Ihara zeta function, coefficients of Maclaurin series, and Ramanujan graphs

TL;DR

This paper explores the relationship between Ramanujan graphs, Ihara zeta functions, and Maclaurin series coefficients, establishing equivalences and bounds that deepen understanding of spectral properties of these graphs.

Contribution

It introduces a new characterization of Ramanujan graphs via the positivity of coefficients in a Maclaurin series expansion of a zeta function variant.

Findings

Equivalence between Ramanujan property and positivity of series coefficients.

Derived Hasse–Weil bound for Ramanujan graphs.

Established functional equation for the zeta function variant.

Abstract

Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever $$ |\lambda|\leq 2q^{\frac{1}{2}} $$ for all nontrivial eigenvalues $\lambda$ of $X$. We consider the variant $\Xi(u)$ of the Ihara zeta function $Z(u)$ of $X$ defined by \begin{gather*} \Xi(u)^{-1} = \left\{ \begin{array}{ll} (1-u)(1-qu)(1-q^{\frac{1}{2}} u)^{2n-2}(1-u^2)^{\frac{n(q-1)}{2}} Z(u) \qquad &\hbox{if $X$ is nonbipartite}, (1-q^2u^2) (1-q^{\frac{1}{2}} u)^{2n-4} (1-u^2)^{\frac{n(q-1)}{2}+1} Z(u) \qquad &\hbox{if $X$ is bipartite}. \end{array} \right. \end{gather*} The function $\Xi(u)$ satisfies the functional equation $\Xi(q^{-1} u^{-1})=\Xi(u)$. Let $\{h_k\}_{k=1}^\infty$ denote the number sequence given by $$ \frac{d}{du}\ln \Xi(q^{-\frac{1}{2}}u) =\sum_{k=0}^\infty h_{k+1} u^k. $$ In this paper we establish the equivalence of the following statements: (i) $X$ is Ramanujan; (ii) $h_k\geq 0$ for all $k\geq 1$; (iii) $h_{k}\geq 0$ for infinitely many even $k\geq 2$. Furthermore we derive the Hasse--Weil bound for the Ramanujan graphs.

Equations

$$|\lambda |\le 2q^{\frac{1}{2}}$$

$$\displaystyle\Xi(u)^{-1}=\left\{\begin{array}[]{ll}(1-u)(1-qu)(1-q^{\frac{1}{2}}u)^{2n-2}(1-u^{2})^{\frac{n(q-1)}{2}}Z(u)&\hbox{if $X$ is nonbipartite},\\ (1-q^{2}u^{2})(1-q^{\frac{1}{2}}u)^{2n-4}(1-u^{2})^{\frac{n(q-1)}{2}+1}Z(u)&\hbox{if $X$ is bipartite}.\end{array}\right.$$

$$\frac{d}{du}\ln \Xi (q^{-\frac{1}{2}}u)=\sum _{k=0}^{\infty }{h}_{k+1}{u}^k.$$

$$\sum _{n=1}^{\infty }\frac{1}{n^s}.$$

$$\xi (s)=\frac{1}{2}s(s-1){\pi }^{-\frac{s}{2}}\Gamma \left(\frac{s}{2}\right)\zeta (s)$$

$$\xi (1-s)=\xi (s).$$

$$\frac{d}{dz}\ln \xi \left(\frac{1}{1-z}\right)=\sum _{k=0}^{\infty }{\lambda }_{k+1}{z}^k.$$

$$\exp \left(\sum _{k=1}^{\infty }\frac{N_k}{k}{u}^k\right).$$

$$|\lambda |\le 2q^{\frac{1}{2}}$$

$$\displaystyle\Xi(u)^{-1}=\left\{\begin{array}[]{ll}(1-u)(1-qu)(1-q^{\frac{1}{2}}u)^{2n-2}(1-u^{2})^{\frac{n(q-1)}{2}}Z(u)&\hbox{if $X$ is nonbipartite},\\ (1-q^{2}u^{2})(1-q^{\frac{1}{2}}u)^{2n-4}(1-u^{2})^{\frac{n(q-1)}{2}+1}Z(u)&\hbox{if $X$ is bipartite}.\end{array}\right.$$

$$\Xi (q^{-1}{u}^{-1})=\Xi (u).$$

$$\frac{d}{du}\ln \Xi (q^{-\frac{1}{2}}u)=\sum _{k=0}^{\infty }{h}_{k+1}{u}^k.$$

$$|\lambda|\leq\left\{\begin{array}[]{ll}(1+\sqrt[k]{4n-7})q^{\frac{1}{2}}&\hbox{if $X$ is nonbipartite},\\ (1+\sqrt[k]{2n-7})q^{\frac{1}{2}}&\hbox{if $X$ is bipartite}\end{array}\right.$$

$$|N_k-q^k-1|$$

$$|N_k-n(q-1)-q^k-1|$$

$$|N_k-n(q-1)-2q^k-2|$$

$$Z(u{)}^{-1}=(1-u^2{)}^{\frac{n(q-1)}{2}}\prod _{\lambda \in \mathrm{Spec}(X)}1-\lambda u+q{u}^2.$$

$$\displaystyle\Xi^{-1}(u)=\left\{\begin{array}[]{ll}\displaystyle\frac{(1-u)(1-qu)(1-q^{\frac{1}{2}}u)^{2n-2}}{\prod\limits_{\lambda\in{\rm Spec}(X)}1-\lambda u+qu^{2}}&\hbox{if $X$ is nonbipartite},\\ \displaystyle\frac{(1-u^{2})(1-q^{2}u^{2})(1-q^{\frac{1}{2}}u)^{2n-4}}{\prod\limits_{\lambda\in{\rm Spec}(X)}1-\lambda u+qu^{2}}&\hbox{if $X$ is bipartite}.\end{array}\right.$$

$$\displaystyle{\rm Spec}^{*}(X)=\left\{\begin{array}[]{ll}{\rm Spec}(X)\setminus\{q+1\}&\hbox{if $X$ is nonbipartite},\\ {\rm Spec}(X)\setminus\{\pm(q+1)\}&\hbox{if $X$ is bipartite}.\end{array}\right.$$

$$\Xi (u)=\prod _{\lambda \in {\mathrm{Spec}}^{\ast }(X)}\frac{1-\lambda u+q{u}^2}{(1-q^{\frac{1}{2}}u{)}^2}.$$

$$\displaystyle h_{k}=\left\{\begin{array}[]{ll}2(n-1)+q^{\frac{k}{2}}+q^{-\frac{k}{2}}-q^{-\frac{k}{2}}N_{k}&\hbox{for all odd $k$},\\ 2(n-1)+q^{\frac{k}{2}}+q^{-\frac{k}{2}}-q^{-\frac{k}{2}}\left(N_{k}-n(q-1)\right)&\hbox{for all even $k$}.\end{array}\right.$$

$$\displaystyle h_{k}=\left\{\begin{array}[]{ll}2(n-2)&\hbox{for all odd $k$},\\ 2(n-2+q^{\frac{k}{2}}+q^{-\frac{k}{2}})-q^{-\frac{k}{2}}\left(N_{k}-n(q-1)\right)&\hbox{for all even $k$}.\end{array}\right.$$

$$\ln Z(u)=\sum _{k=1}^{\infty }\frac{N_k}{k}{u}^k.$$

$$x{T}_k(x)=T_{k+1}(x)+T_{k-1}(x)for\; allk\ge 1$$

$$h_k=2|{\mathrm{Spec}}^{\ast }(X)|-\sum _{s\in q^{-1/2}{\mathrm{Spec}}^{\ast }(X)}{T}_k(s).$$

$$\ln \Xi (q^{-\frac{1}{2}}u)$$

$$=\sum _{s\in S}\ln (1-su+u^2)+2|S|\sum _{k=1}^{\infty }\frac{u^k}{k}.$$

$$\ln (1-su+u^2)$$

$$\ln \Xi (q^{-\frac{1}{2}}u)=\sum _{k=1}^{\infty }\frac{2|S|-{\sum }_{s\in S}{T}_k(s)}{k}{u}^k.$$

$$\displaystyle N_{k}=\left\{\begin{array}[]{ll}q^{\frac{k}{2}}\sum_{s\in q^{-1/2}{\rm Spec}(X)}T_{k}(s)&\hbox{for all odd $k$},\\ n(q-1)+q^{\frac{k}{2}}\sum_{s\in q^{-1/2}{\rm Spec}(X)}T_{k}(s)&\hbox{for all even $k$}.\end{array}\right.$$

$$\displaystyle N_{k}=\left\{\begin{array}[]{ll}\displaystyle\sum_{i=0}^{\frac{k-1}{2}}(-q)^{i}\left({k-i\choose i}+{k-i-1\choose i-1}\right)C_{k-2i}&\hbox{for all odd $k$},\\ \displaystyle n(q-1)+\sum_{i=0}^{\frac{k}{2}}(-q)^{i}\left({k-i\choose i}+{k-i-1\choose i-1}\right)C_{k-2i}&\hbox{for all even $k$}.\end{array}\right.$$

Full text

[TABLE]

for all nontrivial eigenvalues $\lambda$ of $X$. We consider the variant $\Xi(u)$ of the Ihara zeta function $Z(u)$ of $X$ defined by

[TABLE]

The function $\Xi(u)$ satisfies the functional equation $\Xi(q^{-1}u^{-1})=\Xi(u)$. Let $\{h_{k}\}_{k=1}^{\infty}$ denote the number sequence given by

[TABLE]

In this paper we establish the equivalence of the following statements: (i) $X$ is Ramanujan; (ii) $h_{k}\geq 0$ for all $k\geq 1$; (iii) $h_{k}\geq 0$ for infinitely many even $k\geq 2$. Furthermore we derive the Hasse–Weil bound for the Ramanujan graphs.

Keywords: Hasse–Weil bound, Ihara zeta function, Li’s criterion, Ramanujan graphs.

MSC2020: 05C50, 11M26.

1. Introduction

The motivation of this paper originates from developing a graph theoretical counterpart of the following sufficient and necessary condition for the Riemann hypothesis. Recall that the Riemann zeta function $\zeta(s)$ is the analytic continuation of

[TABLE]

The negative even integers are trivial zeros of $\zeta(s)$ and the Riemann hypothesis asserts that the real part of every nontrivial zero of $\zeta(s)$ is $\frac{1}{2}$. The Riemann xi function $\xi(s)$ is a variation of $\zeta(s)$ defined by

[TABLE]

where $\Gamma(s)$ is the Gamma function. The function $\xi(s)$ satisfies the functional equation

[TABLE]

Let $\{\lambda_{k}\}_{k=1}^{\infty}$ denote the number sequence given by

[TABLE]

Li’s criterion states that the Riemann hypothesis holds if and only if $\lambda_{k}\geq 0$ for all $k\geq 1$ [4].

Let $X$ denote an undirected graph of finite order allowing loops and multiple edges. We endow two opposite orientations on all edges of $X$ called the oriented edges of $X$. Given a vertex $x$ of $X$ the valency of $x$ is the number of oriented edges with the initial vertex $x$. If the valency of $x$ is equal to a constant $k$ for all vertices $x$ of $X$ then $X$ is said to be $k$-regular. The adjacency matrix $A$ of $X$ is a square matrix indexed by the vertices of $X$ whose $(x,y)$-entry is defined as the number of oriented edges from $x$ to $y$ for all vertices $x,y$ of $X$. Since $A$ is symmetric $A$ is diagonalizable with real eigenvalues. The eigenvalues of $A$ are also called the eigenvalues of $X$. A walk is a nonempty finite sequence of oriented edges which joins vertices. The length of a walk is the number of oriented edges in the walk. The graph $X$ is said to be connected if there exists a walk from $x$ to $y$ for any two distinct vertices $x,y$ of $X$. A cycle is meant to be a walk from a vertex to itself. If all cycles on $X$ have even length then $X$ is called bipartite. A walk is said to have backtracking if an oriented edge is immediately followed by its opposite orientation in the walk. A cycle is said to be geodesic if all shifted cycles are backtrackless. For all $k\geq 1$ let $N_{k}$ denote the number of geodesic cycles on $X$ of length $k$. The Ihara zeta function $Z(u)$ of $X$ is the analytic continuation of

[TABLE]

For the rest of this paper, we always assume that $X$ is a connected $(q+1)$-regular undirected graph of finite order $n$ with $q\geq 1$ and $n\geq 3$. In this case, the eigenvalues of $X$ with absolute value $q+1$ are called trivial eigenvalues and the poles of $Z(u)$ with values $\pm 1$ and $\pm q^{-1}$ are called trivial poles. The graph $X$ is said to be Ramanujan whenever

[TABLE]

for all nontrivial eigenvalues $\lambda$ of $X$ [5]. The graph $X$ is Ramanujan if and only if all nontrivial poles of $Z(u)$ have the same absolute value $q^{-\frac{1}{2}}$, which is similar to the Riemann hypothesis by writing $u=q^{-s}$ [3, 8]. We define the function $\Xi(u)$ by

[TABLE]

The function $\Xi(u)$ is considered as the Ihara xi function which satisfies the functional equation

[TABLE]

If we set $u=q^{-s}$, then this becomes a functional equation relating $1-s$ and $s$ similar to the Riemann xi function.

Definition 1.1**.**

Let $\{h_{k}\}_{k=1}^{\infty}$ denote the number sequence given by

[TABLE]

The main results of this paper are as follows:

Theorem 1.2**.**

If there is a positive even integer $k$ with $h_{k}\geq 0$ then

[TABLE]

for all nontrivial eigenvalues $\lambda$ of $X$.

Theorem 1.3**.**

The following are equivalent:

1. (i)

$X$* is Ramanujan.* 2. (ii)

$h_{k}\geq 0$* for all $k\geq 1$.* 3. (iii)

$h_{k}\geq 0$* for infinitely many even $k\geq 2$.*

Theorem 1.4**.**

1. (i)

If $X$ is nonbipartite, then $X$ is Ramanujan if and only if

[TABLE] 2. (ii)

If $X$ is bipartite, then $X$ is Ramanujan if and only if

[TABLE]

Theorem 1.2 is an improvement of the implication from Theorem 1.3(iii) to Theorem 1.3(i). The equivalence of Theorem 1.3(i), (ii) is an analogue of Li’s criterion. In [7, §1.3] it was shown that if $X$ is nonbipartite Ramanujan then $N_{k}=q^{k}+O(nq^{\frac{k}{2}})$; if $X$ is bipartite Ramanujan then $N_{k}=2q^{k}+O(nq^{\frac{k}{2}})$ for even $k$. Theorem 1.4 strengthens the above necessary conditions for $X$ as Ramanujan. Furthermore, Theorem 1.4 is an analogue of the Hasse–Weil bound [9].

The paper is organized as follows: In §2 we give some preliminaries on $Z(u)$ and $\Xi(u)$. In §3 we derive three formulae for $\{h_{k}\}_{k=1}^{\infty}$. In §4 we prove Theorems 1.2–1.4. In §5 we discuss the behavior of $\{h_{2k}\}_{k=1}^{\infty}$ when $X$ is not Ramanujan.

2. The Ihara zeta and xi functions

Let ${\rm Spec}(X)$ denote the spectrum of $X$; that is the multiset of all eigenvalues of $X$ with geometric multiplicities. Since $X$ is a connected $(q+1)$-regular undirected graph, the value $q+1\in{\rm Spec}(X)$ with multiplicity one. Ihara’s theorem [2] states that $Z(u)$ is a rational function of the form

[TABLE]

Substituting (5) into (4) yields that

[TABLE]

Let ${\rm Spec}^{*}(X)$ denote the multiset of all nontrivial eigenvalues of $X$ with geometric multiplicities. Recall that ${\rm Spec}(X)$ is symmetric with respect to [math] when $X$ is bipartite. Hence

[TABLE]

Combined with (8) we obtain that

[TABLE]

Proposition 2.1**.**

$\Xi(u)$* satisfies the functional equation $\Xi(q^{-1}u^{-1})=\Xi(u)$.*

Proof.

It is routine to verify the proposition by using (12). ∎

3. Formulae for $h_{k}$

Recall the sequence $\{h_{k}\}_{k=1}^{\infty}$ from Definition 1.1. In this section we give two combinatorial formulae for $\{h_{k}\}_{k=1}^{\infty}$ and a formula for $\{h_{k}\}_{k=1}^{\infty}$ in terms of the Chebyshev polynomials.

Proposition 3.1**.**

1. (i)

If $X$ is nonbipartite then

[TABLE] 2. (ii)

If $X$ is bipartite then

[TABLE]

Proof.

Taking logarithm on (1) yields that

[TABLE]

Evaluate $h_{k}$ by using (4) and (13) directly. ∎

Let $\{T_{k}(x)\}_{k=0}^{\infty}$ denote the polynomials defined by

[TABLE]

with $T_{0}(x)=2$ and $T_{1}(x)=x$ [1, §2.3]. Note that $\frac{1}{2}T_{k}(2x)$ is the $k$th Chebyshev polynomial of the first kind for all $k\geq 0$ [6].

Lemma 3.2** (​​[1, 6]).**

$T_{k}(x+x^{-1})=x^{k}+x^{-k}$* for all $k\geq 0$.*

Given a multiset $S$ of numbers and a constant $c$, we let $cS$ denote the multiset consisting of $cs$ for all $s\in S$.

Proposition 3.3**.**

For all $k\geq 1$ the following equation holds:

[TABLE]

Proof.

Let $S=q^{-\frac{1}{2}}{\rm Spec}^{*}(X)$. Applying (12) yields that

[TABLE]

Let $s\in S$ be given. Write $s=\alpha+\alpha^{-1}$ for some nonzero complex number $\alpha$. Then

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It follows from Lemma 3.2 that $T_{k}(s)=\alpha^{k}+\alpha^{-k}$. By the above comments we have

[TABLE]

Now the proposition follows by taking differential on both sides of the above equation. ∎

For convenience we define ${-1\choose-1}=1$ and ${k\choose-1}=0$ for all $k\geq 0$.

Lemma 3.4** (​​[1, 6]).**

$T_{k}(x)=\displaystyle\sum_{i=0}^{\lfloor\frac{k}{2}\rfloor}(-1)^{i}\left({k-i\choose i}+{k-i-1\choose i-1}\right)x^{k-2i}$* for all $k\geq 0$.*

For all $k\geq 1$ let $C_{k}$ denote the number of the cycles on $X$ of length $k$ and define $C_{0}=n$.

Lemma 3.5**.**

$\sum\limits_{s\in q^{-1/2}{\rm Spec}(X)}T_{k}(s)=q^{-\frac{k}{2}}\displaystyle\sum_{i=0}^{\lfloor\frac{k}{2}\rfloor}(-q)^{i}\left({k-i\choose i}+{k-i-1\choose i-1}\right)C_{k-2i}$* for all $k\geq 0$.*

Proof.

Evaluate the left-hand side by using Lemma 3.4 along with the fact $C_{k}=\sum_{\lambda\in{\rm Spec}(X)}\lambda^{k}$ for all $k\geq 0$. ∎

Combining Propositions 3.1 and 3.3 yields that

[TABLE]

As far as we know, the formula (16) was first given in [7, Lemma 4] and it was given a proof of [7, Lemma 4] without using (5). By Lemma 3.5 and (16) we have the following lemma:

Lemma 3.6**.**

The following equation holds:

[TABLE]

Proposition 3.7**.**

1. (i)

If $X$ is nonbipartite then

[TABLE] 2. (ii)

If $X$ is bipartite then

[TABLE]

Proof.

Combine Proposition 3.1 and Lemma 3.6. ∎

4. Proof of the main results

In this section we show Theorems 1.2–1.4.

Lemma 4.1**.**

For all $k\geq 1$ the coefficient $h_{k}\geq 0$ if and only if

[TABLE]

Proof.

Immediate from Proposition 3.3. ∎

Lemma 4.2** (​​[6]).**

$T_{k}(2\cos\theta)=2\cos k\theta$* for all $k\geq 0$ and all real numbers $\theta$.*

Lemma 4.3**.**

If $s$ is a real number with $|s|\leq 2$ then $|T_{k}(s)|\leq 2$ for all $k\geq 0$.

Proof.

Immediate from Lemma 4.2. ∎

Lemma 4.4**.**

If $X$ is Ramanujan then $|T_{k}(s)|\leq 2$ for all $k\geq 0$ and all $s\in q^{-1/2}{\rm Spec}^{*}(X)$.

Proof.

Since $X$ is Ramanujan if and only if $|s|\leq 2$ for all $s\in q^{-1/2}{\rm Spec}^{*}(X)$, the lemma is immediate from Lemma 4.3. ∎

Lemma 4.5** (​​[6]).**

$T_{k}(x)=\displaystyle{2\left(\frac{x}{2}\right)^{k}\sum_{i=0}^{\lfloor\frac{k}{2}\rfloor}{k\choose 2i}\left(1-\left(\frac{x}{2}\right)^{-2}\right)^{i}}$* for all $k\geq 0$.*

Lemma 4.6**.**

If $s$ is a real number with $|s|>2$ then $T_{k}(s)>0$ for all even $k\geq 0$.

Proof.

Immediate from Lemma 4.5. ∎

Proposition 4.7**.**

Let $S$ denote a nonempty finite multiset consisting of real numbers. If there is a positive even integer $k$ with

[TABLE]

then $|s|\leq 1+\sqrt[k]{4|S|-3}$ for all $s\in S$.

Proof.

For convenience let

[TABLE]

Suppose on the contrary that there is a real number $t\in S$ with

[TABLE]

Using (19) yields that $1-\left(\frac{t}{2}\right)^{-2}>1-(1-\varepsilon)^{2}=2\varepsilon-\varepsilon^{2}$. Since $0<\varepsilon<1$ we have $\varepsilon^{2}<2\varepsilon-\varepsilon^{2}$. Combined with Lemma 4.5 this implies

[TABLE]

By Lemmas 4.3 and 4.6 we have $T_{k}(s)\geq-2$ for all $s\in S\setminus\{t\}$.

Combining (17) with the above comments yields that

[TABLE]

It leads to

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Since $k$ is even the inequality (19) implies that

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Combining (20) and (21) we see that

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Using the setting (18) it is routine to verify that both sides of (22) are equal, a contradiction. The proposition follows. ∎

Proof of Theorem 1.2. By Lemma 4.1, when $X$ is nonbipartite the result follows by applying Proposition 4.7 with $S$ replaced by $q^{-1/2}{\rm Spec}^{*}(X)$.

Note that the number of zeros in ${\rm Spec}^{*}(X)$ is even if the regular graph $X$ is bipartite. Since $k$ is even it follows from Lemma 3.2 that $T_{k}(x)$ is an even function. Combined with Lemma 4.1, when $X$ is bipartite the result follows by applying Proposition 4.7 with $S$ chosen as the multiset of all positive numbers and a half number of zeros in $q^{-1/2}{\rm Spec}^{*}(X)$. $\square$

Proof of Theorem 1.3. (i) $\Rightarrow$ (ii): Combine Lemmas 4.1 and 4.4.

(ii) $\Rightarrow$ (iii): It is obvious.

(iii) $\Rightarrow$ (i): Immediate from Theorem 1.2. $\square$

Lemma 4.8**.**

If $X$ is Ramanujan then

[TABLE]

for all $k\geq 1$.

Proof.

By Proposition 3.3 and Lemma 4.4 the coefficient $h_{k}\leq 4|{\rm Spec}^{*}(X)|$ for all $k\geq 1$. Hence the lemma follows by (11). ∎

Proof of Theorem 1.4. Combine Theorem 1.3(i), (ii) and Proposition 3.1 along with Lemma 4.8. $\square$

5. Behavior of $h_{2k}$

We end this paper with a remark on $\{h_{2k}\}_{k=1}^{\infty}$ under the assumption that $X$ is not Ramanujan. Let $S$ denote the set of all nonzero complex numbers $\alpha$ with $\alpha+\alpha^{-1}\in q^{-1/2}{\rm Spec}^{*}(X)$. For those $s\in q^{-1/2}{\rm Spec}^{*}(X)$ with $|s|\leq 2$ the corresponding numbers $\alpha\in S$ have the same absolute value $1$. By the assumption that $X$ is not Ramanujan there exists an $s\in q^{-1/2}{\rm Spec}^{*}(X)$ with $|s|>2$ and the corresponding numbers $\alpha\in S$ are real and $\alpha\not=\pm 1$. Let

[TABLE]

Since the function $f(x)=x+x^{-1}$ is strictly increasing on $(1,\infty)$, it follows that

[TABLE]

Using Lemma 3.2 yields that

[TABLE]

for all $\alpha\in S$. It follows from Proposition 3.3 that $h_{2k}$ is asymptotic to $-m\mu^{2k}$ as $k$ approaches to $\infty$, where $m$ is the number of $s\in q^{-1/2}{\rm Spec}^{*}(X)$ with $|s|=\mu+\mu^{-1}$. Therefore the following result holds:

Theorem 5.1**.**

If $X$ is not Ramanujan then

[TABLE]

where $\max$ is over all nontrivial eigenvalues $\lambda$ of $X$.

Acknowledgements

The research is supported by the Ministry of Science and Technology of Taiwan under the project MOST 106-2628-M-008-001-MY4.