Counting Paths in Graphs

Abstract

We give a simple combinatorial proof of a formula that extends a result by Grigorchuk (rediscovered by Cohen) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of `bumps' on paths in a graph: in a $d$-regular (not necessarily transitive) non-oriented graph let the series $G(t)$ count all paths between two fixed points weighted by their length $t^{length}$, and $F(u,t)$ count the same paths, weighted as $u^{numberofbumps}{t}^{length}$. Then one has $$F(1-u,t)/(1-u^2{t}^2)=G(t/(1+u(d-u)t^2))/(1+u(d-u)t^2).$$ We then derive the circuit series of `free products' and `direct products' of graphs. We also obtain a generalized form of the Ihara-Selberg zeta function.

Cited by in corpus (1)

• The coefficients of the reduced Bartholdi zeta function