Fundamental Cycle

Author

Eric W. Weisstein
February 7, 2024

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

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

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

Length == circuit rank

DeleteCases[Monitor[{gn=#,GraphData[#,"CircuitRank"],Length@FindFundamentalCycles[GraphData@#]}&/@GraphData[;;50],gn],{_,x_,x_}]//Timing

Example

g=StyleGraphs[GraphData["CuboctahedralGraph"]]

tree=FindSpanningTree[GraphData["CuboctahedralGraph"]]//StyleGraphs

(cyc=FindFundamentalCycles[GraphData["CuboctahedralGraph"]])[[All,All,1]]

GraphicsGrid[Join[{{g,tree}},Partition[
HighlightGraph[g,Style[#,Directive[Red,Thick]]]&/@cyc,
UpTo[5]]],ImageSize->500]

GraphicsRow[{
g,
tree,
GraphicsGrid[Partition[
HighlightGraph[g,Style[#,Directive[Red,Thick]]]&/@cyc,
UpTo[5]],ImageSize->All]}]

GraphData["CuboctahedralGraph",{"M","V","C"}]

{1,-1,1}.%