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.

翻译：已知有限图上非回溯游走枚举的生成函数是一个参数的有理矩阵值函数；该函数也与Ihara定理和图上的zeta函数等图论结果密切相关。在[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]中，针对简单图（即无向、无权重且无自环的图）研究了生成函数的收敛半径，并证明了该半径依赖于图中的环数。本文利用多项式与有理矩阵理论中的技术，将这些结果大幅推广至一般图（可能是有向图和/或加权图），研究了相应生成函数的收敛半径。对于有向无权图，我们给出了收敛半径的类似刻画，证明其取决于图的无向化中的环数。对于加权图，我们首次给出了收敛半径的精确公式，改进了先前仅给出下界的结果。最后，我们还考虑了无权有向图上的回溯降权游走，并在此情形下证明了Ihara定理的一个版本。