Spectral and Combinatorial Properties of Some Algebraically Defined Graphs

Let $k\ge 3$ be an integer, $q$ be a prime power, and $\mathbb{F}_q$ denote the field of $q$ elements. Let $f_i, g_i\in\mathbb{F}_q[X]$, $3\le i\le k$, such that $g_i(-X) = -\, g_i(X)$. We define a graph $S(k,q) = S(k,q;f_3,g_3,\cdots,f_k,g_k)$ as a graph with the vertex set $\mathbb{F}_q^k$ and edges defined as follows: vertices $a = (a_1,a_2,\ldots,a_k)$ and $b = (b_1,b_2,\ldots,b_k)$ are adjacent if $a_1\ne b_1$ and the following $k-2$ relations on their components hold: $$ b_i-a_i = g_i(b_1-a_1)f_i\Bigl(\frac{b_2-a_2}{b_1-a_1}\Bigr)\;,\quad 3\le i\le k. $$ We show that graphs $S(k,q)$ generalize several recently studied examples of regular expanders and can provide many new such examples.


Introduction and Motivation
All graphs in this paper are simple, i.e., undirected, with no loops and no multiple edges. See, e.g., Bollobás [4] for standard terminology. Let Γ = (V, E) be a graph with vertex set V and edge set E. For a subset of vertices A of V , ∂A denotes the set of edges of Γ with one endpoint in A and the other endpoint in V \ A.
The Cheeger constant h(Γ) (also known as edge-isoperimetric number or expansion ratio) of Γ, is defined by h(Γ) := min |∂A| |A| : A ⊆ V, 0 < |A| ≤ 1 2 |V | . The graph Γ is d-regular if each vertex is adjacent to exactly d others. An infinite family of expanders is an infinite family of regular graphs whose Cheeger constants are uniformly bounded away from 0. More precisely, for n ≥ 1, let Γ n = (V n , E n ) be a sequence of graphs such that each Γ n is d n -regular and |V n | → ∞ as n → ∞. We say that the members of the sequence form a family of expanders if the corresponding sequence h(Γ n ) is bounded away from zero, i.e. there exists a real number c > 0 such that h(Γ n ) ≥ c for all n ≥ 1. In general, one would like the valency sequence (d n ) n≥1 to be growing slowly with n, and ideally, to be bounded above by a constant.
The adjacency matrix A = A(Γ) of a graph Γ = (V, E) has its rows and columns labeled by V and A(x, y) equals the number of edges between x and y (i.e. 0 or 1). When Γ is simple, the matrix A is symmetric and therefore, its eigenvalues are real numbers. For j between 1 and the order of Γ, let λ j = λ j (G) denote the j-th eigenvalue of A. For an arbitrary graph Γ, it is hard to find or estimate h(Γ), and often it is done by using the second-largest eigenvalue λ 2 (Γ) of the adjacency matrix of Γ. If Γ is a connected d-regular graph, then 1 2 d − λ 2 ≤ h(Γ) ≤ d 2 − λ 2 2 . The lower bound was proved by Dodziuk [9] and independently by Alon-Milman [1] and by Alon [2]. In both [1] and [2], the upper bound on h(Γ), namely 2d(d − λ 2 ) was provided. Mohar in [20] improved the upper bound to the one above. See [5,10,12], for terminology and results on spectral graph theory and connections between eigenvalues and expansion properties of graphs. The difference d − λ 2 which is present is both sides of this inequality above, also known as the spectral gap of Γ, provides an estimate on the expansion ratio of the graph. In particular, for an infinite family of d-regular graphs Γ n , the sequence h(Γ n ) is bounded away from zero if and only if the sequence d − λ 2 (Γ n ) is bounded away from zero. A d-regular Alon and Boppana [22] proved that this bound is asymptotically best possible for any infinite family of d-regular graphs and their results imply that for any infinite family of d-regular For the rest of the paper, let q = p e , where p is a prime and e is a positive integer.
For a sequence of prime powers (q m ) m≥1 , we always assume that q m = p em m , where p m is a prime and e m ≥ 1. Let F q be the finite field of q elements and F k q be the cartesian product of k copies of F q . Clearly, F k q is a vector space of dimension k over F q . For 2 ≤ i ≤ k, let h i be an arbitrary polynomial in 2i − 2 indeterminants over F q . We define the bipartite graph BΓ k = BΓ(q; h 2 , . . . , h k ), k ≥ 2, as follows. The vertex set of BΓ k is the disjoint union of two copies of F k q , one denoted by P k and the other by L k . We define edges of BΓ k by declaring vertices p = (p 1 , p 2 , . . . , p k ) ∈ P k and l = (l 1 , l 2 , . . . , l k ) ∈ L k to be adjacent if the following k − 1 relations on their coordinates hold: The graphs BΓ k were introduced by Lazebnik and Woldar [15], as generalizations of graphs introduced by Lazebnik and Ustimenko in [14] and [16]. For surveys on these graphs and their applications, see [15] and Lazebnik, Sun and Wang [13]. An important basic property of graphs BΓ k (see [15]) is that for every vertex v of BΓ k and every α ∈ F q , there exists a unique neighbor of v whose first coordinate is α.
This implies that each BΓ k is q-regular, has 2q k vertices and q k+1 edges.
The spectral and combinatorial properties of three specializations of graphs BΓ k has received particular attention in recent years. Cioabă, Lazebnik and Li [7] determined the complete spectrum of the Wenger graphs W k (q) = BΓ(q; h 2 , . . . , h k+1 ) with Cao, Lu, Wan, Wang and Wang [6] determined the eigenvalues of the linearized Wenger graphs L k (q) = BΓ(q; h 2 , . . . , h k+1 ) with h i = p p i−2 1 l 1 , 2 ≤ i ≤ k + 1, and Yan and Liu [24] determined the multiplicities of the eigenvalues the linearized Wenger graphs. Moorhouse, Sun and Williford [21] studied the spectra of graphs D(4, q) = BΓ(q; p 1 l 1 , p 1 l 2 , p 2 l 1 ), and in particular, proved that the second largest eigenvalues of these graphs are bounded from above by 2 √ q (so D(4, q) is 'close' to being Ramanujan).
Let V 1 and V 2 denote the partite sets or color classes of the vertex set of a bipartite graph Γ. The distance-two graph of Γ on V 1 is the graph having V 1 as its vertex set with the adjacency defined as follows: two vertices x = y ∈ V 1 are adjacent if there exists a vertex z ∈ V 2 adjacent to both x and to y in Γ (which is equivalent of saying that x and y are at distance two in Γ). If Γ is d-regular and contains no 4-cycles, There is simple connection between the eigenvalues of Γ and the eigenvalues of Γ (2) (see, e.g., [7]): every eigenvalue λ of Γ (2) with multiplicity m, corresponds to a pair of eigenvalues ± √ λ + d of Γ, each with multiplicity m (or a single eigenvalue 0 of multiplicity 2m in case λ = −d).
This relation between the spectra of q-regular bipartite graph Γ and its q(q − 1)regular distance-two graph Γ (2) has been utilized in each of the papers [7,6,21] in order to find or to bound the second-largest eigenvalue of Γ, and then use this information to claim the expansion property of Γ. In each of these cases, Γ (2) turned out to be a Cayley graph of a group, that allowed to use representation theory to compute its spectrum. In [7,6] the group turned out be abelian, as in [21] it was not for odd q.
The main motivation behind the construction below is to directly generalize the defining systems of equations for W (2) k (q) and of L (2) k (q), thereby obtaining a family of q(q − 1)-regular Cayley graphs of an abelian group. The adverb directly used in the previous sentence was to stress that the graphs we build are not necessarily distance-two graphs of q-regular bipartite graphs Γ. Examples when they are not will be discussed in Remark 1 of Section 7.

Main Results
In this section, we define the main object of this paper, the family of graphs S(k, q) and we describe our main results. Let k be an integer, k ≥ 3. Let f i , g i ∈ F q [X], 3 ≤ i ≤ k, be 2(k − 2) polynomials of degrees at most q − 1 such that g i (−X) = − g i (X) for each i. We define S(k, q) = S(k, q; f 3 , g 3 , · · · , f k , g k ) as the graph with the vertex set F k q and edges defined as follows: a = (a 1 , a 2 , . . . , a k ) is adjacent to Clearly, the requirement g i (−X) = − g i (X) is used for the definition of the adjacency in S(k, q) to be symmetric. One can easily see that S(k, q) is a Cayley graph with the underlying group G being the additive group of the vector space F k q with generating set In order to present our results, we need a few more notation. For any α ∈ F q , let T r(α) = α + α p + · · · + α p e−1 be the trace of α over F p . It is known that T r(α) ∈ F p .
For any element β ∈ F p , let β * denote the unique integer such that 0 ≤ β * < p and the residue class of β * in F p is β. For any complex number c, the expression c β will mean c β * . Let ζ p = exp( 2π p i) be a complex p-th root of unity. For every f ∈ F q [X], We are ready to state the main results of this paper.
The following theorem describes the spectrum of the graphs S(k, q).
For a fixed k ≥ 3, the theorem below provides sufficient conditions for the graphs S(k, q) to form a family of expanders.
Theorem 2.2. Let k ≥ 3, (q m ) m≥1 be an increasing sequence of prime powers, and , and for all m ≥ 1, at least one of the following two conditions is satisfied: 2. The polynomials f 3,m , . . . , f k,m are F q -linearly independent, and there exists The following two theorems demonstrate that for some specializations of S(k, q), we can obtain stronger upper bounds on their second largest eigenvalues.
For large k, specifically, when Similarly to Theorem 2.3, when choosing f i (X) = X p i−2 , the same f functions as in k (q), we obtain the following upper bounds for the second largest eigenvalue.
Theorem 2.4. Let q be an odd prime power with q ≡ 2 mod 3, and 3 ≤ k ≤ e + 2.
For large k, specifically, when The paper is organized as follows. In Section 3, we present necessary definitions and results concerning finite fields used in the proofs. In Section 4, we prove Theorem 2.1.
In Section 5, we study some sufficient conditions on f i and g i for the graph S(k, q) to be connected and have large eigenvalue gap, and prove Theorem 2.2. In Section 6, we prove Theorem 2.3 and Theorem 2.4. We conclude the paper with several remarks in Section 7.

Background on finite fields
For definitions and theory of finite fields, see Lidl and Niederreiter [18].
is a polynomial of degree one or less, then For a general f ∈ F q [X], no explicit expression for the exponential sum ε f exists.
The following theorem provides a good upper bound for the exponential sum ε f .
and g(−X) = − g(X). Then ε g is a real number.
Proof. We have that Since ζ β p + ζ −β p ∈ R for any β ∈ F p , it follows that ε g ∈ R.
4 Spectra of the graphs S(k, q) The proof we present here is based on the same idea as the one in [7]. Namely, computing eigenvalues of Cayley graphs by using the method suggested in Babai [3].
The original completely different (and much longer) proof of Theorem 2.1 that used circulants appears in Sun [23]. Proof of Theorem 2.1. As we mentioned in Section 2, S(k, q) is a Cayley graph with the underlying group G being the additive group of the vector space F k q , and connection set Since G is an abelian group, it follows that the irreducible C-representations of G are linear (see [11], Ch. 2). They are given by where w = (w 1 , · · · , w k ) ∈ F k q and v = (v 1 , · · · , v k ) ∈ F k q .
Using Theorem 4.1, we conclude that the spectrum of S(k, q) is a multiset formed by all λ w , w = (w 1 , · · · , w k ) ∈ F k q , of the form:

Connectivity and expansion of the graphs S(k, q)
It is hard to get a closed form of λ w in (3) for arbitrary f i and g i . But if the degrees of the polynomials f i and g i satisfy some conditions, we are able to show that the components of the graphs S(k, q) have large eigenvalue gap. For these f i and g i , we find sufficient conditions such that the graphs S(k, q) are connected, and hence form a family of expanders.
From now on, for any graph S(k, q; f 3 , g 3 , · · · , f k , g k ), we let d g = max 3≤i≤k deg(g i ) and We also assume that d f ≥ 1 and d g ≥ 1. For each i, 3 ≤ i ≤ k, let c i,j be the coefficient of X j in the polynomial g i , for any j, 1 ≤ j ≤ d g , i.e.
For any w = (w 1 , · · · , w k ) in F k q , let N w be the number of u's in F q satisfying the following system and let S w be the set of all u's in F q such that the following inequality holds for If d g = 1, then system (4) contains only the first equation, and S w = ∅.
Lemma 5.1. Let k ≥ 3. If 1 ≤ d g < p, then for any w = (w 1 , · · · , w k ) in F k q , the eigenvalue λ w of S(k, q) in (3) is at most Moreover, λ w = q(q − 1) if and only if N w = q.
Proof. Let w = (w 1 , . . . , w k ) ∈ F k q . Using Theorem 2.1, we have If u satisfies (4), then z u = q − 1. If u ∈ S w , then z u is an exponential sum of a polynomial of degree at least 2 and at most d g . By the assumption of the theorem that d g < p and Weil's bound in Theorem 3.2, it follows that Finally, for the remaining q − N w − |S w | elements u ∈ F q , we have If d g = 1, then system (7) contains only the first inequality. In both cases, we have z u = −1. Therefore, we have Let us now prove the second statement of the lemma. It is clear that if N w = q, then |S w | = 0 and λ w = q(q − 1). For the rest of this proof, we assume that N w < q, and show that λ w < q(q − 1).
For e = 1, we consider the following two cases: q = p = 2 and q = p ≥ 3.
If q = p ≥ 3, then, as λ w is a real number and |z u | ≤ p − 1, we have and λ w = p(p − 1) if and only if z u = p − 1 for all u ∈ F p . The latter condition is equivalent to for any a ∈ F * p . Therefore, the polynomial which is over F p , has p distinct roots in F p and is of degree at most d g , d g < p.
Hence, it must be zero polynomial, and so N p = p, a contradiction. Hence, λ w < p(p − 1).
Let (q m ) m≥1 be an increasing sequence of prime powers. For a fixed k, k ≥ 3, we consider an infinite family of graphs S(k, q m ; f 3,m , g 3,m , · · · , f k,m , g k,m ). Hence, Proof. For any w ∈ F k qm , the eigenvalue λ w of S(k, q m ) is at most by Lemma 5.1.
It is clear that for any w ∈ F k qm , system (4) has either N w = q m solutions or at most d (m) f solutions with respect to u. If N w = q m , then λ w = q m (q m − 1) by Lemma 5.1.
f . Therefore, we have As an immediate corollary from Theorem 5.2, we have the following theorem. Our next theorem provides a sufficient condition for the graph S(k, q) to be connected.
Theorem 5.4. For k ≥ 3, let S(k, q) = S(k, q; f 3 , g 3 , · · · , f k , g k ) and 1 ≤ d g < p. If at least one of the following two conditions is satisfied, then S(k, q) is connected.
1. The polynomials 1, X, f 3 , . . . , f k are F q -linearly independent, and g i contains a linear term for each i, 3 ≤ i ≤ k.
2. The polynomials f 3 , . . . , f k are F q -linearly independent, and there exists some Proof. First, notice that the number of components of S(k, q) is equal to the multiplicity of the eigenvalue q(q − 1). By Lemma 5.1, this multiplicity is equal to {w ∈ F k q : N w = q} . As the equality N w = q is equivalent to the statement that system (4) (with respect to u) has q solutions, the set {w ∈ F k q : N w = q} is a subspace of F k q .
We are ready to prove Theorem 2.2. Let (q m ) m≥1 be an increasing sequence of prime powers such that b q 1 ≥ k.
Let f 3,m , . . . , f k,m be such that 1, X, f 3,m , . . . , f k,m are F q -linearly independent and Let g 3,m , . . . , g k,m be such that g i,m (−X) = −g i,m (X) for each i, the coefficient of X in g i,m is non-zero and 1 ≤ d (m) g < min(p m , c qm ). Then the graphs S(k, q m ) = S(k, q m ; f 3,m , g 3,m , · · · , f k,m , g k,m ), m ≥ 1, form a family of expanders.
6 Spectra of the graphs S(k, q) for g i (X) = X 3 In this section, we provide some specializations of the graphs S(k, q) for g i (X) = X 3 , 3 ≤ i ≤ k, and bound or compute their eigenvalues. Our goal is to prove Theorems

and 2.4.
Lemma 6.1. Let q be an odd prime power with q ≡ 2 mod 3 and k ≥ 3. Suppose that g i (X) = X 3 for any i, 3 ≤ i ≤ k. For any w ∈ F k q , let T w be the number of u ∈ F q such that f 3 (u)w 3 + · · · + f k (u)w k = 0. Then λ w is either q(T w − 1) or at Proof. By (3), we have the following, Case 1: For w of the form (0, 0, w 3 , · · · , w k ), we have: Since q ≡ 2 mod 3, it follows that gcd(q − 1, 3) = 1, and a → a 3 defines a bijection of F q . Therefore the above term Case 2: For those w of the form (w 1 , 0, w 3 , · · · , w k ) with w 1 = 0, we have, Case 3: For those w of the form w = (w 1 , w 2 , w 3 , · · · , w k ) with w 2 = 0, we have If F (−w 1 /w 2 ) = 0, then the number of u ∈ F q such that F (u) = 0 and w 1 + uw 2 = 0 is 1, and hence, Now assume that F (−w 1 /w 2 ) = 0. Then, w 1 + uw 2 = 0 if F (u) = 0. Then the first double sum in (8) has no terms, the second double sum in (8) is equal to T w (−1), and splitting the double sum in (9) into two double sums, we obtain: As q ≡ 2 mod 3, we have gcd(3, q) = 1. By Theorem 3.2, M q ≤ 2 √ q, and the lemma is proven.
In this case, we are able to determine their second largest eigenvalues.
Proof of Theorem 2.3. Since 3 ≤ k ≤ q + 1, it follows that X 2 , X 3 , · · · , X k−1 are F q -linearly independent, and hence S(k, q) is connected by Theorem 5.4. For any w = (w 1 , · · · , w k ) ∈ F k q , let F (X) = X 2 w 3 + X 3 w 4 + · · · + X k−1 w k = X 2 (w 3 + Xw 4 + · · · + X k−3 w k ), which implies that T w (defined in the statement of Lemma 6.1) is either q or between 1 and k − 2. By Lemma 6.1, we have that if λ w is not q(T w − 1), then it is at most (q − T w )M q ≤ (q − 1)M q . Therefore, we obtain: Moreover, if k ≥ 4, then the above inequality becomes equality. Indeed, for any w ∈ F k q of the form w = (0, w 2 , 0, w 4 , 0, · · · , 0), where w 2 , w 4 = 0, the following holds: This implies that max w=(0,w 2 ,0,w 4 ,0,··· ,0) w 2 ,w 4 =0 Therefore, we have λ 2 (S(k, q)) = max{q(k − 3), (q − 1)M q }. As q ≡ 2 (mod 3), by Proof of Theorem 2.4. Since 3 ≤ k ≤ e + 2, it follows that X p , · · · , X p k−2 are F q -linearly independent, and hence S(k, q) is connected by Theorem 5.4. For any Since a → a p defines a bijection on F q , it implies that T w (defined here as the number of roots of F (X) in F q ), is either q or at most p k−3 . The statement of the theorem then follows from Lemma 6.1.

Concluding remarks
In this section, we make some remarks on several specializations of S(k, q) considered in Section 6.
Remark 1. As we mentioned in Section 1, for every q-regular bipartite graph Γ, every eigenvalue of Γ (2) should be at least −q. For graphs S(3, q; x 2 , x 3 ) for prime q between 5 and 19, and for graphs S(4, q; x 2 , x 3 , x 3 , x 3 ) for prime q between 5 and 13, our computations show that their smallest eigenvalues are strictly less than −q.
This implies that these graphs are not distance two graphs of any q-regular bipartite graphs.
Remark 2. In Section 6, we discussed the graphs S(k, q) with g i (X) = X 3 . Now assume that n ≥ 1, and g i (X) = X 2n+1 for all i, 3 ≤ i ≤ k. For these graphs, Lemma 6.1 can be generalized as follows: Let q be an odd prime power with q ≡ 1 mod (2n + 1) and (2n + 1, q) = 1. For any w ∈ F k q , let N w be the number of u ∈ F q such that w 3 f 3 (u) + · · · + w k f k (u) = 0. Then λ w is either q(N w − 1) or at most 2n(q − N w ) √ q.
Interestingly, in the case when f i (X) = X i−1 and g i (X) = X 3 for each i ≥ 3, we actually have equality in the inequality above for (q, k) whenever k < q − 1 q M q + 2 (immediate from Theorem 2.3).

Acknowledgment
The work of the first author was supported by the NSF grant DMS-1600768, and of the second author by the Simons Foundation grant #426092.