Counting Paths in Graphs

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.

[1]  Harry Kesten,et al.  Symmetric random walks on groups , 1959 .

[2]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[3]  S. A. Amitsur On the characteristic polynomial of a sum of matrices , 1980 .

[4]  Michael Doob,et al.  Spectra of graphs , 1980 .

[5]  David E. Muller,et al.  Context-free languages, groups, the theory of ends, second-order logic, tiling problems, cellular automata, and vector addition systems , 1981 .

[6]  Joel M. Cohen Cogrowth and amenability of discrete groups , 1982 .

[7]  Wolfgang Woess,et al.  Cogrowth of groups and simple random walks , 1983 .

[8]  David E. Muller,et al.  Groups, the Theory of Ends, and Context-Free Languages , 1983, J. Comput. Syst. Sci..

[9]  C. Reutenauer,et al.  A formula for the determinant of a sum of matrices , 1987 .

[10]  B. Mohar,et al.  A Survey on Spectra of infinite Graphs , 1989 .

[11]  A short proof of the Grigorchuk-Cohen cogrowth theorem , 1989 .

[12]  Noncommutative random variables and spectral problems in free product $C^*$-algebras , 1990 .

[13]  D. I. Cartwright Singularities of the Green function of a random walk on a discrete group , 1992 .

[14]  Sam Northshield,et al.  Cogrowth of Regular Graphs , 1992 .

[15]  H. Bass THE IHARA-SELBERG ZETA FUNCTION OF A TREE LATTICE , 1992 .

[16]  C. Champetier Cocroissance Des Groupes À Petite Simplification , 1993 .

[17]  W. L. Paschke Lower bound for the norm of a vertex-transitive graph , 1993 .

[18]  Wolfgang Woess,et al.  Random Walks on Infinite Graphs and Groups — a Survey on Selected topics , 1994 .

[19]  Récurrence des marches aléatoires et ergodicité du flot géodésique sur les graphes réguliers. , 1996 .

[20]  Rostislav Grigorchuk,et al.  On problems related to growth, entropy, and spectrum in group theory , 1997 .

[21]  D. Zeilberger,et al.  A Combinatorial Proof of Bass’s Evaluations of the Ihara-Selberg Zeta Function for Graphs , 1998, math/9806037.

[22]  L. Saloff‐Coste RANDOM WALKS ON INFINITE GRAPHS AND GROUPS (Cambridge Tracts in Mathematics 138) , 2001 .