A connected (vertex-)induced subgraph of 
is a vertex-induced
subgraph of 
that is connected.
The number of undirected 
-cycles in a graph 
is given by
where the sum is over connected induced subgraphs 
of 
, 
denotes the number of neighbors of 
in 
(namely vertices 
of 
which are not in 
and such that there exists at least one edge from 
to a vertex of 
), 
denotes the matrix trace,
and 
is the 
th
matrix power of the adjacency matrix of the graph 
(Giscard et al. 2016).
See also
Connected Graph, Graph Cycle, Induced Subgraph, Vertex-Induced Subgraph
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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/pdf/1612.05531.pdf.
Cite this as:
Weisstein, Eric W. "Connected Induced Subgraph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConnectedInducedSubgraph.html