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Ihara zeta functions for some simple graph families

Maize Chico, Thomas W. Mattman, Alex Richards

TL;DR

This work studies the Ihara zeta function $\zeta_G(u)$ for graphs, presenting a streamlined coefficient calculus via linear subgraphs of the oriented line graph and proving that $\zeta_G(u)^{-1}$ is even for bipartite graphs. It classifies rank-two graphs into three families, derives explicit zeta inverses for these families, and proves that the zeta function is a complete invariant for rank-two graphs. The authors provide closed-form zeta formulas for five graph families (including $K_n$, $K_{m,n}$, and Möbius ladders) and extend to small-order graphs, with a precise method to count spanning trees from $\zeta_G(u)$. Overall, the paper connects determinant formulas, graph structure, and exact zeta expressions to illuminate the discriminative power of Ihara zeta functions in low-rank graph families.

Abstract

The reciprocal of the Ihara zeta function of a graph is a polynomial invariant introduced by Ihara in 1966. Scott and Storm gave a method to determine the coefficients of the polynomial. Here we simplify their calculation and determine the zeta function for all graphs of rank two. We verify that it is a complete invariant for such graphs: If $G_1$ and $G_2$ are of rank two, then $G_1$ and $G_2$ are isomorphic if and only if they have the same Ihara zeta function. We observe that the reciprocal of the zeta function is an even polynomial if the graph is bipartite. We also determine the zeta function for several graph families: complete graphs, complete bipartite graphs, Möbius ladders, cocktail party graphs, and all graphs of order five or less. We use the special value $u=1$ to count the spanning trees for these families.

Ihara zeta functions for some simple graph families

TL;DR

This work studies the Ihara zeta function for graphs, presenting a streamlined coefficient calculus via linear subgraphs of the oriented line graph and proving that is even for bipartite graphs. It classifies rank-two graphs into three families, derives explicit zeta inverses for these families, and proves that the zeta function is a complete invariant for rank-two graphs. The authors provide closed-form zeta formulas for five graph families (including , , and Möbius ladders) and extend to small-order graphs, with a precise method to count spanning trees from . Overall, the paper connects determinant formulas, graph structure, and exact zeta expressions to illuminate the discriminative power of Ihara zeta functions in low-rank graph families.

Abstract

The reciprocal of the Ihara zeta function of a graph is a polynomial invariant introduced by Ihara in 1966. Scott and Storm gave a method to determine the coefficients of the polynomial. Here we simplify their calculation and determine the zeta function for all graphs of rank two. We verify that it is a complete invariant for such graphs: If and are of rank two, then and are isomorphic if and only if they have the same Ihara zeta function. We observe that the reciprocal of the zeta function is an even polynomial if the graph is bipartite. We also determine the zeta function for several graph families: complete graphs, complete bipartite graphs, Möbius ladders, cocktail party graphs, and all graphs of order five or less. We use the special value to count the spanning trees for these families.
Paper Structure (8 sections, 21 theorems, 91 equations, 4 figures, 4 tables)

This paper contains 8 sections, 21 theorems, 91 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Proof of Theorems \ref{['thm:Main']} and \ref{['thm:even']}
  3. Families of rank two graphs
  4. Rank two graphs with equal zeta function
  5. Matrix determinant formulas
  6. Ihara zeta function of five graph families
  7. Graphs of small order
  8. Spanning trees

Key Result

Theorem 1.5

For a graph $G$, the reciprocal of the Ihara zeta function, $\zeta_G(u)^{-1}$, is a polynomial with terms $c_k u^k$ for $0 \leq k \leq 2 |E|$. The constant term is $c_0 = 1$, and $c_k = 0$ when $\mathscr{L}_k(L^o G) = \emptyset$. Otherwise,

Figures (4)

  • Figure 1: The oriented line graph $L^o G_{3,4}$
  • Figure 2: The oriented line graph $L^o G_{5,6,2}$
  • Figure 3: The oriented line graph $L^o G_{4,5,1}$
  • Figure 4: The oriented line graph $L^o H_{4,3,2}$

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 2.1
  • proof
  • ...and 41 more