Discrete trace formulas and holomorphic functional calculus for the adjacency matrix of regular graphs

Abstract: We study the holomorphic functional calculus for the adjacency matrices on possibly infinite regular graphs. More precisely, we show an expansion of $h(A)$ in terms of non-backtracking matrices, where $A$ is the adjacency matrix and $h$ is a holomorphic function on the interior of a particular ellipse containing the spectrum of $A$. Furthermore, we prove related trace formulas for regular graphs. Our results provide a simple and direct approach to the study of resolvent, heat and Schr\"odinger equations, Ihara zeta functions, and combinatorial enumeration problems on regular graphs.