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While the adjacency matrix depends on the vertex labeling, its spectrum is a Graph Invariant . SPECIFIC RESULTS Let ''M'' be the Adjacency Matrix of a graph. The Characteristic Polynomial of the graph is p Given a particular polynomial, it is not known if a corresponding adjacency matrix can be deduced. Two graphs are said to be Isospectral if the adjacency matrices of the graphs have the same eigenvalues. Isospectral graphs need not be Isomorphic , but isomorphic graphs are always isospectral, because the characteristic polynomial is a Topological Invariant of the graph. The Ihara Zeta Function of the graph is given by : and is another topological invariant of the graph. The Ihara zeta function of a ''k''-regular connected graph satisfies the Riemann Hypothesis if and only if the graph is a Ramanujan Graph . A graph is ''k''-regular if every vertex has the same number of incoming and outgoing arcs. The Perlis theorem states that : where ''n''M(''k'') is the number of closed paths (with no backtracking or repetition) of length ''k''. The Ihara-Hashimoto-Bass Theorem relates the zeta function to the Euler Characteristic of the graph. The study of the topological invariants of the Cayley Graph is known as Geometric Group Theory . SEE ALSO EXTERNAL LINKS
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