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.