The spectral geometry of \(k\)-regular groups (Q1803634)

A \(k\)-regular graph is a graph with the property that the degree of each vertex is equal to \(k\). One way to produce such graphs is via coset spaces of finitely generated groups. In general we can associate to any graph a discrete Laplacian. This paper studies the relationship between the spectral properties of the Laplacian on a \(k\)-regular graph and \(\Gamma\) and its (combinatorial) geometry. In particular let \(N(\Gamma,m)\) denote the number of closed paths in \(\Gamma\) of length \(m\). The first result relates the first eigenvalue of the Laplacian to the asymptotic behaviour of \(N(\Gamma,m)\) as \(m\) goes to \(\infty\). This is part of a more general picture. The author considers the function \[ f_ \Gamma(x) = \sum_ m mN(\Gamma,m)x^ m. \] This function turns out to be rational and the paper contains a description of it in terms of the eigenvalues of the Laplacian.