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Fig. 2 | Applied Network Science

Fig. 2

From: Non-backtracking cycles: length spectrum theory and graph mining applications

Fig. 2

Conjugate homotopy classes in a graph. Left: In an undirected graph G we label some directed edges a,b,c,d and e. We use the red node as basepoint to construct homotopy classes. Center: Three homotopy classes [h1],[h2],[g] each with a sequence of edges that form a cycle of minimal length. Right: Proof that [h1] and [h2] are conjugate via [g]. Each time an edge is followed by its inverse we remove them as they are homotopically trivial. By doing so, we reduce the cycle [g−1][h1][g] to one that is homotopic to it and has minimal-length, namely [h2]

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