Topological Index

Author

Eric W. Weisstein
March 10, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Examples

Only mention of this index in Devellers and Balaban appears to be a throwaway reference on p. 31

"The index based similarly on the determinant of the adjacency-plus-distance matrix:

TI = det|A + D|

has, like MTI, a fairly low degeneracy.

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

TopologicalIndex[g]

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

Co-TI

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

Checks

DeleteCases[{#,GraphData[#,"TopologicalIndex"]/.Missing["NotApplicable"]->Indeterminate,TopologicalIndex[GraphData[#]]}&/@GraphData[;;10],{_,x_,x_}]//Timing

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

DeleteCases[{#,GraphData[#,"TopologicalIndex"]/.Missing["NotApplicable"]->Indeterminate,TopologicalIndex[GraphData[#]]}&/@GraphData[;;200],{_,x_,x_}]//Timing

DeleteCases[{#,GraphData[#,"TopologicalIndex"]/.Missing["NotApplicable"]->Indeterminate,TopologicalIndex[GraphData[#]]}&/@GraphData[;;500],{_,x_,x_}]//Timing

DeleteCases[{#,GraphData[#,"TopologicalIndex"]/.Missing["NotApplicable"]->Indeterminate,TopologicalIndex[GraphData[#]]}&/@GraphData[;;1000],{_,x_,x_}]//Timing

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