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Fixing incorrect capital in section heading; see Wikipedia:Manual of Style |
Small fixes in both TeX and non-TeX mathematical notation. In TeX, use a backslash in \det, \log, \max, \cos, etc. to prevent italicization and to get proper spacing. |
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The [[characteristic polynomial]] of the graph is
:''p''<sub>''M''</sub>(''t'') = [[determinant|det]](''M''
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
:<math>\zeta_M(t) = \frac{1}{\det (I-tM)}</math>
and is another topological invariant of the graph.
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The Perlis theorem states that
:<math>t \frac{d}{dt} \log \zeta_M(t) = \sum_{k=1}^\infty n_M(k) t^k</math>
where ''n''<sub>M</sub>(''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.
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