Zeta functions of weighted graphs and covering graphs

Abstract

ABSTRACT. We nd a condition for weights on the edges of a graph which insures that the Ihara zeta function has a 3-term determinant formula. Then we investigate the locations of poles of abelian graph coverings and compare the results with random covers. We discover that the zeta function of the random cover satis es an approximate Riemann hypothesis while that of the abelian cover does not.

Ramanujan graphs arising as weighted Galois covering graphs

“…Several publications on covering graph theory have dealt with both the covering and base graphs having only unweighted edges. See [1,5,6,8,13,15,16,17]. We refer to this as standard unweighted covering graph theory.…”

“…See [2,4,8,14] among others. While authors have considered zeta functions of both weighted and standard covering graphs, the definitions we present here are different in the use of weights.…”

Looking into a graph theory mirror of number theoretic zetas

“…1) Do experiments on di¤erences between properties of zetas of weighted or quantum and unweighted graphs. See Horton et al [7], [8]. In particular, consider the connections with random matrix theory.…”

“…For example, if you set h(u) = u (m+1) ; for m = 1; 2; 3;... , the explicit formula (9) says N m is the sum of the mth powers of the eigenvalues of W 1 ; which is clear from formulas (5) and (8).…”

“…See Angel et al [1], Horton et al [7], [8]. When a graph is regular, the Riemann hypothesis is equivalent to the statement that the graph is Ramanujan and thus a good expander.…”