Connected Induced Subgraph
A connected (vertex-)induced subgraph of G is a vertex-induced subgraph of G that is connected.
The number of undirected k-cycles in a graph G is given by
where the sum is over connected induced subgraphs H of G, N(H) denotes the number of neighbors of H in G (namely vertices v of G which are not in H and such that there exists at least one edge from v to a vertex of H), Tr(A) denotes the matrix trace, and A_H^k is the kth matrix power of the adjacency matrix of the graph H (Giscard et al. 2016).
See also
Connected Graph, Cube Polynomial, Graph Cycle, Induced Subgraph, Vertex-Induced SubgraphExplore with Wolfram|Alpha
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References
Giscard, P.-L.; Kriege, N.; and Wilson, R. C. "A General Purpose Algorithm for Counting Simple Cycles and Simple Paths of Any Length." 16 Dec 2016. https://arxiv.org/abs/1612.05531.Cite this as:
Weisstein, Eric W. "Connected Induced Subgraph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConnectedInducedSubgraph.html