The Eigenvalues of the Graphs $D(4,q)$

TL;DR

This paper determines the eigenvalues of the graphs D(4,q), revealing they are excellent expanders close to Ramanujan graphs, which enhances understanding of their spectral and expansion properties.

Contribution

The paper computes the spectrum of D(4,q) graphs, the smallest open case, showing they are nearly Ramanujan expanders with specific eigenvalue bounds.

Findings

Eigenvalues other than ±q are bounded by 2√q

Graphs are q-regular on 2q^4 vertices

Graphs are nearly Ramanujan expanders

Abstract

The graphs $D(k,q)$ have connected components $CD(k,q)$ giving the best known bounds on extremal problems with {\em forbidden\/} even cycles, and are denser than the well-known graphs of Lubotzky, Phillips, Sarnak and Margulis. Despite this, little about the spectrum and expansion properties of these graphs is known. In this paper we find the spectrum for $k=4$, the smallest open case. For each prime power $q$, the graph $D(4,q)$ is $q$-regular graph on $2q^4$ vertices, all of whose eigenvalues other than $\pm q$ are bounded in absolute value by $2\sqrt{q}$. Accordingly, these graphs are good expanders, in fact very close to Ramanujan.

Equations

$$h(\Gamma )=\min \left\{\frac{|\partial F|}{|F|}:F\subset V\text{ and }|F|⩽\frac{|V|}{2}\right\}.$$

$${\textstyle{\frac{1}{2}}}\bigl{(}q-\lambda_{2}(\Gamma)\bigr{)}\leqslant h(\Gamma)\leqslant\sqrt{2q(q-\lambda_{2}(\Gamma))}\,;$$

$$\cdots \to D(5,q)\to D(4,q)\to D(3,q)\to D(2,q)$$

$$(x^2-q^2)(x^2-q{)}^{q-1}{x}^{2q(q-1)}$$

$$(x^2-q^2)(x^2-2q{)}^{q(q-1{)}^2/2}(x^2-q{)}^{2q(q-1)}{x}^{q^3-2q^2+3q-2}$$

$$p_2+{\ell }_2=p_1{\ell }_1,p_3+{\ell }_3=p_1{\ell }_{2}and{p}_4+{\ell }_4=p_2{\ell }_1.$$

$$p_1\ne p_1^{\prime },(p_1-p_1^{\prime })(p_4-p_4^{\prime })=(p_2-p_2^{\prime }{)}^{2}and{p}_3-p_3^{\prime }=p_2{p}_1^{\prime }-p_1{p}_2^{\prime }.$$

$$B:=\left[\begin{array}[]{cc}0&B_{1}\\ B_{1}^{T}&0\end{array}\right]$$

$$\varphi (x)=(x-q(q-1))(x+q{)}^{(q-1)(q^2-q+1)}{x}^{3q(q-1)}(x-q{)}^{q(q-1{)}^2}\tilde{\varphi }(x)$$

$$\varphi (x)=\det (x{I}_{|G|}-A)=\prod _{i=1}^k\det [x{I}_{n_i}-{\pi }_i(S){]}^{n_i}.$$

$$tr:F\to {\mathbb{F}}_p,tr(a)=a+a^p+a^{p^2}+\cdots +a^{p^{e-1}}.$$

$${\epsilon }_f=\sum _{a\in F}{\zeta }^{tr[f(a)]}\in \mathbb{Z}[\zeta ].$$

$$\varepsilon_{f}=\left\{\begin{array}[]{ll}0,&\hbox{if $b\neq 0$;}\\ q\zeta^{tr(c)},&\hbox{otherwise.}\end{array}\right.$$

$$\sum_{a\in F}a^{k}=\left\{\begin{array}[]{ll}-1,&\hbox{if $k=(q-1)k_{1}$ for some integer $k_{1}\geqslant 1$;}\\ 0,&\hbox{otherwise}.\end{array}\right.$$

$$n_0=\frac{1}{2}(q-1)(q^2-q+2);n_1=2q(q-1);n_2=\frac{1}{2}q(q-1{)}^2$$

$$n_0=\frac{1}{3}(q-1)(q^2+8);n_1=\frac{1}{2}(q-1{)}^2(q+4);n_3=\frac{1}{6}(q-1{)}^2(q-2)$$

$$n_1=(q-1{)}^2+\frac{1}{2}q(q-1{)}^2+(q-1{)}^2=\frac{1}{2}(q-1)(q^2+4).$$

$$n_0=q^3-1-n_1-n_3=\frac{1}{2}(q-1)(q^2+8).$$

$$g=g(t,u,v,w)=\left[\begin{array}[]{ccccc}1&t&u&v{+}tu&w\\ &1&0&-u&0\\ &&1&t&0\\ &&&1&0\\ &&&&1\end{array}\right].$$

$$(1,p_1,p_2,p_3,p_4)\mapsto (1,p_1,p_2,p_3,p_4)g(t,u,v,w).$$

$$P\mapsto Pg$$

$$P=P(p_1,p_2,p_3,p_4)=(1,p_1,p_2,p_3,p_4).$$

$$S$$

$$=\{g(t,rt,-r{t}^2,r^{2}t):r,t\in F,t\ne 0\}.$$

$$[g(t,u,v,w),g(t^{\prime },u^{\prime },v^{\prime },w^{\prime })]=g(0,0,2t^{\prime }u-2t{u}^{\prime },0).$$

$$g(t,u,v,w)g(t^{\prime },u^{\prime },v^{\prime },w^{\prime })=g(t+t^{\prime },u+u^{\prime },v+v^{\prime },w+w^{\prime }).$$

$$\chi_{\alpha,\beta,\gamma,\eta}\bigl{(}g(t,u,v,w)\bigr{)}=(-1)^{tr(\alpha t+\beta u+\gamma v+\eta w)},\quad\alpha,\beta,\gamma,\eta\in F.$$

$$\varphi (x)$$

$$\displaystyle{}=\bigl{(}x-q(q-1)\bigr{)}(x-3q)^{q(q-1)^{2}(q-2)/24}(x-q)^{q(q-1)^{2}(q+4)/4}$$

$$\times x^{(q-1)(q^3+8q+3)/3}(x+q{)}^{3q(q-1{)}^2(q+2)/8}.$$

$$\varphi (x)$$

$$\displaystyle{}=(x-q(q-1))(x+q)^{q-1}\mskip-12.0mu\prod_{\genfrac{}{}{0.0pt}{}{\alpha,\beta,\gamma,\eta\in F}{(\beta,\gamma,\eta)\neq(0,0,0)}}\mskip-12.0mu\Bigl{(}x-\sum_{\genfrac{}{}{0.0pt}{}{r,t\in F}{t\neq 0}}(-1)^{tr(\alpha t+\beta rt+\gamma rt^{2}+\eta r^{2}t)}\Bigr{)}.$$

Full text

1 Introduction

Let $\Gamma$ be a graph with vertex set $V$. (All our graphs are undirected and have no loops or multiple edges. See e.g. [6, 3] for standard terminology and theory of graphs.) Given a set of vertices $F\subset V$, we define $\partial F$ to be the set of vertices in $V\smallsetminus F$ which are adjacent to some vertex of $F$. The isoperimetric constant of $\Gamma$ is defined to be

[TABLE]

An infinite family of $d$-regular graphs whose isoperimetric constants are uniformly bounded away from 0 is an expander family. The best known general bounds on $h(\Gamma)$ are expressed in terms of the spectrum of $\Gamma$, i.e. the multiset of eigenvalues of its adjacency matrix. In particular, if $\Gamma$ is $q$-regular with second-largest eigenvalue $\lambda_{2}(\Gamma)<\lambda_{1}=q$, then

[TABLE]

see e.g. [9, Prop.1.84]. (The second-largest eigenvalue is denoted differently in some sources, including [9].) Thus to certify an infinite family of $q$-regular graphs as an expander family, we require a uniform lower bound on the spectral gap $q-\lambda_{2}(\Gamma)$. A $q$-regular connected graph $\Gamma$ is Ramanujan if $\lambda_{2}(\Gamma)\leqslant 2\sqrt{q-1}$; by the Alon-Boppana Theorem (see e.g. [9, Ch.3]) this bound is asymptotically best possible for any infinite family of $q$-regular graphs.

In searching for good families of explicitly defined graphs with good expansion, a particularly promising infinite family of graphs is the sequence

[TABLE]

defined by Lazebnik and Ustimenko [10] for each prime power $q$. Each graph $D(k,q)$ in this sequence is bipartite $q$-regular on $2q^{k}$ vertices having girth${}\geqslant k+4$ (or $k+5$, when $k$ is even); and each connecting map ‘$\to$’ is a graph-theoretic cover (see [12, Sec.3B]). The graphs $D(k,q)$ are connected for $k\leqslant 5$ and $q$ odd; see [11]. The covering property ensures that the girth of $D(k,q)$ is weakly increasing as $k\to\infty$, and the spectrum of $D(k,q)$ embeds in that of $D(k+1,q)$; see [12, Sec.3C].

The graphs $CD(k,q)$ are important in the study of Turán type problems on even cycles, giving better lower bounds on the maximum number of edges in graphs of girth $g\geqslant 6$ than the well-known Ramanujan graphs of Lubotzky, Phillips and Sarnak [15]. Similarly, the graphs $LD(q,r)$ of Alon et al. [1], another expander family with fixed degree, have girth 3 (after removing loops). By comparison, therefore, one might expect the graphs $CD(k,q)$ to have very good expansion properties.

However, little is known about the eigenvalues of these graphs. In fact, to date only the spectrum of $D(2,q)$ and $D(3,q)$ are known, their characteristic polynomials being

[TABLE]

and

[TABLE]

respectively; see [13, Sec.5]. In particular, these graphs are Ramanujan. However, Reichard [19] and Thomason [20] independently showed by computer that the graphs $D(4,q)$ are not Ramanujan for certain $q$, refuting the claim of [21]; see also the final note in this paper where we investigate this question more closely. The same statements apply to $D(k,q)$ for all $k\geqslant 4$, since the spectrum of $D(4,q)$ is embedded in that of $D(k,q)$ for $k\geqslant 4$.

It was later claimed in [22] that the eigenvalues of $D(k,q)$ other than $\pm q$ are bounded by $2\sqrt{q}$. However, a flaw was later found in the argument, leaving the problem open; see the Math Review MR2048644 for [22]. To date, we have not found any counterexample to this statement, so we list it as a conjecture. Following [10], we denote by $CD(k,q)$ a connected component of $D(k,q)$; and we note that $CD(4,q)=D(4,q)$ whenever $q\notin\{2,4\}$.

Conjecture 1.1** (Ustimenko)**

For all $(k,q)$, $CD(k,q)$ has second largest eigenvalue less than or equal to $2\sqrt{q}$.

In this paper we verify Conjecture 1.1 for $k=4$:

Theorem 1.2

The second largest eigenvalue of $CD(4,q)$ is less than or equal to $2\sqrt{q}$.

This implies that these graphs are very close to Ramanujan. Our proof is given in Section 5 for even $q$, and in Section 6 for odd $q$. A more explicit determination of the spectrum is given in Section 7 for prime values $q=p$.

Our approach is similar to [4], in that we first realize the halved (point) graph of $D(k,q)$ as a Cayley graph of a certain $p$-group $G$. Unlike the situation for the Wenger graphs in [4], or the graphs $D(2,q)$ and $D(3,q)$, our group $G$ is nonabelian whenever $q$ is odd, thus requiring more extensive use of the representation theory of $G$. Finally, our bounds on eigenvalues are obtained using Weil’s bound for exponential sums over ${\mathbb{F}}_{q}$, or over Galois rings of characteristic 3 in the case $q=3^{e}$.

2 The Graphs $D(4,q)$ and their Point Collinearity Graphs $\Gamma(4,q)$

Throughout, we take $F={\mathbb{F}}_{q}$ where $q$ is a prime power. The graph $D(4,q)$ is bipartite and $q$-regular with $2q^{4}$ vertices. These include $q^{4}$ vertices $P=P(p_{1},p_{2},p_{3},p_{4})$ called points, and $q^{4}$ vertices $L=L(\ell_{1},\ell_{2},\ell_{3},\ell_{4})$ called lines, where all coordinates are in $F$; and the point $P$ and line $L$ (with coordinates as above) are incident iff

[TABLE]

Also denote by $\Gamma=\Gamma(4,q)$ the point collinearity graph of $D(4,q)$, i.e. the graph whose vertices are the points of $D(4,q)$, two points being adjacent in $\Gamma$ iff they are distinct but collinear in $D(4,q)$; see e.g. [3, Sec.14.2.2]. One checks that two vertices $P(p_{1},p_{2},p_{3},p_{4}),P(p_{1}^{\prime},p_{2}^{\prime},p_{3}^{\prime},p_{4}^{\prime})$ are adjacent in $\Gamma$ (i.e. distinct and collinear in $D(4,q)$) iff

[TABLE]

The adjacency matrix of $\Gamma$ has the form $A=B_{1}B_{1}^{T}-qI_{q^{4}}$ where $B_{1}$ is a $q^{4}\times q^{4}$ matrix for which

[TABLE]

is the adjacency matrix of $D(4,q)$ (with the first $q^{4}$ rows and columns indexed by points, and the last $q^{4}$ rows and columns indexed by lines). Note that $\Gamma$ is a $q(q-1)$-regular graph on $q^{4}$ vertices. The spectra of $A$ and $B$ are in direct relationship. Indeed, elementary methods yield the following, which is also implicit in [4, 13]:

Lemma 2.1

Denote the characteristic polynomial of $A$, the adjacency matrix of $\Gamma(4,q)$, by $\phi(x)=\det(xI_{q^{4}}-A)$. Then the characteristic polynomial of $B$, the adjacency matrix of $D(4,q)$, is $\det(xI_{2q^{4}}-B)=\phi(x^{2}-q)$.\qed

Equivalently, every eigenvalue $\lambda$ of $A$, with multiplicity $m$, corresponds to a pair of eigenvalues $\pm\sqrt{q+\lambda}$ of $B$, each with multiplicity $m$ (or a single eigenvalue [math] of multiplicity $2m$ in case $\lambda=-q$). The remainder of this paper is devoted to proving

Theorem 2.2

The graph $\Gamma=\Gamma(4,q)$ is connected except for $q\in\{2,4\}$, when the graph has 4 connected components. When $q$ is odd, the adjacency matrix $A$ of $\Gamma$ has characteristic polynomial $\phi(x)=\det(xI_{q}-A)$ of the form

[TABLE]

where all roots of $\widetilde{\phi}(x)\in{\mathbb{Z}}[x]$ have the form $\lambda=-q+\varepsilon^{2}$ where $|\varepsilon|\leqslant 2\sqrt{q}$. Each such value $\varepsilon$ lies the ring {\mathbb{Z}}\bigl{[}2\cos\frac{2\pi}{p}\bigr{]}, or {\mathbb{Z}}\bigl{[}2\cos\frac{2\pi}{9}\bigr{]} if $p=3$.

A complete determination of $\phi(x)$ is given in Theorem 5.1 when $q$ is even, and in Theorem 7.6 when $q=p$ is prime. Now using Lemma 2.1 we obtain

Theorem 2.3

The graph $D(4,q)$ has eigenvalues $\pm q$, each of multiplicity 1 (unless $q\in\{2,4\}$ when each of the eigenvalues $\pm q$ has multiplicity 4). All remaining eigenvalues have the form $\pm\varepsilon$ where $|\varepsilon|\leqslant 2\sqrt{q}$.

Once again, the eigenvalues $\varepsilon$ of Theorem 2.3 are cyclotomic integers satisfying the conclusion of Theorem 2.2. In Theorem 2.2 the multiplicity of the eigenvalue 0 may actually exceed $3q(q-1)$; in particular this happens whenever $q\equiv 2\mod 3$. We find explicit formulas for the actual eigenvalues, by expressing the ‘error’ terms $\varepsilon$ as exponential sums defined over finite fields (or over the Galois ring $GR(9,e)$ of order $9^{e}=q^{2}$ and characteristic 9, in the case $q=3^{e}$). This leads to our bound $|\varepsilon|\leqslant 2\sqrt{q}$, using the Hasse-Davenport-Weil bound when $q=p^{e}$, $p\geqslant 5$; or the analogous bound of Kumar, Helleseth and Calderbank [8] in the case $p=3$.

Our strategy for proving this result (see [2] for details) is to first realize $\Gamma$ as a Cayley graph Cay$(G,S)$ for a nonabelian group $G$ of order $q^{4}$, and connection set $S\subset G$. (Thus $\Gamma$ has vertices labeled by elements of $G$; and two vertices $g,g^{\prime}\in G$ are adjacent in $\Gamma$ iff $g^{\prime}g^{-1}\in S$). Since our graph $\Gamma$ is undirected and connected with no loops or multiple edges, we will have $\langle S\rangle=G$, $1\notin S$, and $g\in S$ iff $g^{-1}\!\in S$. We then determine the number $k$ of conjugacy classes of $G$, and a complete set (up to equivalence) of irreducible ordinary representations $\pi_{i}:G\to GL_{n_{i}}({\mathbb{C}})$ for $i=1,2,\ldots,k$. For each $i$, we compute the complex $n_{i}\times n_{i}$ matrix $\pi_{i}(S):=\sum_{g\in S}\pi_{i}(g)$.

Theorem 2.4** ([2, 5]; see also [9])**

The characteristic polynomial of $A$, the adjacency matrix of $\Gamma$, is given by

[TABLE]

Note that this gives $\sum_{i=1}^{k}n_{i}^{2}=|G|$ eigenvalues (counting according to their respective multiplicities) as required. In those cases where $G$ is abelian, the eigenvalues are simply the character values $\chi_{i}(S)$. A similar simplification is possible when $S$ is a union of conjugacy classes of $G$, but this does not apply in our case. When $G$ is nonabelian and the full matrices of the representations $\pi_{i}$ are not explicitly known, determining the eigenvalues of $\pi_{i}(S)$ from the character values alone may require substantial additional work (see [2]); but for us, the group $G$ is sufficiently nice that explicit descriptions of the full matrices of the representations $\pi_{i}$ are easily available, making our job much easier.

3 Background on Finite Fields

General results on finite fields can be found in [14]. Let $F={\mathbb{F}}_{q}$ be a field of order $q=p^{e}$ where $e\geqslant 1$ and $p$ is prime. The absolute trace map is

[TABLE]

We also fix a primitive $p$-th root of unity $\zeta=\zeta_{p}\in{\mathbb{C}}$; here it suffices to assume that $\zeta=e^{2\pi i/p}$. We define the exponential sum of an arbitrary function $f:F\to F$ as the cyclotomic integer

[TABLE]

Lemma 3.1

For every polynomial of the form $f(t)=bt+c\in F[t]$ we have

[TABLE]

Proof 1

See [14, Ch.5].\qed

Lemma 3.2

Let $k$ be a non-negative integer. Then

[TABLE]

Proof 2

See [14, p.271].\qed

Lemma 3.3

• (i)

Let $n_{k}$ be the number of nonzero polynomials $a_{2}t^{2}+a_{1}t+a_{0}\in F[t]$ having exactly $k$ distinct roots in $F$. Then

[TABLE]

and $n_{k}=0$ otherwise. Here $n_{0}+n_{1}+n_{2}=q^{3}-1$; and for $k=0$ we include $\frac{1}{2}q(q-1)^{2}$ irreducible quadratics and $q-1$ nonzero constant polynomials.

• (ii)

For $q$ even, let $n_{k}$ be the number of nonzero polynomials $a_{3}t^{3}+a_{1}t+a_{0}\in F[t]$ having exactly $k$ distinct nonzero roots in $F$. Then

[TABLE]

and $n_{k}=0$ otherwise. Here $n_{0}+n_{1}+n_{3}=q^{3}-1$.

Proof 3

Every nonzero polynomial of degree${}\leqslant 2$ with a single root has the form $a_{1}(t-t_{1})$ or $a_{2}(t-t_{1})^{2}$, giving $n_{1}=2q(q-1)$. Every nonzero polynomial of degree $2$ having two distinct roots has the form $a_{1}(t-t_{1})(t-t_{2})$ with $a_{2}\neq 0$ and $t_{1}\neq t_{2}$; and there are $n_{2}=\frac{1}{2}q(q-1)^{2}$ such polynomials. This leaves $n_{0}=q^{3}-1-n_{1}-n_{2}=\frac{1}{2}(q-1)(q^{2}-q+2)$, and the remaining assertions of (i) follow.

Now suppose $q$ is even, and consider a nonzero polynomial $f(t)=a_{3}t^{3}+a_{1}t+a_{0}\in F[t]$. If $f(t)=a_{3}(t+t_{1})(t+t_{2})(t+t_{3})$ then $t_{1}+t_{2}+t_{3}=0$; so in characteristic 2, the number of distinct nonzero roots must be 0, 1 or 3. There are $n_{3}=\frac{1}{6}(q-1)^{2}(q-2)$ nonzero polynomials of the form $f(t)=a_{3}(t+t_{1})(t+t_{2})(t+t_{1}+t_{2})$ where $t_{1},t_{2}$ are nonzero and distinct. There are $(q-1)^{2}$ cubics of the form $a_{3}t(t+t_{1})^{2}$ where $t_{1}\neq 0$; and by (i), there are $\frac{1}{2}q(q-1)^{2}$ cubics of the form (t+t_{1})\bigl{(}a_{3}t^{2}+a_{3}t_{1}t+\frac{a_{0}}{t_{1}}\bigr{)} for which $t_{1}\neq 0$ and the quadratic factor is irreducible. These, together with the $(q-1)^{2}$ polynomials $a_{1}t+a_{0}$ having $a_{1},a_{0}\neq 0$, give

[TABLE]

This leaves

[TABLE]

One checks that this includes $\frac{1}{3}(q-1)(q^{2}+2)$ irreducible cubics of the required form, together with $q-1$ polynomials of the form $a_{1}t$ with $a_{1}\neq 0$, and $q-1$ nonzero constant polynomials.\qed

4 A Regular Group of Automorphisms of $\Gamma$

For all $t,u,v,w\in F$ we define the matrix

[TABLE]

These $q^{4}$ matrices form a subgroup $G<GL_{5}(F)$ acting regularly on points via

[TABLE]

which can be written simply as

[TABLE]

after a slight abuse of notation by which we identify

[TABLE]

One checks that this action preserves collinearity of points, and so gives a group of automorphisms of $\Gamma$ which is regular on the vertices. Thus $\Gamma$ is a Cayley graph Cay$(G,S)$ for the set of $q(q-1)$ elements

[TABLE]

The commutator of two typical elements of $G$ is

[TABLE]

At this point we must consider separately the cases $q$ even and $q$ odd, for which $G$ is abelian or nonabelian, respectively.

5 The case $q$ even

In this section we suppose $q$ is even, so that

[TABLE]

In this case $G$ is elementary abelian, with $q^{4}$ irreducible linear characters

[TABLE]

Theorem 5.1

Suppose $q$ is even. Then the characteristic polynomial of $A$, the incidence matrix of $\Gamma$, is

[TABLE]

The graph $\Gamma$ is connected for $q\geqslant 8$; while for $q\in\{2,4\}$, $\Gamma$ has 4 connected components.

Proof 4

By Theorem 2.4 and Lemma 3.1, we have

[TABLE]

Now using the fact that the map $F\to F$, $r\mapsto r^{2}$ is an automorphism (in particular bijective and trace-preserving),

[TABLE]

After re-indexing via $(\beta,\gamma,\eta)\mapsto(\beta^{1/2},\gamma^{1/2},\eta)$,

[TABLE]

If the polynomial $f(t)=\gamma t^{3}+\beta t+\eta\in F[t]$ has a unique nonzero root $t_{1}\in F$, then the map $F\to{\mathbb{F}}_{2}$, $\alpha\mapsto tr(\alpha t_{1})$ takes each of the values in $\{0,1\}$ exactly $\frac{q}{2}$ times, in which case

[TABLE]

Similarly, if $f(t)$ (as above) has three distinct nonzero roots $t_{1},t_{2},t_{3}\in F$, then $t_{1}+t_{2}+t_{3}=0$ and the map $F\to{\mathbb{F}}_{2}^{3}$, $\alpha\mapsto(tr(\alpha t_{1}),tr(\alpha t_{2}),tr(\alpha t_{3}))$ attains each of the triples $(0,0,0),(1,1,0),(1,0,1),(0,1,1)$ exactly $\frac{q}{4}$ times, in which case

[TABLE]

Thus

[TABLE]

where $n_{k}$ is given by Lemma 3.3(ii). Simplification yields the formula claimed for $\phi(x)$. Now we simply read off the multiplicity of the largest eigenvalue to obtain the number of connected components of $\Gamma$ (see e.g. [3, Prop.1.3.8]).\qed

6 The case $q$ is odd

Here and for the remainder of this paper, we take $q$ to be odd. From the general formula for commutators in $G$ given at the end of Section 4, we deduce the commutator subgroup and centre

[TABLE]

also the centralizer of a noncentral element (i.e. with $(t,u)\neq(0,0)$) is a subgroup

[TABLE]

of order $q^{3}$. So $G$ has $q^{3}+q^{2}-q$ conjugacy classes ($q^{2}$ of size 1, and $q^{3}-q$ of size $q$). There are $|G/G^{\prime}|=q^{3}$ linear characters of $G$, given by

[TABLE]

where $\alpha,\beta,\gamma\in F$. As in Section 3, $\zeta=\zeta_{p}$ is a complex $p$-th root of unity and $tr:F\to{\mathbb{F}}_{p}$ is the trace map. The remaining irreducible characters of $G$ may be found by inducing linear characters of a subgroup of order $q^{3}$ (thus yielding monomial representations of degree $q$); but guided by a little hindsight, we will instead directly exhibit the missing representations and show that they are irreducible and distinct. For each pair $\alpha,\beta\in F$ with $\alpha\neq 0$, we define $M_{\alpha,\beta}:G\to GL_{q}({\mathbb{C}})$ by

[TABLE]

using the Kronecker delta notation $\delta_{i,j}=0$ or $1$ according as $i,j\in F$ either differ or coincide. It is routine to check that $M_{\alpha,\beta}(g)M_{\alpha,\beta}(g^{\prime})=M_{\alpha,\beta}(gg^{\prime})$ for all $g,g^{\prime}\in G$, and $M_{\alpha,\beta}(g(0,0,0,0))=I_{q}$; so $M_{\alpha,\beta}$ is a representation of degree $q$. The associated character is found to be

[TABLE]

using the fact that $\alpha\neq 0$. These $q^{2}-q$ characters of $G$ are irreducible and inequivalent since

[TABLE]

These are also distinct from the characters $\chi_{\alpha,\beta,\gamma}$ and so we have the complete list of $q^{3}+q^{2}-q$ irreducible characters of $G$.

Now by Theorem 2.4, the adjacency matrix $A$ of $\Gamma$ has characteristic polynomial

[TABLE]

Those eigenvalues of $A$ obtained from the linear characters of $G$ are

[TABLE]

where $m_{\alpha,\beta,\gamma}$ is the number of values $r\in F$ such that $\alpha+\beta r+\gamma r^{2}=0$. By Lemma 3.3(ii), the first $q^{3}$ factors of $\phi(x)$ are

[TABLE]

thus the characteristic polynomial $\phi(x)=\det(xI_{q^{4}}-A)$ has the form

[TABLE]

Now for $\alpha\neq 0$,

[TABLE]

where ‘$*$’ denotes conjugate-transpose, and we have introduced the $q\times q$ complex matrices

[TABLE]

We first treat the cases $\beta=0\neq\alpha$ for which we obtain

[TABLE]

Denoting by $\{e_{r}\}_{r\in F}$ the standard basis of ${\mathbb{C}}^{F}={\mathbb{C}}^{q}$, we find a new basis consisting of eigenvectors of $M_{\alpha,0}(S)$ as follows:

$\frac{1}{2}(q-1)$ eigenvectors of the form $e_{r}+e_{-r}$ where $0\neq r\in F$, each with eigenvalue $q$;

• 2.

$\frac{1}{2}(q-1)$ eigenvectors of the form $e_{r}-e_{-r}$ as $r$ ranges over a set of representatives of the distinct nonzero pairs $\{r,-r\}$ in $F$. Each such vector has eigenvalue $-q$;

• 3.

$M_{\alpha,0}e_{0}=0$.

After including the factors

[TABLE]

we update our formula for the characteristic polynomial of $A$ as

[TABLE]

Finally we describe the remaining $q^{2}(q-1)^{2}$ eigenvalues of $A$ arising from $M_{\alpha,\beta}(S)$ for $\alpha\beta\neq 0$.

Lemma 6.1

For any nonzero elements $c,d\in F$ the matrix $M_{\alpha,\beta}(S)$ is similar to $M_{c^{2}d\alpha,cd^{2}\beta}(S).$

Proof 5

Re-indexing rows and columns of $U_{\alpha,\beta}$ via $(i,j)\mapsto(ci,dj)$, we see that

[TABLE]

where $P_{c}$ and $P_{d}$ are $q\times q$ permutation matrices, and so $M_{c^{2}d\alpha,cd^{2}\beta}=P_{c}M_{\alpha,\beta}P_{c}^{T}$.\qed

Corollary 6.2

If $q\not\equiv 1\mod 3$ then for all nonzero $\alpha,\beta\in F$, $M_{\alpha,\beta}(S)$ is unitarily similar to $M_{1,1}(S)$. If $q\equiv 1\mod 3$ then there are at most three similarity classes of matrices $M_{\alpha,\beta}(S)$ with $\alpha\beta\neq 0$, represented by $M_{1,1}(S)$, $M_{1,\omega}(S)$ and $M_{1,\omega^{2}}(S)$ where $\omega\in F$ is a primitive root.

Proof 6

If $q\not\equiv 1\mod 3$, then every element of $F$ has a cube root in $F$; so let $c\in F$ be any cube root of $\beta/\alpha^{2}$ and take $d=\alpha c/\beta$. Then $M_{\alpha,\beta}(S)$ is similar to $M_{1,1}(S)$ by Lemma 6.1. The second conclusion follows similarly.\qed

If $q\equiv 2\mod 3$ and $\alpha\beta\neq 0$, then $M_{\alpha,\beta}(S)$ is similar to $M_{3,3}$, since by Corollary 6.2, both matrices are similar to $M_{1,1}$. In this case U_{3,3}=\bigl{[}\zeta^{tr(3i^{2}j-3ij^{2})}\bigr{]}_{i,j\in F} is unitarily similar to

[TABLE]

where $D$ is a diagonal matrix with diagonal entries $\zeta^{tr(i^{3})}$ for $i\in F$. In this case the $q$ vectors v_{c}=\bigl{(}\zeta^{tr(ci)}\bigr{)}_{i\in F} for $c\in F$ form a basis of ${\mathbb{C}}^{F}$ consisting of eigenvectors of $\tilde{U}$; indeed

[TABLE]

so that $\tilde{U}v_{c}=\varepsilon_{f}v_{c}$ where

[TABLE]

Note that $\varepsilon_{f}\in{\mathbb{Z}}[\zeta]$ satisfies $\overline{\varepsilon_{f}}=\varepsilon_{f}$ since $f(-x)=-f(x)$; thus \varepsilon_{f}\in{\mathbb{Z}}[\zeta+\overline{\zeta}]={\mathbb{Z}}\bigl{[}2\cos\frac{2\pi}{p}\bigr{]} (see [23, Prop.2.16]). Also $M_{3,3}=D^{*}\tilde{U}\tilde{U}^{*}D-qI_{q}$ has eigenvalues $\varepsilon_{f}^{2}-q$. The Weil bound (see e.g. [24], [14, p.223]) gives $|\varepsilon_{f}|\leqslant 2\sqrt{q}$ as required. The all-ones eigenvector $v_{0}$ has eigenvalue $\sum_{j\in F}\zeta^{tr(j^{3})}=0$ since $f(x)=x^{3}$ is a permutation of $F$; so for $q\equiv 2\mod 3$ we obtain

[TABLE]

When $q\equiv 1\mod 3$ we work just a little harder. Let $a=\frac{1}{3\alpha\beta}$ and use the identity

[TABLE]

to see that $U_{\alpha,\beta}=N\tilde{U}N^{\prime}$ where

[TABLE]

and the unitary matrices $N$ and $N^{\prime}$ are given by

[TABLE]

Now the vectors $v_{c}$ (as above) are eigenvectors of $\tilde{U}$ since

[TABLE]

with corresponding eigenvalue $\varepsilon_{f}$ where $f(t)=at^{3}+ct\in F[t]$. Now $M_{\alpha,\beta}=N\tilde{U}\tilde{U}^{*}N^{*}-qI_{q}$ has eigenvalues $\varepsilon_{f}^{2}-q$, and

[TABLE]

since as $\alpha$ and $\beta$ range over the nonzero elements of $F$, the coefficient $a=\frac{1}{3\alpha\beta}$ falls in each of the three multiplicative cosets of the cubes equally often, and we recall Corollary 6.2. So for $q\equiv 1\mod 3$ we obtain

[TABLE]

As before, \varepsilon_{f}\in{\mathbb{Z}}\bigl{[}2\cos\frac{2\pi}{p}\bigr{]} and $|\varepsilon_{f}|\leqslant 2\sqrt{q}$.

Finally, suppose $q=3^{e}$ so that $p=3$. By the Lemma, $U_{\alpha,\beta}$ is unitarily similar to $U_{1,1}$. Unlike the cases $q\not\equiv 0\mod 3$, in this case the eigenvalues of $U_{1,1}$ do not lie in ${\mathbb{Q}}[\zeta]$; rather they lie in ${\mathbb{Q}}[\xi]$ where we abbreivate $\xi=\zeta_{9}=e^{2\pi i/9}$, chosen so that $\xi^{3}=\zeta=\zeta_{3}$. This can be seen even in the case $q=3$ where

[TABLE]

whose eigenvalues are

[TABLE]

Likewise, the eigenvalues of $M_{1,1}$ lie in ${\mathbb{Q}}[\xi]$ but not in ${\mathbb{Q}}[\zeta]$. However we see that the eigenvalues of $U_{1,1}$ are expressible as exponential sums defined over Galois rings; see e.g. [8, 18]. Let $R=GR(9,e)$ be the Galois ring of order $9^{e}=q^{2}$ and characteristic 9. The ring $R$ enjoys the following properties:

$R$ is a commutative ring with a maximal ideal $3R$ consisting of all zero divisors in $R$, and the quotient ring is $R/3R\cong F={\mathbb{F}}_{q}$.

• 2.

The units of $R$ form a multiplicative group $R^{\times}$ consisting of all elements not in $3R$. This group has a multiplicative subgroup $\{1,\beta,\beta^{2},\ldots,\beta^{q-2}\}$ of order $q-1$.

• 3.

Every element $x\in R$ has a unique 3-adic expansion $x=x_{0}+3x_{1}$ where $x_{0},x_{1}\in{\mathcal{T}}$. where we define ${\mathcal{T}}=\{0,1,\beta,\beta^{2},\ldots,\beta^{q-2}\}$. In particular, ${\mathcal{T}}$ is a set of representatives of the cosets $R/3R\cong{\mathbb{F}}_{q}$.

• 4.

The trace map $tr:R\to{\mathbb{Z}}/9{\mathbb{Z}}$ is defined by

[TABLE]

where $x_{0},x_{1}\in{\mathcal{T}}$. After reducing both domain and range modulo $3R$, this gives the usual absolute trace map ${\mathbb{F}}_{q}\to{\mathbb{F}}_{3}$.

After replacing $F={\mathbb{F}}_{q}$ by ${\mathcal{T}}$ as index set for entries of our vectors and matrices, we may rewrite our basis of ${\mathbb{C}}^{q}$ as v_{c}=\bigl{(}\xi^{3tr(ci)}\bigr{)}_{i\in{\mathcal{T}}} for $c\in{\mathcal{T}}$, and

[TABLE]

which is unitarily similar to

[TABLE]

after conjugating by the unitary diagonal matrix D=\bigl{[}\xi^{tr(i^{3})}\delta_{i,j}\bigr{]}_{i,j\in{\mathcal{T}}}\,. Now

[TABLE]

so that $\tilde{U}v_{c}=\varepsilon_{f}v_{c}$ where

[TABLE]

The ‘weighted degree’ of $f(t)$, as defined in [8, p.458], is $d=3$; and as shown in [8], the Weil bound $|\varepsilon_{f}|\leqslant(d-1)\sqrt{q}=2\sqrt{q}$ holds. Once again by [23, Prop.2.16] we have \varepsilon_{f}\in{\mathbb{Z}}\bigl{[}2\cos\frac{2\pi}{9}\bigr{]}. With the notation above, we have

[TABLE]

7 Exact Spectra over Prime Fields

It is possible to refine Theorem 2.2 to express $\phi(x)$ precisely. Here, however, we state such a result (Theorem 7.6 below) only for the case $q=p$ is prime; in the general case $q=p^{e}$ with $e\geqslant 2$, counting multiplicities is more technical and Theorem 2.2 is probably adequate for any intended applications. In the following we denote $f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(r)=\{a\in F:f(a)=r\}$.

Lemma 7.1

Let $f,g:F\to F$ be two functions where the field $F={\mathbb{F}}_{p}$ has prime order $p$. Then $\varepsilon_{f}=\varepsilon_{g}$ iff $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(r)|=|g{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(r)|$ for all $r\in F$.

Proof 7

If $\varepsilon_{f}=\varepsilon_{g}$ then

[TABLE]

Since the minimal polynomial of $\zeta$ over ${\mathbb{Q}}$ is the cyclotomic polynomial $1+x+x^{2}+\cdots+x^{p-1}\in{\mathbb{Z}}[x]$, there exists $n\in{\mathbb{Z}}$ such that $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(r)|-|g{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(r)|=n$ for all $r\in F$. Since $\sum_{r\in F}|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(r)|=\sum_{r\in F}|g{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(r)|=p$, we must have $n=0$. The converse is clear.\qed

Lemma 7.2

Let $f(t)\in F[t]$ and $\tilde{f}(t)=f(\lambda t)\in F[t]$ where $0\neq\lambda\in F$. Then $\varepsilon_{\tilde{f}}=\varepsilon_{f}$.

Proof 8

Straightforward.\qed

Let $C$ be the set of $p(p-1)$ cubic polynomials of the form $f(t)=at^{3}+ct\in F[t]$. By Lemma 7.2, the corresponding exponential sums $\varepsilon_{f}$ for $f\in C$ are not all distinct. We next find a set of representatives $\tilde{C}\subset C$ giving rise to distinct exponential sums; that is, for each $f\in C$ there is a unique $g\in\tilde{C}$ such that $\varepsilon_{f}=\varepsilon_{g}$.

Lemma 7.3

Let $f(t)=at^{3}+ct\in F[t]$ where the field $F={\mathbb{F}}_{p}$ has prime order $p\geqslant 5$, and $a\neq 0$.

• (i)

For $p\equiv 2\mod 3$, we have $\varepsilon_{f}=0$ iff $c=0$. We may take $\tilde{C}=\{t^{3}+\tilde{c}t:\tilde{c}\in F\}$; and $\varepsilon_{f}=\varepsilon_{t^{3}+\tilde{c}t}$ iff $\tilde{c}=a^{-1/3}c$. Here $|\tilde{C}|=p$.

• (ii)

For $p\equiv 1\mod 3$, we have $\varepsilon_{f}\neq 0$. Let $\omega\in F$ be a primitive element, i.e. a generator of the multiplicative group $F^{\times}$. We may take $\tilde{C}=\{t^{3},\omega t^{3},\omega^{2}t^{3}\}\cup\{\tilde{a}t^{3}+t:0\neq\tilde{a}\in F\}$. Here $|\tilde{C}|=p+2$. If $c\neq 0$ then $\varepsilon_{f}=\varepsilon_{\tilde{a}t^{3}+1}$ where $\tilde{a}=a/c^{3}$. If $c=0$ then $\varepsilon_{f}=\varepsilon_{\omega^{i}t^{3}}$ where $i\in\{0,1,2\}$ is uniquely determined by $(a/\omega^{i})^{(p-1)/3}=1$.

Proof 9

(i) First suppose $p\equiv 2\mod 3$, so that $p=3m-1$ where $m\geqslant 2$. In this case every element $a\in F$ has a unique cube root $a^{1/3}=a^{2m-1}$. By Lemma 3.2,

[TABLE]

If $\varepsilon_{f}=\varepsilon_{g}$ where $g(t)=t^{3}+\tilde{c}t$ then $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=|g{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|$ for all $s\in F$ by Lemma 7.1, so $-ma^{m-1}c=-m\tilde{c}$ and $\tilde{c}=a^{m-1}c=a^{-1/3}c$; but conversely, if $\tilde{c}=a^{-1/3}c$ then $f(t)=g(a^{1/3}t)$ so by Lemma 7.2, $\varepsilon_{f}=\varepsilon_{g}$.

If $c=0$ then $f(t)=at^{3}$ defines a permutation of $F$, so $\varepsilon_{f}=0$.

(ii) Now suppose $p\equiv 1\mod 3$, and write $p=3m+1$ where $m\geqslant 2$. Again by Lemma 3.2,

[TABLE]

and

[TABLE]

If $\varepsilon_{f}=\varepsilon_{g}$ where $g(t)=\tilde{a}t^{3}+\tilde{c}t$, then as in (i), it follows that $\tilde{a}^{m}=a^{m}$, $\tilde{a}^{m-1}\tilde{c}^{3}=a^{m-1}c^{3}$ and $\tilde{a}c^{3}=a\tilde{c}^{3}$. We consider two cases:

Suppose $c=0$; then $\tilde{c}=0$. We may write $a=\omega^{3d+i}$ where $d\in{\mathbb{Z}}$ and $i\in\{0,1,2\}$, so that $(\omega^{i})^{m}=a^{m}$. Now if $g\in\tilde{C}$ satisfies $\varepsilon_{g}=\varepsilon_{f}$, we must have $g(t)=\omega^{i}t^{3}$. Conversely, $\omega^{i}t^{3}=f(\omega^{-d}t)$; so $g(t)=\omega^{i}t^{3}$ is the unique $g\in\tilde{C}$ satisfying $\varepsilon_{g}=\varepsilon_{f}$.

• 2.

Suppose $c\neq 0$; then $\tilde{c}\neq 0$. If $g\in\tilde{C}$ satisfies $\varepsilon_{g}=\varepsilon_{f}$, we must have $g(t)=\tilde{a}t^{3}+t$ where $\tilde{a}=c^{-3}a$. Conversely, $c^{-3}at^{3}+t=f(c^{-1}t)$; so $g(t)=c^{-3}at^{3}+t$ is the unique $g\in\tilde{C}$ satisfying $\varepsilon_{g}=\varepsilon_{f}$.\qed

In order to determine the exact spectrum of $\Gamma(4,p)$, we need to know not only when the values $\varepsilon_{f}$ are distinct, but actually when the values $\varepsilon_{f}^{2}$ are distinct. As preparation, we need the following.

Lemma 7.4

Let $F={\mathbb{F}}_{p}$ where $p\geqslant 5$ is prime, and suppose there exists a polynomial of the form $f(x)=x^{3}+cx\in F[t]$ having $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|\leqslant 2$ for all $s\in F$. Then one of the following holds:

• (i)

$p\equiv 2\mod 3$, and $c=0$;

• (ii)

$p=5$* and $c\in\{2,3\}$; or*

• (iii)

$p=7$* and $c\in\{1,2,4\}$.*

Proof 10

We will assume $p\geqslant 11$ since the cases $p\in\{5,7\}$ may be easily checked by explicit computation. First observe that $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=2$ for at most two values of $s\in F$; this is because for any such value of $s$, $f(t)-s=t^{3}+ct-s$ and $f^{\prime}(t)=3t^{2}+c$ have a linear factor in common, forcing $s^{2}=-4c^{3}/27$.

For each $k\in\{0,1,2,3\}$, let $n_{k}=|\{s\in F\,:\,|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=k\}|$. By hypothesis, $n_{3}=0$; and we have just shown that $n_{2}\leqslant 2$. Elementary counting arguments give $n_{0}+n_{1}+n_{2}=p=n_{1}+2n_{2}$. Exactly three possibilities must be considered.

Case (i): $(n_{0},n_{1},n_{2})=(0,p,0)$. In this case $f$ is a permutation polynomial; but then $\varepsilon_{f}=0$ and Lemma 7.3 gives $p\equiv 2\mod 3$, and $f(t)=t^{3}$.

Case (ii): $(n_{0},n_{1},n_{2})=(1,p{-}2,1)$. Here $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(a)|=2$ and $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(b)|=0$ for some $a,b\in F$, and $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=1$ for all other values $s\in F$. Since $f$ has degree $3<p-1$, $0=\sum_{t\in F}f(t)=a-b$, contradicting Lemma 3.2.

Case (iii): $(n_{0},n_{1},n_{2})=(2,p{-}4,2)$. Here there exist distinct values $a_{0},a_{1}$, $a_{2},a_{3}\in F$ such that

[TABLE]

Since $\deg(f(t)^{d})=3d<p-1$ for $d\in\{0,1,2,3\}$, Lemma 3.2 gives

[TABLE]

This gives a nontrivial linear dependence between four columns of the nonsingular Vandermonde matrix \bigl{[}a_{i}^{j}\,:\,0\leqslant i,j\leqslant 3\bigr{]}, a contradiction.\qed

Corollary 7.5

Let $F={\mathbb{F}}_{p}$ be a field of prime order $p\geqslant 5$, and suppose $\varepsilon_{g}=-\varepsilon_{f}$ where $f(t)=t^{3}+ct\in F[t]$ and $g(t)=t^{3}+\tilde{c}t\in F[t]$. If $\tilde{c}\neq c$ then we must have $p=5$ and $\{c,\tilde{c}\}=\{2,3\}$.

Proof 11

If $\varepsilon_{f}=-\varepsilon_{g}$ then

[TABLE]

and arguing as in the proof of Lemma 7.1, we must have $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|+|g{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=2$ for all $s\in F$. In particular, $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|\leqslant 2$ for all $s\in F$. By Lemma 7.4, $p\in\{5,7\}$. For $p=7$ the only cubics of the form $f(t)=t^{3}+ct$ satisfying $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|\leqslant 2$ for all $s\in F$, have $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=1,0,2,1,1,2,0$ for $s=0,1,2,3,4,5,6$ respectively; and no pair of such cubics can satisfy $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|+|g{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=2$ for all $s\in F$.

This leaves only the case $p=5$ and the pair of cubics $f(t)=t^{3}+2t$, $g(t)=t^{3}+3t$ where $|f{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=1,0,2,2,0$ and $|g{}^{\scriptscriptstyle-1}{\mskip-2.0mu}(s)|=1,2,0,0,2$ for $s=0,1,2,3,4$ respectively.\qed

Theorem 7.6

Let $p$ be an odd prime, and let $A$ be the adjacency matrix of $\Gamma(4,p)$, with characteristic polynomial $\phi(x)=\det(xI_{p^{4}}-A)$.

• (i)

For $p=3$, we have $\phi(x)=x^{18}(x-6)(x-3)^{12}(x+3)^{14}(x^{3}-9x-9)^{12}$.

• (ii)

For $p=5$, we have $\phi(x)=x^{220}(x-20)(x-5)^{80}(x+5)^{164}(x^{2}-5x-25)^{80}$.

• (iii)

For $5<p\equiv 2\mod 3$, we have

[TABLE]

with $p+3$ distinct roots and multiplicities as indicated by the exponents.

• (iv)

For $p\equiv 1\mod 3$, we have

[TABLE]

with $p+6$ distinct roots and multiplicities as indicated by the exponents.

Proof 12

For $p=3$ we take $R={\mathbb{Z}}/9{\mathbb{Z}}$, $\xi=e^{2\pi i/9}$ and ${\mathcal{T}}=\{0,1,8\}$ in the notation of Section 4, and

[TABLE]

We compute

[TABLE]

and

[TABLE]

so (i) follows.

Conclusion (iii) follows immediately from the previous results; and for $p=5$, the same reasoning yields

[TABLE]

where we need only to check for coincidence of roots. Straightforward computations show that $\varepsilon_{t^{3}+t}=3+\zeta^{2}+\zeta^{3}=\frac{5-\sqrt{5}}{2}$ and $\varepsilon_{t^{3}+4t}=2-\zeta^{2}-\zeta^{3}=\frac{5-\sqrt{5}}{2}$, whence

[TABLE]

similarly, $\varepsilon_{t^{3}\pm 2t}=\pm(1+2\zeta^{2}+2\zeta^{3})=\mp\sqrt{5}$, giving

[TABLE]

This yields (ii), and conclusion (iv) follows similarly from the previous results.\qed

As an example, the spectrum of $D(4,13)$ contains $\pm\varepsilon$ where $\varepsilon=\varepsilon_{4t^{3}}=1+6\cos\frac{8\pi}{13}+6\cos\frac{12\pi}{13}\approx-6.9533$. Compare $|\varepsilon|$ with $2\sqrt{12}\approx 6.9282$ and $2\sqrt{13}\approx 7.2111$ to see that while $D(4,13)$ is an expander, it is not quite Ramanujan. Similar conclusions are found for other values including $p=19,37$.

Acknowledgements

This research was supported in part by NSF grant DMS-1400281. The second author is grateful to her doctoral supervisor Dr. Felix Lazebnik for providing direction in this research.