Ihara zeta function

In this post I will describe the Ihara zeta function and some of its basic properties.

The Ihara function presented by Ihara in 1966, can be considered as an analogue of the Dedekind zeta functions defined over a number field. So in a way it carries the ‘good’ properties of the classical zeta functions and as a result it has an Euler product and an analytic continuation to a meromorphic function which satisfies a functional equation. Generally speaking, each graph is associated with the Ihara zeta function, which is defined over the primes walks of the associated graph. The graphs are considered to be finite, though work of Grigorchuk and Zuk has been made towards the infinite graph direction. So lets begin to formalize the Ihara zeta function.
We consider graphs of the form { G(V,E) } where { V } is the set of vertices and { E } is the set of edges of the graph. When we are saying that the graph { G } is finite,we imply that the { V,E } sets are finite. Next we consider two maps the one is { e \in E \mapsto ( o(e),t(e)) \in V \times V } and the other is { e \in E \mapsto \overline{e} \in E } which satisfy the following conditions { \overline{\overline{e}}=e , o( \overline{e} ) = t(e) } for every { e \in E }. We shall call the vertex { o(e) } the origin of { e } and { t(e) } the terminus of { e }. A graph is called simple if { E \subset \lbrace (u,v) \in V \times V : u \neq v \rbrace } and { o(u,v) = u }, { t(u,v) = v } and { \overline{(u,v)}=(v,u) }. We will only consider simple graphs. If an edge connects two vertices { u,v } , then we will say that { u } and { v } are adjacent and we will use the symbol { u \sim v }. A path of length { m } in the graph G from vertex {u_0 } to vertex { u_m } we will be considered as a sequence of vertices { \lbrace u_0, u_1 , \dots , u_m \rbrace } with the property { u_{i+1} \sim u_i } for { i=1, \dots, m-1 }. The length of a path { C } will be denoted as { |C| } and will be considered to be the number of edges between the vertices of the path. A path is closedif { u_0 = u_m } and a graph is said to be connected if there is a path between to pairs of any distinct vertices of the graph.
Having these definitions in mind we shall define proper closed paths in the graph.

Definition 1 {Proper closed paths} i) Backtracking: A path in { G } has backtracking if { u_{i-1}=u_{i+1} } for some { i \in \lbrace 1, \dots, m-1 \rbrace }. A path with no backtracking is also called proper. Denote by { \mathcal{C} } the set of proper closed paths.
ii) A proper closed path is said to have tail if there exists a { k \in \lbrace 1, \dots , [m/2] -1 \rbrace } such as { u_j = u_{m-j} }, for { j =1, \dots,k } . Denote with { \mathcal{C}^{tail} } the set of all closed proper paths with tail in { G } , and with { \mathcal{C}^{notail} } the set of all tail-less proper closed paths in { G } which are also called reduced closed paths. Note that { \mathcal{C}= \mathcal{C}^{tail} \cup \mathcal{C}^{notail} } with { \mathcal{C}^{tail} \cap \mathcal{C}^{notail} = \emptyset }.
iii) A reduced closed path is primitive if it is not obtained by going { n \leq 2 } times around another closed path.

Definition 2 {Cycles} Given closed paths { C=(u_0,u_1, \dots, u_m = u_0 ) }, { D = (w_0, w_1, \dots, w_m = w_0 ) }, we say that { D} and { C } are equivalent and write { C \sim_{o} D } if there exists { k } such as { w_j = w_{j+k} } for all { j }, where the addition is taken mod { m }. The equivalence class of { C } is denoted by { [C]_o }. An equivalence class is also called a cycle. Denote finally by { \mathcal{R} } the set of reduced cycles and by { \mathcal{P} \subset \mathcal{R} } the subset of primitive reduced cycles which are also called prime cycles.

We are finally ready to define the Ihara zeta function of the graph { G }.

Definition 3 (Ihara zeta function)

{Z_G(u) := \prod_{ C \in \mathcal{P}} (1 - u^{|C|})^{-1} }.

A very interesting property of the Ihara zeta function which was discovered by Bass, is that it can be written as the inverse of a polynomial.

{ Z_G(u) = \dfrac{1}{(1-u^2)^{e-n}det(I-Au+Qu^22)} } where { A } is the adjacency matrix of { G } , { D } is a diagonal matrix with elements { D_{ii} } equal to the degree of vertex { i } , { Q = D -I } and { n=|V| , e= |E| } the number of vertices and edges of the graph { G } respectively.

Another interesting property is that we can consider the graph { G } to have minimum degree at least 2. This means the minimum number of edges for very vertex in the graph { G } should be two or more. This is because Ihara zeta function depends only on the primitive prime walks and vertices with degree 0 add no nonzero length paths to the graph. Similarly vertices with degree 1 add either zero length paths either paths with tail. So { Z_G(u) } remains unchanged and as a final remark it is of no harm to consider from now on graphs { G } to have minimum degree at least 2.

I will continue with some lemmas and theorems which indicate how Ihara zeta function determines the structure of the graph.

Lemma 4 If { G } has minimum degree at least 2, then the number of edges of { G } is given by { e = \dfrac{1}{2}deg(Z_G(u)^{-1}) }.

Proof:

Firstly { det Q \neq 0 } since { G } has minimum degree at least 2. By the previous formula given by Bass we have that the degree of { Z_G(u)^{-1} } is { 2n -2n+2e = 2e }.

\Box

I shall discuss further properties in another post.

References

[1] H. Bass. The Ihara-Selberg zeta function of a tree lattice. Int. J. Math., 3:717 – 797, 1992.

[2] Cooper Yaim, Properties Determined by the Ihara zeta function of a Graph, The electronic journal of combinatorics 16 (2009), # R84

[3] Guid Daniele, Isola Tomasso, Lapidus Michel, Ihara zeta function for periodic simple graphs, http://arxiv.org/abs/math/0605753v3

[4] Y. Ihara. On discrete subgroups of the two by two projective linear group over p-adic fields. J. Math. Soc. Japan, 18:219 – 235, 1966.

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