In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Ihara zeta function is a
zeta function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of the Airy function
* ...
associated with a finite
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
. It closely resembles the
Selberg zeta function
The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function
: \zeta(s) = \prod_ \frac
where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...
, and is used to relate closed walks to the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of the
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
. The Ihara zeta function was first defined by
Yasutaka Ihara in the 1960s in the context of
discrete subgroups of the two-by-two
p-adic
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...
.
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
suggested in his book ''Trees'' that Ihara's original definition can be reinterpreted graph-theoretically. It was
Toshikazu Sunada
is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor emeritus of both Meiji University and Tohoku University. He is also distinguished professor of emeritus at Meiji in recogni ...
who put this suggestion into practice in 1985. As observed by Sunada, a
regular graph
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree o ...
is a
Ramanujan graph In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. AMurty's survey papernotes, Ramanu ...
if and only if its Ihara zeta function satisfies an analogue of the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
.
Definition
The Ihara zeta function is defined as the analytic continuation of the infinite product
The product in the definition is taken over all prime
closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flo ...
s
of the graph
, where geodesics which differ by a
cyclic rotation are considered equal. A ''closed geodesic''
on
(known in graph theory as a "
closed walk") is a finite sequence of vertices
such that
:
:
The integer
is the ''length''
of
. The closed geodesic
is ''prime'' if it cannot be obtained by repeating a closed geodesic
times, for an integer
.
This graph-theoretic formulation is due to Sunada.
Ihara's formula
Ihara (and Sunada in the graph-theoretic setting) showed that for regular graphs the zeta function is a rational function.
If
is a
-regular graph with
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
then
[Terras (1999) p. 677]
:
where
is the
circuit rank
In graph theory, a branch of mathematics, the circuit rank, cyclomatic number, cycle rank, or nullity of an undirected graph is the minimum number of edges that must be removed from the graph to break all its cycles, making it into a tree or fo ...
of
. If
is connected and has
vertices,
.
The Ihara zeta-function is in fact always the reciprocal of a
graph polynomial:
:
where
is Ki-ichiro Hashimoto's edge adjacency operator.
Hyman Bass
Hyman Bass (; born October 5, 1932). The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the c ...
References
External links
*Directory page at University of MichiganAuthor profilein the database zbMATH
{{DEFAUL ...
gave a determinant formula involving the adjacency operator.
Applications
The Ihara zeta function plays an important role in the study of
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
s,
spectral graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix ...
, and
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
, especially
symbolic dynamics In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (e ...
, where the Ihara zeta function is an example of a
Ruelle zeta function.
[Terras (2010) p. 29]
References
*
*
*
*
*
* {{cite book , title=Zeta Functions of Graphs: A Stroll through the Garden , volume=128 , series=Cambridge Studies in Advanced Mathematics , first=Audrey , last=Terras , authorlink=Audrey Terras , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, year=2010 , isbn=0-521-11367-9 , zbl=1206.05003
Zeta and L-functions
Algebraic graph theory