Graph Zeta Functions 1 INTRODUCTION
Zeta Functions of Finite Graphs and Coverings, III
A. A. Terras Math. Dept., U.C.S.D., La Jolla, CA 92093-0112 E-mail: [email protected] and
H. M. Stark Math. Dept., U.C.S.D., La Jolla, CA 92093-0112
Version: March 13, 2006
A graph theoretical analog of Brauer-Siegel theory for zeta functions of number …elds is developed using the theory of Artin L-functions for Galois coverings of graphs from parts I and II. In the process, we discuss possible versions of the Riemann hypothesis for the Ihara zeta function of an irregular graph.
1. INTRODUCTION
In our previous two papers [12], [13] we developed the theory of zeta and L-functions of graphs and covering graphs. Here zeta and L-functions are reciprocals of polynomials which means these functions have poles not zeros. Just as number theorists are interested in the locations of the zeros of number theoretic zeta and L-functions, thanks to applications to the distribution of primes, we are interested in knowing the locations of the poles of graph-theoretic zeta and L-functions. We study an analog of Brauer-Siegel theory for the zeta functions of number …elds (see Stark [11] or Lang [6]). As explained below, this is a necessary step in the discussion of the distribution of primes. We will always assume that our graphs X are …nite, connected, rank 1 with no danglers (i.e., degree 1 vertices). Let us recall some of the de…nitions basic to Stark and Terras [12], [13]. If X is any connected …nite undirected graph with vertex set V and (undirected) edge set E, we orient its edges arbitrarily and obtain 2 E = 2m oriented edges. We always use the following oriented edge labelling j j
1 1 e1; e2; ; em; em+1 = e ; :::; e2m = e : (1) 1 m “Primes” [C] in X are equivalence classes of closed backtrackless tailless primitive paths C. Write C = a1a2 as, where aj is an oriented edge of X. The length of C is (C) = s. Back- 1 1 trackless means that ai+1 = ai , for all i. Tailless means that as = a1 : The equivalence class [C] is the set 6 6
[C] = a1a2 as; a2a3 asa1; ::: ; asa1 as 1 : f g [C] is primitive means C = Dm, for any integer m 2 and path D in X. 6 Here rX will denote the rank of the fundamental group of X: We have rX 1 = E V : j j j j Then rX is the number of edges deleted from X to form a spanning tree. We will call such deleted edges "cut" edges, since there should be no confusion with the other meaning of cut edge. Next let us de…ne an unrami…ed …nite covering graph Y over X (written Y=X) in the case that the graphs have no loops or multiple edges. In this case, Y covers X means that
1 Graph Zeta Functions 1 INTRODUCTION there is a covering map : Y X such that is an onto graph map and for each x X 1 ! 2 and each y (x); the set of points adjacent to y in Y is mapped by 1-1, onto the set of points adjacent2 to x in X. We always consider connected coverings Y of the connected graph X obtained by viewing the d sheets of Y as copies of a spanning tree in X: If the graphs have loops and multiple edges, one must be a little more precise about the de…nition of covering graph. See Stark and Terras [13]. A d-sheeted (unrami…ed) graph covering Y=X is normal i¤ there are d graph auto- morphisms : Y Y such that (y) = (y); for all y Y: Then Gal(Y=X), the Galois ! 2 group of Y=X, is the set of all these 0s: Recall that the Ihara zeta function of X is de…ned at u C, for u su¢ ciently small, by 2 j j 1 (u) = 1 u(C) ; (2) X Y[C] where [C] runs over the primes of X. As a power series in the complex variable u,
1 n X (u) = anu ; (3) n=0 X where each coe¢ cient an 0: Thus, by a classic theorem of Landau, both the series (3) and the product (2) will converge absolutely in a circle u < R with a singularity (pole of order 1 for connected X) at u = R. j j
Definition 1. RX is the radius of the largest circle of convergence of the Ihara zeta function and !X = 1=RX .
When X is a (q + 1)-regular graph, RX = 1=q and !X = q. As with the Dedekind zeta 1 function, X (u) has a meromorphic continuation to the entire complex u-plane, but now X (u) 1 is entire. Thus our interest lies with the poles of X (u): In general, X (u) is a polynomial which is essentially the characteristic polynomial of our edge matrix WX (which we de…ne next) and the largest eigenvalue of WX is !X :
Definition 2. We de…ne the 0,1 edge matrix WX by orienting the edges of X and labeling them as in (1). Then WX is the 2m 2m matrix with ij entry 1 if edge ei feeds into ej provided 1 that ej = e , and ij entry 0 otherwise. 6 i Recall from Stark and Terras [13] that
1 (u) = det (I WX u) : (4) X From this one can derive Ihara’sformula
1 2 rX 1 2 (u) = (1 u ) det I AX u + QX u ; (5) X where rX is the rank of the fundamental group of X , AX is the adjacency matrix of X, QX is the diagonal matrix whose jth diagonal entry is ( 1+ degree of jth vertex). Kotani and Sunada [5] show that, if q + 1 is the maximum degree of X and p + 1 is the minimum degree of X, then every non-real pole u of X (u) satis…es the inequality
1=2 1=2 q u p : (6) j j
Moreover, they show that every pole u of X (u) satis…es
RX u 1: j j
2 Graph Zeta Functions 1 INTRODUCTION
Another result of Kotani and Sunada [5] says that
1 1 q RX p ; p !X q: (7)
In particular, we can think of !X + 1 as a zeta function average vertex degree of X which is precisely q + 1 when X is a (q + 1)-regular graph. However Example 2 below shows that for irregular graphs, !X + 1 is in general neither the arithmetic nor geometric mean of all the vertex degrees. The main term in the graph analog of the prime number theorem is a power of !X . Further terms come from the same power of the reciprocals of the other poles of X (u) and thus the …rst step in discussing the error term is to locate the poles of X (u) with u very near to RX . In number theory, there is a known zero free region of a Dedekind zeta functionj j which can be explicitly given except for the possibility of a single …rst order real zero within this region. This possible exceptional zero has come to be known as a "Siegel zero" and is closely connected with the famous Brauer-Siegel Theorem. There is no known example of a Siegel zero for Dedekind zeta functions. 1 Since X (u) is a polynomial with a …nite number of zeros, given X there is an > 0 such that any pole of (u) in the region RX u < RX + must lie on the circle u = RX . This X j j j j gives us the graph theoretic analog of a "pole free region", u < RX + ; the only exceptions j j lie on the circle u = RX . We will show that X (u) is a function of u with = X a positive integer from De…nitionj j 5 below. This will imply there is a fold symmetry in the poles of th X (u); i.e., u = "R is also a pole of X (u); for all roots of unity ": Any further poles of (u) on u = R will be called “Siegel poles” of (u): Thus if = 1; any pole u = R of X j j X 6 X (u) with u = R will be called a Siegel pole. In numberj j …elds, a Siegel zero "deserves" to arise already in a quadratic extension of the base …eld. This has now been proved in many cases (see Stark [8]). Our initial motivation for this paper was to carry over these results to zeta functions of graphs. This was accomplished, essentially by the same representation theoretic methods that were used for Dedekind zeta functions, and is presented in Theorem 2 below. In the process, we were led to study possible extensions of the meaning "Ramanujan graph" and the "Riemann hypothesis for graph zeta functions" for irregular graphs. We discuss these possibilities in Section 2. A key reduction in the location of Siegel poles leads us to our …rst theorem which is purely combinatorial and is of independent interest. For this, we need three de…nitions.
Definition 3. X = g:c:d: (C) C = closed backtrackless tailless path on X : f j g Note that X is even if and only if X is bipartite. Definition 4. A vertex of X having degree 3 is called a node of X. A graph X of rank 2 always has at least one node. Definition 5. If X has rank 2 P = backtrackless path in X such that the = g:c:d: (P ) : X initial and terminal vertices are both nodes
When a path P in the de…nition of X is closed, the path will be backtrackless but may have a tail. However, in Section 4 we will give an equivalent de…nition of X which does not involve paths with tails. The equivalent de…nition has the added advantage that it is visibly a …nite calculation. The relation between X and X is given by the following result which will be proved in Section 4.
Theorem 1. Suppose X has rank 2. Then either X = X or X = 2X :
3 Graph Zeta Functions 2 RAMANUJAN GRAPHS AND THE RIEMANN HYPOTHESIS.
It is easy to see that if Y is a covering graph of X (of rank 2) we have Y = X since they are the g:c:d:s of the same set of numbers. Therefore X is a covering invariant. Because of this, Theorem 1 gives us the important
Corollary 1. If Y is a covering of a graph X of rank 2 then
Y = X or 2X :
For a cycle graph X the ratio Y =X can be arbitrarily large. As we have stated, we consider the Brauer Siegel theorem to be a statement about the location of Siegel zeros (Siegel poles for graph theory zeta functions). The general case, Theorem 3 in Section 5, will be reduced to the more easily stated case where X = 1, where any pole of X (u) on u = R other than u = R; is a Siegel pole, j j
Theorem 2. Suppose X has rank 2; and X = 1. Let Y be a connected covering graph of X and suppose Y (u) has a Siegel pole . Then we have the following facts.
1. The pole is a …rst order pole of (u) and = R is real: Y 2. There is a unique intermediate graph X2 to Y=X with the property that for every inter-
mediate graph X to Y=X (including X2); is a Siegel pole of X (u) if and only if X is intermediate to Y=X2: e e e 3. X2 is either X or a quadratic (i.e., 2-sheeted) cover of X:
In Theorem 2, X is intermediate to Y=X means that Y covers X and X covers X such that the composition of projection maps is consistent. Our goal in thise paper was to take the …rst steps in investigatinge the locationse of the poles of Ihara zeta functions for irregular graphs. Having discovered Theorem 2, we saw that there is a purely graph theoretic equivalent statement which we state for completeness in Section 5 as Theorem 4. Knowing the result, a direct combinatorial proof of Theorem 4 was easily found. We give this proof in Section 6. In Section 6 we will also show that any graph X of rank 2 possesses a covering Y with a Siegel pole and thus the graph X2 of Theorem 2 actually exists for all X: The analog of this result for number …elds is an open question. One does not know any example of a number …eld whose Dedekind zeta function has a Siegel zero. Remark 1. The authors wish to thank the referee for many useful comments.
2. RAMANUJAN GRAPHS AND THE RIEMANN HYPOTHESIS.
Example 1. As just stated, in a (q + 1)-regular graph with q 2; every vertex is a node. Thus = 1: By Theorem 1, when q 2; must then be either 1 or 2 for a regular graph. Consider the cube Y covering the tetrahedron X = K4: Then X = 1; Y = 2; X = 1; Y = 1: The cube is the unique X2 described in Theorem 2 in this case. The Ihara zeta function for the graph Y thus has a Siegel pole. We now consider the twin problems of formulating the de…nitions of Ramanujan graphs and the Riemann hypothesis for irregular graphs. For regular graphs, we know that the two concepts are the same. We begin with Ramanujan graphs. First de…ne two constants associated to the graph X:
Definition 6.
= max spectrum(AX ) ; X fj j j 2 g 0 = max spectrum(AX ); = : X fj j j 2 j j 6 X g
4 Graph Zeta Functions 2 RAMANUJAN GRAPHS AND THE RIEMANN HYPOTHESIS.
Lubotzky [8] has de…ned X to be Ramanujan if
0 X : X where X is the spectral radius of the adjacency operator on the universal covering tree of X. Hoory [3] has proved that if dX denotes the average degree of the vertices of X, then
X 2 dX 1: This yields an easy test that detects some Ramanujanq graphs, namely a graph X is certainly Ramanujan if we have the Hoory inequality
0 2 dX 1: (8) X Another possible de…nition of Ramanujan forq X irregular would be the following. We say X satis…es the naive Ramanujan inequality if
0 2 1: (9) X X Note that p dX : X This is easily seen using the fact that X is the maximum value of the Rayleigh quotient Af; f = f; f ; while dX is the value when f is the vector all of whose entries are 1. h i h i Now we turn to possible versions of the Riemann hypothesis. We will …rst make some comments about the number …eld situation in order to explain our choices of potential Riemann hypotheses. In number theory, given a number …eld K, the full Generalized Riemann Hypothesis (GRH) for the Dedekind zeta function K (s) corresponding to a number …eld K is equivalent to saying RH-I. K (s) = 0 for 1=2 < Re(s) 1. It is known that, except6 for a …rst order pole at s = 1, (s) = 0 for Re(s) 1: Because K 6 75 years of e¤ort have failed to prove that Siegel zeros (real zeros of K (s) very near s = 1, where "very near" depends upon K) do not exist (although it is known that given K, K (s) has at most one Siegel zero), researchers in the …eld have privately suggested the possibility of the weaker RH-II. RH-I except possibly for one Siegel zero. More recently, a further weakening has been proposed, the Modi…ed Generalized Riemann Hypothesis (MGRH). RH-III. K (s) = 0 for 1=2 < Re(s) 1 except for possible zeros, arbitrary in number, when s is real. 6 This was introduced because various spectral approaches to GRH would not detect real zeros and so would at best end up proving MGRH. Our …rst potential Riemann Hypothesis for graphs will be most analogous to RH-II. The reason for this can be traced to the de…nition of a Ramanujan graph by Lubotzky, Phillips and Sarnak [9] who wanted to allow nice regular bipartite graphs to be Ramanujan graphs and so de…ned Ramanujan graphs in terms of the second largest absolute value of an eigenvalue of the adjacency matrix of the regular graph, thereby allowing the largest absolute value to come from a plus and minus pair. The Riemann Hypothesis for regular graphs was de…ned in such a way that it was equivalent to a regular graph being a Ramanujan graph. We will carry over this de…nition to irregular graphs and see that thanks to Theorem 2, it will essentially be RH-II above for graphs. s In the (q + 1)-regular case the standard change of variables u = q ; s C; turns X (u) into a Dirichlet series which is zero- and pole-free for Re(s) > 1 and has a …rst2 order pole at s = 1. The Riemann hypothesis for regular graphs was then phrased (e.g. [12], p. 129) as X (u) has
5 Graph Zeta Functions 2 RAMANUJAN GRAPHS AND THE RIEMANN HYPOTHESIS. no poles with 0 < Re(s) < 1 except for Re(s) = 1=2. When X is a regular bipartite graph, X (u) has poles at u = 1=q, but also at u = 1=q. To include irregular graphs, the natural s change of variable is u = !X with !X from De…nition 1: All poles of X (u) are then located 1 in the "critical strip", 0 Re(s) 1 with poles at s = 0 (u = 1) and s = 1 (u = !X = RX ). From this point of view, it is natural to say that the Riemann hypothesis for X should require that X (u) has no poles in the open strip 1=2 < Re(s) < 1: In terms of u, we would then put forward a "graph theory Riemann hypothesis" which says that there are no poles of X (u) strictly between the circles u = RX and u = pRX : Because there is no functional equation for irregular graphs relatingj js to 1 s, thisj j Riemann hypothesis makes no statement regarding poles in the open strip, 0 < Re(s) <