Spacings->-20]

Counts

classes={"Andrasfai",(*"AntelopesTour",*)"Antiprism","Apollonian","Barbell","BipartiteKneser","Book","CocktailParty","Complete","CompleteBipartite","CompleteTripartite","CrossedPrism","Crown","CubeConnectedCycle","Cycle","Cyclotomic","Doob","Fan",(*"FiveleapersTour",*)"FoldedCube","Gear","Grid","HalvedCube","Hamming","Hanoi","Helm","Hypercube","Johnson","KingsTour","Kneser",(*"KnightsTour",*)"Ladder",(*"LadderRung",*)"Lattice","MoebiusLadder","Mycielski","Odd","Pan","Path","PermutationStar","Prism","QueensTour","Square","StackedBook","Star","Sun","Sunlet","Tetrahedral","TorusGrid","Triangular","Web","Wheel"};

TableForm[Outer[GraphData[{##},"MolecularTopologicalIndex"]&,classes,Range[10]]/.GraphData[{c_,n_},a_]:>GraphData[{c,{n,n}},a]/.GraphData[{c_,{n_,n_}},a_]:>GraphData[{c,{n,n,n}},a]/.{_Missing->"?",_GraphData->"X"},TableDepth->2,TableHeadings->{classes,None},TableAlignments->{{Left,Right},Automatic}]//Quiet

Parametrized families

Antiprism

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

seq=Table[GraphData[{"Antiprism",n},"MolecularTopologicalIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

Apollonian

A192792

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

gf=x((36 (2-48 x+421 (x^2)-1584 (x^3)+2096 (x^4)+1960 (x^5)-9573 (x^6)+17670 (x^7)-25056 (x^8)+15552 (x^9)))/((1-x) (1-2 x) (((1-3 x))^2) (1-4 x) (((1-9 x))^2) (1+2 x) (1+2 (x^2))));

seq=CoefficientList[Series[((36 (2-48 x+421 (x^2)-1584 (x^3)+2096 (x^4)+1960 (x^5)-9573 (x^6)+17670 (x^7)-25056 (x^8)+15552 (x^9)))/((1-x) (1-2 x) (((1-3 x))^2) (1-4 x) (((1-9 x))^2) (1+2 x) (1+2 (x^2)))),{x,0,50}],x]

coef=FindLinearRecurrence[seq]

Length@coef

Take[seq,10]

LinearRecurrence[{29,-320,1676,-4109,2537,9182,-26346,43920,-49896,23328},{72,360,2556,22572,219636,2204244,22197420,222257988,2207645892,21754722852},20]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n>0]

Table[((3 (3025+605 (2^(2+n))+3 (((-1))^n) (2^(4+n))+605 (2^(3+2 n))+5 (9^n) (63+44 n)+4 (3^n) (-277+55 n)+35 (2^(2+(n/2))) Cos[((n π)/2)]-15 (2^((7+n)/2)) Sin[((n π)/2)]))/1210),{n,20}]

Butterfly

Entity["Graph",{"Butterfly",n}]["MolecularTopologicalIndex"]

seq=Table[CycleCount[System`ButterflyGraph[n],6],{n,6}]

Butterfly (n,b)

Entity["Graph",{"Butterfly",{n,b}}]["MolecularTopologicalIndex"]

(seq=Table[CycleCount[System`ButterflyGraph[n,b],6],{b,4},{n,5}])//Column//Timing

Circulant

Table[GraphData[#,"MolecularTopologicalIndex"]&/@GraphData["Circulant",n],{n,10}]

Complete

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

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

FindSequenceFunction[seq,n]//Factor

CompleteBipartite m×n

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

DeleteCases[Flatten[Table[{
{m,n},
GraphData[{"CompleteBipartite",{m,n}},"MolecularTopologicalIndex"],
4 m n (-1+m+n)
},{n,20},{m,n}],1],{_,x_,x_}]

CompleteBipartite n×n

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

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

FindSequenceFunction[seq,n]//Factor

CompleteTripartite k×m×n

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

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"MolecularTopologicalIndex"],
4 (-1+k+m+n) (k m+k n+m n)
},{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

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

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

FindSequenceFunction[seq,n]//Factor

CubeConnectedCycle

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

Table[GraphData[{"CubeConnectedCycle",n},"MolecularTopologicalIndex"],{n,3,8}]

FindSequenceFunction[seq,n-2]//Factor

Cycle

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

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

FindSequenceFunction[seq,n-2]//FullSimplify[#,n∈Integers]&

DiagonalIntersection

A293365

Table[GraphData[{"DiagonalIntersection", n},"MolecularTopologicalIndex"],{n,3,14}]

Empty

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

Table[GraphData[{"Empty", n},"MolecularTopologicalIndex"],{n,30}]

DeBruijn

?DeBruijnGraph

Table[System`DeBruijnGraph[m,n],{m,3},{n,3}]

GeneralizedPetersen

Entity["Graph",{"GeneralizedPetersen",{n,k}}]["MolecularTopologicalIndex"]

DeleteCases[Sort@Flatten[Table[{
{n,k},
CycleCount[System`PetersenGraph[n,k],6],
Piecewise[{{n,k==1}},0]
},{n,3,50},{k,(n-1)/2}],1],{_,x_,x_}]

Grid m×n

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

Table[GraphData[{"Grid",{m,n}},"MolecularTopologicalIndex"],{n,10},{m,n}]//Column

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

Grid n×n

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

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

FindSequenceFunction[seq,n]//Factor

Table[(2/3) (2+n) (6-24 n+25 (n^2)-13 (n^3)+4 (n^4)),{n,10}]

Grid k×m×n

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

DeleteCases[Flatten[Table[GraphData[{"Grid",{k,m,n}},"MolecularTopologicalIndex"],{n,5},{m,n},{k,m}],2],{_,x_,x_}]

Grid n×n×n

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

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

FindSequenceFunction[seq,n]//Factor

Table[n (24-66 n+36 (n^2)+4 (n^3)-3 (n^4)-7 (n^5)+6 (n^6)),{n,10}]

Harary

?HararyGraph

Entity["Graph",{"Harary",{k,n}}]["MolecularTopologicalIndex"]

{mkn="Harary"/.GraphData[#,"NotationRules"],GraphData[#,"MolecularTopologicalIndex"]}&/@GraphData["Harary"]

Hypercube

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

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

FindSequenceFunction[seq,n+1]//Factor

Table[(2^n) (1+(2^(-1+n))) (n^2),{n,0,10}]

Keller

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

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

FindSequenceFunction[seq,n]

Knight m×n

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

Table[GraphData[{"Knight",{m,n}},"MolecularTopologicalIndex"],{n,10},{m,n}]//Column

Knight n×n

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

seq=Table[GraphData[{"Knight",{n,n}},"MolecularTopologicalIndex"],{n,10}]

FindSequenceFunction[Drop[seq,3],n-3]

MengerSponge

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

MoebiusLadder

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

seq=Table[GraphData[{"MoebiusLadder",n},"MolecularTopologicalIndex"],{n,3,20}]

FindSequenceFunction[seq,n-2]//Factor

Odd

A192835

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

FindSequenceFunction[seq,n]

FindSequenceFunction[Rest@seq,n-1]

FindGeneratingFunction[seq,x]//Factor

(seq=Table[MolecularTopologicalIndex[CombinatoricaGraphToV8Graph@OddGraph[n]],{n,2,8}])//Timing

b(n)=sum(i=1, n-1, binomial(n-1, i)*binomial(n, i)*min(2*i, 2*(n-i)-1));
a(n)=if(n<2, 0, binomial(2*n-1, n)*n*(n+b(n)))

Table[Sum[Binomial[n-1,i]Binomial[n,i]Min[2i,2(n-i)-1],{i,n-1}],{n,2,7}]

seq=Table[Binomial[2n-1,n]n(n+Sum[Binomial[n-1,i]Binomial[n,i]Min[2i,2(n-i)-1],{i,n-1}]),{n,2,20}]

FindLinearRecurrence[seq]

FindSequenceFunction[seq,n-1]

gf=(x^2)FindGeneratingFunction[seq,x]

Sum[Binomial[n-1,i]Binomial[n,i]Min[2i,2(n-i)-1],{i,n-1}]

PiecewiseExpand[Min[2 i,-1+2 (-i+n)]]

Solve[2i-n==-1/2,i]

Table[Sum[Binomial[n-1,i]Binomial[n,i]2i,{i,(1/4) (2 n-1)}]+Sum[Binomial[n-1,i]Binomial[n,i](-1-2 i+2 n),{i,Ceiling[(1/4) (2 n-1)],n-1}],{n,2,7}]

sum=Sum[Binomial[n-1,i]Binomial[n,i]2i,{i,(1/4) (2 n-1)}]+Sum[Binomial[n-1,i]Binomial[n,i](-1-2 i+2 n),{i,Ceiling[(1/4) (2 n-1)],n-1}]

FullSimplify[sum,Mod[n,2]==0]

FullSimplify[sum,Mod[n,2]==1]

Path

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

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

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

PermutationStar

A192837

Table[GraphData[{"PermutationStar",n},"MolecularTopologicalIndex"],{n,6}]

Construct using:

A007799

a[n_,0]=1;a[n_,1]=n-1;a[n_,2]=(n-1)(n-2);a[n_,k_/;k≥2]:=a[n,k]=(n-1)a[n-1,k-1]+Sum[j a[j,k-3],{j,n-2}];

Sum[j a[j,k-3],{j,n-2}]

Table[a[n,k],{n,9},{k,0,Floor[3(n-1)/2]}]//Column

Take[seq=Table[n!(n-1)(n-1+Sum[k a[n,k],{k,Floor[3(n-1)/2]}]),{n,50}],20]

FindLinearRecurrence[seq]//Head//Timing

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

Prism

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

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

FindSequenceFunction[seq,n-2]//FullSimplify

Rook n×n

A192832

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

FindSequenceFunction[seq,n]//Factor

RookComplement n×n

A292282

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

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

seq=Table[2 (((-1+n))^3) (n^2) (1+n),{n,20}]

coef=FindLinearRecurrence[seq]

Take[seq,Length@coef]

LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,24,576,4320,19200,63000,169344},30]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[-((24 x(1+17 x+33 (x^2)+9 (x^3)))/(((-1+x))^7)),{x,0,20}],x]

Star

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

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

FindSequenceFunction[seq,n]

Transposition

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

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

FindSequenceFunction[seq,n]

Wheel

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

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

FindSequenceFunction[seq,n]//Factor

Co-MTI

List

Table[Total[Length/@Select[Split[Sort[MolecularTopologicalIndex/@ConnectedGraphs[n,"Format"->"V7"]]],Length[#]>1&]],{n,8}]//Timing

Table[Total[Length/@Select[Split[Sort[MolecularTopologicalIndex/@ConnectedGraphs[n,"Format"->"V7"]]],Length[#]>1&]],{n,9}]//Timing

3 Connected

{#[[1]],Length[#]}&/@Select[Split[Sort[MolecularTopologicalIndex/@ConnectedGraphs[3,"Format"->"V7"]]],Length[#]>1&]

4 All

GraphicsGrid[Partition[Graphs[4],5,5,{1,1},{}],Dividers->All]

Partition[MolecularTopologicalIndex/@Graphs[4,"Format"->"V7"],5,5,{1,1},{}]

4 Connected

{#[[1]],Length[#]}&/@Select[Split[Sort[MolecularTopologicalIndex/@ConnectedGraphs[4,"Format"->"V7"]]],Length[#]>1&]

{#[[1,1]],#[[All,2]]}&/@Select[SplitBy[SortBy[{MolecularTopologicalIndex[#],RecognizeGraph[#]}&/@ConnectedGraphs[4,"Format"->"V7"],First],First],Length[#]>1&]

5 All

GraphicsGrid[Partition[Graphs[5],8,8,{1,1},{}],Dividers->All]

Partition[MolecularTopologicalIndex/@Graphs[5,"Format"->"V7"],8,8,{1,1},{}]//Column

5 Connected

{#[[1]],Length[#]}&/@Select[Split[Sort[MolecularTopologicalIndex/@ConnectedGraphs[5,"Format"->"V7"]]],Length[#]>1&]

{#[[1,1]],#[[All,2]]}&/@Select[SplitBy[SortBy[{MolecularTopologicalIndex[#],RecognizeGraph[#]}&/@ConnectedGraphs[5,"Format"->"V7"],First],First],Length[#]>1&]//Column

Special Classes

Andrasfai

l=Table[GraphData[{"Andrasfai",n},"MolecularTopologicalIndex"],{n,10}]

FindSequenceFunction[l,n]//Factor

Table[2 n (-2+3 n) (-1+3 n),{n,30}]

l/4

Simplify[2 n (-2+3 n) (-1+3 n)/4]

Antiprism

l=Table[GraphData[{"Antiprism",n},"MolecularTopologicalIndex"],{n,3,20}]

FindSequenceFunction[l][n-2]//Factor

Table[4 n (8+n+(n^2)),{n,30}]

l/8

Apollonian

l=DeleteCases[Table[GraphData[{"Apollonian",n},"MolecularTopologicalIndex"],{n,10}],_GraphData|_Missing]//Quiet

l/4

FindSequenceFunction[l][n]//Factor

FindLinearRecurrence[l]

CocktailParty

l=Table[GraphData[{"CocktailParty",n},"MolecularTopologicalIndex"],{n,20}]//Quiet

FindSequenceFunction[Drop[l,1],n-1]//Factor

Table[8 (-1+n) n (-1+2 n),{n,30}]

Table[(-1+n) n (-1+2 n)/2,{n,30}]

CompleteBipartite

l=Table[GraphData[{"CompleteBipartite",{n,n}},"MolecularTopologicalIndex"],{n,10}]

FindSequenceFunction[l,n]//Factor

Table[4 (n^2) (-1+2 n),{n,30}]

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

Complete

l=Table[GraphData[{"Complete",n},"MolecularTopologicalIndex"],{n,20}]

FindSequenceFunction[l,n]//Factor

Table[2 (((-1+n))^2) n,{n,30}]

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

CompleteTriipartite

l=DeleteCases[Table[GraphData[{"CompleteTripartite",{n,n,n}},"MolecularTopologicalIndex"],{n,10}],_Missing|_GraphData]//Quiet

FindSequenceFunction[l,n]//Factor

Table[12 (n^2) (-1+3 n),{n,30}]

Table[12 (n^2) (-1+3 n)/24,{n,20}]

CrossedPrism

l=Table[GraphData[{"CrossedPrism",2n},"MolecularTopologicalIndex"],{n,2,10}]//Quiet

FindSequenceFunction[l,n-1]//Factor

Table[12 n (7+2 (n^2)),{n,30}]

Table[12 n (7+2 (n^2))/12,{n,20}]

Crown

l=Table[GraphData[{"Crown",n},"MolecularTopologicalIndex"],{n,3,10}]//Quiet

FindSequenceFunction[l,n-2]//Factor

Table[2 (-1+n) n (-1+4 n),{n,30}]

l/4

CubeConnectedCycle

l=Table[GraphData[{"CubeConnectedCycle",n},"MolecularTopologicalIndex"],{n,3,8}]//Quiet

MyCubeConnectedCycle[n_]:=MakeGraph[Flatten[Outer[List,Range[0,2^n-1],Range[0,n-1]],1],
(#1[[1]]==#2[[1]]&&#2[[2]]==Mod[#1[[2]]+1,n])||(#1[[1]]==#2[[1]]&&#2[[2]]==Mod[#1[[2]]-1,n])||(#1[[2]]==#2[[2]]&&#2[[1]]==BitXor[#1[[1]],2^#1[[2]]])&,Type->Undirected]

l=Table[MolecularTopologicalIndex[MyCubeConnectedCycle[n]],{n,8}]

l={12,288,5544,57408,458400,3339648,21641088,133367808}

l/12

FindSequenceFunction[l,n]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

l=Table[MolecularTopologicalIndex[MyCubeConnectedCycle[n]],{n,3,8}]

FindSequenceFunction[l,n-2]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

l={1,24,462,4784,38200,278304,1803424,11113984}

FindSequenceFunction[l,n]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

Cycle

l=Table[GraphData[{"Cycle",n},"MolecularTopologicalIndex"],{n,3,20}]//Quiet

FindSequenceFunction[l,n-2]//FullSimplify

Table[(1/4) n (15+(((-1))^n)+2 (n^2)),{n,30}]

%/4

Empty

l=Table[GraphData[{"Empty",n},"MolecularTopologicalIndex"],{n,20}]

FoldedCube

l=Table[GraphData[{"FoldedCube",n},"MolecularTopologicalIndex"],{n,2,10}]//Quiet

FindSequenceFunction[l,n-1]

l/4

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

Gear

l=Table[GraphData[{"Gear",n},"MolecularTopologicalIndex"],{n,3,20}]//Quiet

FindSequenceFunction[l,n-2]//Factor

Table[11 n (-2+3 n),{n,30}]

Table[11 n (-2+3 n)/11,{n,20}]

Grid n×n

l=Table[GraphData[{"Grid",{n,n}},"MolecularTopologicalIndex"],{n,2,10}]

l/4

FindSequenceFunction[l,n-1]//Factor//InputForm

Table[(2/3) (2+n) (6-24 n+25 (n^2)-13 (n^3)+4 (n^4)),{n,2,30}]

Grid n×n×n

l=Table[GraphData[{"Grid",{n,n,n}},"MolecularTopologicalIndex"],{n,2,10}]

FindSequenceFunction[l,n-1]//Factor//InputForm

Table[n*(24-66*n+36*n^2+4*n^3-3*n^4-7*n^5+6*n^6),{n,2,30}]

HalvedCube

l=Table[GraphData[{"HalvedCube",n},"MolecularTopologicalIndex"],{n,10}]

FindSequenceFunction[l]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

Hypercube

l=Table[GraphData[{"Hypercube",n},"MolecularTopologicalIndex"],{n,10}]

l/4

FindSequenceFunction[l,n]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

Lattice

l=Table[GraphData[{"Lattice",{n,n}},"MolecularTopologicalIndex"],{n,2,10}]//Quiet

l/48

{0,1,12,60,200,525,1176,2352,4320,7425}

FindSequenceFunction[l,n-1]//Factor

Table[4 (((-1+n))^2) (n^2) (1+n),{n,30}]

MöbiusLadder

l=Table[GraphData[{"MoebiusLadder",n},"MolecularTopologicalIndex"],{n,3,20}]//Quiet

l/4

FindSequenceFunction[l,n-2]//Factor

Table[-(3/2) n (-9+(((-1))^n)-4 n-2 (n^2)),{n,30}]

Mycielski

l=Table[GraphData[{"Mycielski",n},"MolecularTopologicalIndex"],{n,11}]

l/4

FindSequenceFunction[l,n]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

Series[x^2 ((42/(1-6 x))+(30/(1-2 x))+(4/(-1+x))+(28/(-1+3 x))+(36/(-1+4 x))),{x,0,10}]

SeriesCoefficient[x^2 ((42/(1-6 x))+(30/(1-2 x))+(4/(-1+x))+(28/(-1+3 x))+(36/(-1+4 x))),{x,0,n}]

-(1/36) (3 (2^n)-8) (27 (2^n)-14 (3^n)-18)

Table[-(1/36) (-8+3 (2^n)) (-18+27 (2^n)-14 (3^n)),{n,20}]

Odd

l=Table[GraphData[{"Odd",n},"MolecularTopologicalIndex"],{n,6}]

l/4

Table[MolecularTopologicalIndex[OddGraph[n]],{n,7}]//Timing

l={0,24,540,12040,258300,5258484,102834732};

FindSequenceFunction[l,n]

FindSequenceFunction[Rest[l],n-1]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

Pan

l=Table[GraphData[{"Pan",n},"MolecularTopologicalIndex"],{n,3,20}]//Quiet

l=Table[MolecularTopologicalIndex[Graph[{"Pan",n}]],{n,3,20}]

FindSequenceFunction[l,n-2]//FullSimplify

Table[(1/4) (26+27 n+4 (n^2)+2 (n^3)+(((-1))^n) (2+n)),{n,30}]

Path

l=Table[GraphData[{"Path",n},"MolecularTopologicalIndex"],{n,20}]

l/2

f=FindSequenceFunction[l,n]//FullSimplify

FindLinearRecurrence[l]

FindSequenceFunction[l/2,n]

FindGeneratingFunction[l,x]

gf=((2x(2 x-(x^3)+(x^4)))/(((-1+x))^4));

Series[gf,{x,0,10}]

(coef=SeriesCoefficient[gf,{x,0,n}])//InputForm

Table[coef,{n,20}]

Factor[-6+(13*n)/3-n^2+(2*n^3)/3]

Table[If[n==1,0,(1/3) (-18+13 n-3 (n^2)+2 (n^3))],{n,20}]

PermutationStar

l=Table[GraphData[{"PermutationStar",n},"MolecularTopologicalIndex"],{n,6}]//Quiet

l/4

Table[MolecularTopologicalIndex[Graph[{"PermutationStar",n}]],{n,6}]

FindSequenceFunction[l,n]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

Prism

l=Table[GraphData[{"Prism",n},"MolecularTopologicalIndex"],{n,3,20}]//Quiet

l/4

FindSequenceFunction[l,n-2]//FullSimplify

Table[(3/2) n (11+(((-1))^n)+2 n (2+n)),{n,30}]

Square

l=Table[GraphData[{"Square",n},"MolecularTopologicalIndex"],{n,10}]

l/48

FindSequenceFunction[l,n]//FullSimplify

Table[8 (((-1+n))^3) n (1+n),{n,30}]

Star

l=Table[GraphData[{"Star",n},"MolecularTopologicalIndex"],{n,20}]

FindSequenceFunction[l,n]

Sun

l=Table[GraphData[{"Sun",n},"MolecularTopologicalIndex"],{n,3,20}]//Quiet

l/4

FindSequenceFunction[l,n-2]//Factor

Table[4 n (-3+3 n+(n^2)),{n,30}]

Sunlet

l=Table[GraphData[{"Sunlet",n},"MolecularTopologicalIndex"],{n,3,20}]//Quiet

l/2

FindSequenceFunction[l,n-2]//FullSimplify

Table[n (7+(((-1))^n)+2 n (3+n)),{n,30}]

Tetrahedral

l=Table[GraphData[{"Tetrahedral",n},"MolecularTopologicalIndex"],{n,6,10}]//Quiet

l/4

Table[MolecularTopologicalIndex[TetrahedralGraph[n]],{n,6,10}]

FindSequenceFunction[l,n-5]

FindLinearRecurrence[l]

FindGeneratingFunction[l,x]

TorusGrid

l=Table[GraphData[{"TorusGrid",{n,n}},"MolecularTopologicalIndex"],{n,3,10}]//Quiet

l/4

FindSequenceFunction[l,n-2]//FullSimplify

Table[4 n (2+n) (-1+2 n),{n,30}]

Triangular

l=Table[GraphData[{"Triangular",n},"MolecularTopologicalIndex"],{n,2,10}]//Quiet

l/24

FindSequenceFunction[l,n-1]//FullSimplify

Table[(((-2+n))^2) n (-1+(n^2)),{n,30}]

Web

l=Table[GraphData[{"Web",n},"MolecularTopologicalIndex"],{n,3,20}]//Quiet

l/2

FindSequenceFunction[l,n-2]//FullSimplify

Table[n (3 (7+(((-1))^n))+22 n+6 (n^2)),{n,30}]

Wheel

l=Table[GraphData[{"Wheel",n},"MolecularTopologicalIndex"],{n,4,20}]//Quiet

FindSequenceFunction[l,n-3]//FullSimplify

Table[8 (((-1+n))^2),{n,30}]

Checks

DeleteCases[{#,GraphData[#,"MolecularTopologicalIndex"]/.Missing["NotApplicable"]->Indeterminate,MolecularTopologicalIndex[GraphData[#]]}&/@GraphData[;;100],{_,x_,x_}]//Timing

DeleteCases[{#,GraphData[#,"MolecularTopologicalIndex"]/.Missing["NotApplicable"]->Indeterminate,MolecularTopologicalIndex[GraphData[#]]}&/@GraphData[All],{_,x_,x_}]//Timing