Generating functions of non-backtracking walks on weighted digraphs: radius of convergence and Ihara's theorem

TL;DR

This paper extends the understanding of generating functions for non-backtracking walks on weighted directed graphs, providing new formulas for the radius of convergence and a version of Ihara's theorem, linking graph cycles and spectral properties.

Contribution

It introduces exact formulas for the radius of convergence for weighted and directed graphs and generalizes Ihara's theorem to backtracking-downweighted walks.

Findings

Exact formula for radius of convergence in weighted graphs

Characterization of convergence radius based on graph cycles

A generalized Ihara's theorem for backtracking-downweighted walks

Abstract

It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In [P. Grindrod, D. J. Higham, V. Noferini, The deformed graph Laplacian and its application to network centrality analysis, SIAM J. Matrix Anal. Appl. 39(1), 310--341, 2018], the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed unweighted graphs, showing that it depends on the number of cycles in the undirectization of the graph. For weighted graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound. Finally, we consider also backtracking-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case.