Definition: The Ihara zeta function is a mathematical function associated with finite, undirected graphs, introduced by Yasutaka Ihara in the context of discrete subgroups of p-adic Lie groups. It is an analog of the Riemann zeta function and the Selberg zeta function, adapted to graph theory, and it encodes information about the number of closed, non-backtracking, primitive paths (cycles) on a graph.
Overview: The Ihara zeta function plays a significant role in spectral graph theory and number theory. It is defined for finite, connected, undirected graphs without degree-one vertices (though the definition can be extended). The function can be expressed as a formal power series or as a rational function using a determinant formula. The Ihara zeta function has connections with the adjacency matrix and the Laplacian of the graph, particularly through Ihara's determinant formula, which relates the zeta function to a combination of the graph's adjacency matrix and degree matrix.
The poles and zeros of the Ihara zeta function can reveal structural properties of the graph, analogous to how the zeros of the Riemann zeta function relate to prime number distribution. For regular graphs, the Ihara zeta function satisfies a version of the Riemann hypothesis if and only if the graph is Ramanujan, a property related to optimal spectral expansion.
Etymology/Origin: The function is named after Yasutaka Ihara, a Japanese mathematician who introduced a zeta function for discrete subgroups of p-adic groups in the 1960s. The adaptation of this function to finite graphs was developed in later works by Toshikazu Sunada, Ki-ichiro Hashimoto, and others in the 1980s and 1990s. The modern formulation used in graph theory is attributed primarily to Sunada’s reinterpretation of Ihara's work in combinatorial terms.
Characteristics:
- Defined via the Euler product over prime (primitive, non-backtracking) cycles in a graph.
- Expressed as $ Z(u) = \prod_{[P]} (1 - u^{L(P)})^{-1} $, where the product is over equivalence classes of prime cycles $ [P] $ and $ L(P) $ is the length of the cycle.
- For a graph $ G $ with adjacency matrix $ A $ and degree matrix $ D $, Ihara's determinant formula states:
$ Z(u)^{-1} = (1 - u^2)^{\chi(G)} \det(I - uA + u^2(D - I)) $,
where $ \chi(G) $ is the Euler characteristic of the graph. - The reciprocal of the Ihara zeta function is a polynomial when the graph is finite and has no vertices of degree one.
- The function is sensitive to the graph's connectivity, cycles, and spectral properties.
Related Topics:
- Spectral graph theory
- Ramanujan graphs
- Selberg zeta function
- Riemann hypothesis for graphs
- Graph Laplacian
- Automorphic forms
- Covering graphs
- Number theory and zeta functions
The Ihara zeta function is a well-established tool in modern graph theory and arithmetic combinatorics, with applications in coding theory, network analysis, and mathematical physics.