Enumeration of graphs with the same Ihara zeta function

Abstract

We enumerate all connected graphs with minimal vertex degree 2 on at most 11 vertices and determine their Ihara zeta functions. We also count the number of such graphs for which there is another graph with the same zeta function. We then use these graphs to conjecture properties of the graphs determined by the zeta function. In addition, we study switching constructions, proposed by Godsil and McKay, to determine whether they preserve the zeta function. We show that GM switching is not strong enough to preserve the zeta function, but GM* switching, following the notation of Haemers and Spence, does preserve the zeta function.

Distinguishing graphs with zeta functions and generalized spectra

“…In particular, these zeta invariants force strong restrictions on the degree sequence of a graph. Setyadi-Storm [SS13], in their enumeration of pairs of connected md2 graphs with the same Ihara zeta function on n ≤ 11 vertices, found that the Ihara zeta function of a connected md2 graph determines the degree sequence for n ≤ 11, and conjectured this holds for all n, but we give a counterexample to this conjecture on 12 vertices (Example 2.2). Nevertheless, we show that knowing the zeta functions of sufficiently many cones of G algorithmically determines the degree sequence (Lemma 2.3).…”

“…We remark that while computing this data, we discovered some small errors in the tables in [SS13] for n = 11, 12. Namely, in the cases where more than 2 connected md2 graphs have the same Ihara zeta function, [SS13] undercounts the number of pairs. For instance, for n = 10, the entries in the [SS13] tables should be augmented by 1 for m = 20, 21, 24, 25.…”

“…This pair does not occur as the result of GM* switching for the following reason: the matrices A + xD are similar for all x, but there is no matrix P such that P −1 (A 1 + xD 1 )P = (A 2 + xD 2 ) for all x, as must be the case for GM* pairs. (In [SS13], it was mistakenly written that all pairs of connected md2 graphs on n ≤ 11 vertices with the same zeta function, adjacency spectrum, and Laplacian spectra are obtained by GM* switching, but this is false as this example shows. The second author of that paper informed us that the conclusion of that sentence, "are obtained by GM* switching," should be "have the same ϕ ADJ .")…”

“…Namely, in the cases where more than 2 connected md2 graphs have the same Ihara zeta function, [SS13] undercounts the number of pairs. For instance, for n = 10, the entries in the [SS13] tables should be augmented by 1 for m = 20, 21, 24, 25. In each of these cases, there is one triple of connected md2 graphs all with the same zeta function.…”

“…In fact, Setyadi-Storm [SS13], based on their enumeration of zeta functions of connected md2 graphs on ≤ 11 vertices, conjectured that connected md2 graphs with the same Ihara zeta function have identical degree sequences. However, we give a counterexample.…”