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Counting Paths in Graphs
TL;DR: In this article, the authors give a simple combinatorial proof of a formula that extends a result by Grigorchuk relating cogrowth and spectral radius of random walks, and derive the circuit series of ''free products'' and ''direct products'' of graphs.
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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 $u^{number of bumps}t^{length}$. Then one has $$F(1-u,t)/(1-u^2t^2) = G(t/(1+u(d-u)t^2))/(1+u(d-u)t^2).$$ 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.
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References
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Automata, Languages, and Machines
Samuel Eilenberg
- 01 Mar 1974
TL;DR: This book attempts to provide a comprehensive textbook for undergraduate and postgraduate mathematicians with an interest in formal languages and automata, written by Professor Ian Chiswell.
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A combinatorial proof of bass's evaluations of the ihara-selberg zeta function for graphs
Dominique Foata,Doron Zeilberger +1 more
TL;DR: In this paper, the authors derived combinatorial proofs of the main two evaluations of the Ihara-Selberg Zeta function associated with a graph and showed that the first evaluation is an immediate consequence of Amitsur's identity on the characteristic polynomial of a sum of matrices.
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