What are zeta functions of graphs and what are they good for?

Abstract

We discuss zeta functions of finite irregular undirected connected graphs (which may be weighted) and apply them to obtain, for example an analog of the prime number theorem for cycles in graphs. We consider 3 types of zeta functions of graphs: vertex, edge, and path. Analogs of the Riemann hypothesis are also introduced.

Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs

“…We apply Lemma 40 to F and the partition I + = {j ω }, I − = [r]\{j ω }. We obtain an asymptotically positive overlap for any permutation p in (16) such that p(j ω ) = 1.…”

“…where ζ G is the Ihara zeta function of the graph, refer to [28,19,16,29]. It follows that the poles of the Ihara zeta function are the reciprocal of the eigenvalues of B.…”

“…Using the analogy between the Ihara zeta function and the Riemann zeta function, Stark and Terras (see [16]) have defined a graph to satisfy the graph Riemann hypothesis if its nonbacktracking matrix B has no eigenvalues λ such that |λ| ∈ ( √ ρ B , ρ B ), where ρ B is the PerronFrobenius eigenvalue of B. Interestingly, a regular graph G is Ramanujan if and only if it satisfies the graph Riemann hypothesis (see [26] and [16]). Thus the graph Riemann hypothesis can also be seen as a generalization of the notion of Ramanujan graphs to the non-regular case, phrased in terms of non-backtracking spectra rather than on spectra of universal covers as in the definition of Lubotzky [21].…”

“…Indeed, in this case, for some bounded continuous function f , By Lemma 40 applied to F , if ω = 1, we obtain an positive overlap with I ± = J ± and the permutation p in (16) equal to the identity. If ω = −1, we obtain an positive overlap with I ± = J ∓ and any permutation p in (16) such that p(J ± ) = J ∓ , Case 3: general case.…”

Zeta functions of weighted graphs and covering graphs

“…The idea of weights associated to an edge of a graph is a natural one in applications such as electrical networks: resistance, capacitance in a wire corresponding to an edge; random walks: probability to move along a given edge; quantum graphs: each edge is an interval with a given length and a Schrödinger equation (see [5]). We considered the Ihara zeta function .u; X; / of a ( nite) weighted graph X with weight in our paper [8]. In Section 2 of the present paper, the rst object is to see whether there is a condition for edge weights of a graph which will allow us to prove a weighted analog of the Ihara three term determinant formula for the zeta function.…”