Discrete trace formulas and holomorphic functional calculus for the adjacency matrix of regular graphs
Yulin Gong, Wenbo Li, Shiping Liu
TL;DR
This work develops a unified holomorphic functional-calculus framework for the adjacency operator of $(q+1)$-regular graphs (finite or infinite) via an ellipse domain $\,\Omega(\rho)\,$ containing the spectrum, yielding a germ expansion of $h(q^{-1/2}A)$ in Chebyshev-type bases and a discrete pre-trace/trace theory anchored by non-backtracking matrices $A_r$. The central result expresses $h(q^{-1/2}A)$ as $\int h(x) d\mu_q(x) I +\sum_{r\ge1} q^{-r/2} a_{r,q}(h) A_r$ with $a_{r,q}(h)=(1+q^{-1})^{-1}\int_{-2}^{2} h(x) X_{r,q}(x) d\mu_q(x)$, enabling explicit decompositions and trace formulas. These tools yield new, streamlined proofs of walk counting, the Ihara–Bass theorem, and explicit solutions to the heat and Schrödinger equations on regular graphs, including lattices, and connect spectral data to non-backtracking combinatorics via harmonic analysis on regular trees. The framework extends to infinite graphs, provides concrete kernel formulas with Bessel functions, and offers a versatile analytic approach with potential impact on Ramanujan graphs and expander graph theory.
Abstract
We provide a unified method to study the adjacency matrices of regular graphs (including infinite ones) using holomorphic functional calculus. By applying this calculus on a specific ellipse that contains the spectrum, we derive an expansion of $h(A)$ using non-backtracking matrices. This framework allows us to systematically obtain discrete trace formulas that link spectral theory with graph combinatorics. To show how this method works, we give new proofs for several well-known problems, such as walk counting, the Ihara-Bass formula, and solutions to the heat and Schrödinger equations on graphs.