Spectral and Combinatorial Properties of Some Algebraically Defined Graphs

08/25/2017 ∙ by Sebastian M. Cioabă, et al. ∙ University of Delaware 0

Let k> 3 be an integer, q be a prime power, and F_q denote the field of q elements. Let f_i, g_i∈F_q[X], 3< i< k, such that g_i(-X) = - g_i(X). We define a graph S(k,q) = S(k,q;f_3,g_3,...,f_k,g_k) as a graph with the vertex set F_q^k and edges defined as follows: vertices a = (a_1,a_2,...,a_k) and b = (b_1,b_2,...,b_k) are adjacent if a_1 b_1 and the following k-2 relations on their components hold: b_i-a_i = g_i(b_1-a_1)f_i(b_2-a_2/b_1-a_1) , 3< i< k. We show that graphs S(k,q) generalize several recently studied examples of regular expanders and can provide many new such examples.

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1 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 be a graph with vertex set and edge set . For a subset of vertices of , denotes the set of edges of with one endpoint in and the other endpoint in . The Cheeger constant (also known as edge-isoperimetric number or expansion ratio) of , is defined by The graph is -regular if each vertex is adjacent to exactly 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 , let be a sequence of graphs such that each is -regular and as . We say that the members of the sequence form a family of expanders if the corresponding sequence is bounded away from zero, i.e. there exists a real number such that for all . In general, one would like the valency sequence to be growing slowly with , and ideally, to be bounded above by a constant. For examples of families of expanders, their theory and applications, see Davidoff, Sarnak and Valette [8], Hoory, Linial and Wigderson [10], and Krebs and Shaheen [12].

The adjacency matrix of a graph has its rows and columns labeled by and equals the number of edges between and (i.e. 0 or 1). When is simple, the matrix

is symmetric and therefore, its eigenvalues are real numbers. For

between and the order of , let denote the -th eigenvalue of . For an arbitrary graph

, it is hard to find or estimate

, and often it is done by using the second-largest eigenvalue of the adjacency matrix of . If is a connected -regular graph, then . 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 , namely 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 which is present is both sides of this inequality above, also known as the spectral gap of , provides an estimate on the expansion ratio of the graph. In particular, for an infinite family of -regular graphs , the sequence is bounded away from zero if and only if the sequence is bounded away from zero. A -regular connected graph is called Ramanujan if . Alon and Boppana [22] proved that this bound is asymptotically best possible for any infinite family of -regular graphs and their results imply that for any infinite family of -regular connected graphs , For functions , we write if as .


For the rest of the paper, let , where is a prime and is a positive integer. For a sequence of prime powers , we always assume that , where is a prime and . Let be the finite field of elements and be the cartesian product of copies of . Clearly,

is a vector space of dimension

over . For , let be an arbitrary polynomial in indeterminants over . We define the bipartite graph , , as follows. The vertex set of is the disjoint union of two copies of , one denoted by and the other by . We define edges of by declaring vertices and to be adjacent if the following relations on their coordinates hold:

(1)

The graphs 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 (see [15]) is that for every vertex of and every , there exists a unique neighbor of whose first coordinate is . This implies that each is -regular, has vertices and edges.

The spectral and combinatorial properties of three specializations of graphs has received particular attention in recent years. Cioabă, Lazebnik and Li [7] determined the complete spectrum of the Wenger graphs with , . Cao, Lu, Wan, Wang and Wang [6] determined the eigenvalues of the linearized Wenger graphs with , , 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 , and in particular, proved that the second largest eigenvalues of these graphs are bounded from above by (so is ‘close’ to being Ramanujan).

Let and denote the partite sets or color classes of the vertex set of a bipartite graph . The distance-two graph of on is the graph having as its vertex set with the adjacency defined as follows: two vertices are adjacent if there exists a vertex adjacent to both and to in (which is equivalent of saying that and are at distance two in ). If is -regular and contains no 4-cycles, then is a -regular simple graph. There is simple connection between the eigenvalues of and the eigenvalues of (see, e.g., [7]): every eigenvalue of with multiplicity , corresponds to a pair of eigenvalues of , each with multiplicity (or a single eigenvalue of multiplicity in case .

This relation between the spectra of -regular bipartite graph and its -regular distance-two graph has been utilized in each of the papers [7, 6, 21] in order to find or to bound the second-largest eigenvalue of , and then use this information to claim the expansion property of . In each of these cases, 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

.

The main motivation behind the construction below is to directly generalize the defining systems of equations for and of , thereby obtaining a family of -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 -regular bipartite graphs . Examples when they are not will be discussed in Remark 1 of Section 7.

2 Main Results

In this section, we define the main object of this paper, the family of graphs and we describe our main results. Let be an integer, . Let , , be polynomials of degrees at most such that for each . We define as the graph with the vertex set and edges defined as follows: is adjacent to if and the following relations on their coordinates hold:

(2)

Clearly, the requirement is used for the definition of the adjacency in to be symmetric. One can easily see that is a Cayley graph with the underlying group being the additive group of the vector space with generating set

This implies that is vertex transitive of degree .

Note that for and , , is the distance-two graph of the Wenger graphs on lines and for and , , is the distance-two graph of the linearized Wenger graphs on lines.

In order to present our results, we need a few more notation. For any , let be the trace of over . It is known that . For any element , let denote the unique integer such that and the residue class of in is . For any complex number , the expression will mean . Let be a complex -th root of unity. For every , we call the exponential sum of .

We are ready to state the main results of this paper.

The following theorem describes the spectrum of the graphs .

Theorem 2.1.

Let . Then the spectrum of is the multiset , where

(3)

For a fixed , the theorem below provides sufficient conditions for the graphs to form a family of expanders.

Theorem 2.2.

Let , be an increasing sequence of prime powers, and let

Set and . Suppose , , , and for all , at least one of the following two conditions is satisfied:

  1. The polynomials are -linearly independent, and has linear term for all , .

  2. The polynomials are -linearly independent, and there exists some , , such that each polynomial , , contains a term with .

Then is connected and .

The following two theorems demonstrate that for some specializations of , we can obtain stronger upper bounds on their second largest eigenvalues.

Theorem 2.3.

Let be an odd prime power with , and . Let and for each , . Then is connected, and

where .

For large , specifically, when ,

Similarly to Theorem 2.3, when choosing , the same functions as in , we obtain the following upper bounds for the second largest eigenvalue.

Theorem 2.4.

Let be an odd prime power with , and . Let and for each , . Then is connected, and

where .

For large , 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 and for the graph 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.

3 Background on finite fields

For definitions and theory of finite fields, see Lidl and Niederreiter [18].

Lemma 3.1 ([18], Ch.5).

If is a polynomial of degree one or less, then

For a general , no explicit expression for the exponential sum exists. The following theorem provides a good upper bound for the exponential sum .

Theorem 3.2 (Hasse-Davenport-Weil Bound, [18], Ch.5).

Let be a polynomial of degree . If , then

Lemma 3.3.

Suppose that and . Then is a real number.

Proof.

We have that

Since for any , it follows that . ∎

4 Spectra of the graphs

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].

Theorem 4.1 ([3]).

Let be a finite group and such that and . Let be a representative set of irreducible -representations of . Suppose that the multiset is the spectrum of the complex matrix . Then the spectrum of the Cayley graph Cay is the multiset formed as the union of copies of for .

Proof of Theorem 2.1.

As we mentioned in Section 2, is a Cayley graph with the underlying group being the additive group of the vector space , and connection set

Since is an abelian group, it follows that the irreducible -representations of are linear (see [11], Ch. 2). They are given by

where and .

Using Theorem 4.1, we conclude that the spectrum of is a multiset formed by all , , of the form:

5 Connectivity and expansion of the graphs

It is hard to get a closed form of in (3) for arbitrary and . But if the degrees of the polynomials and satisfy some conditions, we are able to show that the components of the graphs have large eigenvalue gap. For these and , we find sufficient conditions such that the graphs are connected, and hence form a family of expanders.

From now on, for any graph , we let and . We also assume that and . For each , , let be the coefficient of in the polynomial , for any , , i.e.

For any in , let be the number of ’s in satisfying the following system

(4)

and let be the set of all ’s in such that the following inequality holds for some , ,

(5)

If , then system (5) contains only the first equation, and .

Lemma 5.1.

Let . If , then for any in , the eigenvalue of in (3) is at most

(6)

Moreover, if and only if .

Proof.

Let . Using Theorem 2.1, we have

where

If satisfies (5), then . If , then is an exponential sum of a polynomial of degree at least 2 and at most . By the assumption of the theorem that and Weil’s bound in Theorem 3.2, it follows that

Finally, for the remaining elements , we have

(7)

If , then system (5) contains only the first inequality. In both cases, we have . Therefore, we have

Let us now prove the second statement of the lemma. It is clear that if , then and . For the rest of this proof, we assume that , and show that .

If , then as . Therefore, .

For , we consider the following two cases: and .

If , then as , and hence . Therefore, .

If , then, as is a real number and , we have

and if and only if for all . The latter condition is equivalent to

for all . For , if and only if . This implies that

for any . Therefore, the polynomial

which is over , has distinct roots in and is of degree at most , . Hence, it must be zero polynomial, and so , a contradiction. Hence, . ∎

Let be an increasing sequence of prime powers. For a fixed , , we consider an infinite family of graphs . Hence, when . Let and , for each . In what follows we present conditions on and which imply that the components of these graphs have large eigenvalue gaps.

Theorem 5.2.

Let be an increasing sequence of prime powers. Suppose that and for all . Let be the largest eigenvalue of which is not for any . Then

Proof.

For any , the eigenvalue of is at most

by Lemma 5.1.

It is clear that for any , system (5) has either solutions or at most solutions with respect to . If , then by Lemma 5.1. If , then . Therefore, we have

As an immediate corollary from Theorem 5.2, we have the following theorem.

Theorem 5.3.

Let be an increasing sequence of prime powers. Suppose that , and for all . Let be the largest eigenvalue of which is not for any . Then

Our next theorem provides a sufficient condition for the graph to be connected.

Theorem 5.4.

For , let and If at least one of the following two conditions is satisfied, then is connected.

  1. The polynomials are -linearly independent, and contains a linear term for each , .

  2. The polynomials are -linearly independent, and there exists some , , such that each polynomial , , contains a term with .

Proof.

First, notice that the number of components of is equal to the multiplicity of the eigenvalue . By Lemma 5.1, this multiplicity is equal to . As the equality is equivalent to the statement that system (5) (with respect to ) has solutions, the set is a subspace of .

Let , , and . Let denote the dimension of the subspace generated by . Then, we have,

It is clear that if one of the two conditions in the statement of the theorem is satisfied, then