Eigenvalues of graphs and spectral Moore theorems

04/20/2020
by   Sebastian M. Cioabă, et al.
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In this paper, we describe some recent spectral Moore theorems related to determining the maximum order of a connected graph of given valency and second eigenvalue. We show how these spectral Moore theorems have applications in Alon-Boppana theorems for regular graphs and in the classical degree-diameter/Moore problem.

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§1 Introduction

Our graph theoretic notation is standard (see [4, 5]). Let be an undirected graph with vertex set and edge set . Given , the distance equals the minimum length of a path between and if such a path exists or otherwise. If is connected, then all the distances between its vertices are finite and the diameter of is defined as the maximum of , where the maximum is taken over all pairs . The pairwise distance and the diameter of a connected graph can be calculated efficiently using breadth first search. The Moore or degree-diameter problem is a classical problem in combinatorics (see [31]).

Problem 1.

Given and , what is the maximum order of a connected -regular graph of diameter ?

There is a well known upper bound for known as the Moore bound which is obtained as follows. If is a connected -regular graph of diameter , then for any given vertex in and any , the number of vertices at distance from is at most . Therefore,

(1)

We denote by the right hand-side of the above inequality. When , it is straightforward to note that and the maximum is attained by the cycle on vertices. For , it is easy to see that and the maximum is attained by the complete graph on vertices.

For , a classical result of Hoffman and Singleton [20] gives that equals the Moore bound only when (attained by the cycle ), (the Petersen graph), (the Hoffman-Singleton graph) or possibly . The existence of a -regular graph with diameter on vertices is a well known open problem in this area (see [10, 24, 28]). For and , Damerell [11] and independently, Bannai and Ito [2] proved that there are no graphs attaining the Moore bound (1).

The adjacency matrix is the matrix whose -th entry equals the number of edges between and . This matrix is a real symmetric matrix and if is simple (no loops nor multiple edges), then is a symmetric matrix. Let be a given integer. We will use the following family of orthogonal polynomials:

(2)
(3)

for any . Let . The polynomials form a sequence of orthogonal polynomials with respect to the positive weight

on the interval (see [23, Section 4]). The polynomials in are called Geronimus polynomials [18].

For any vertices and of and any non-negative integer , the entry of the matrix equals the number of walks of length between and . A walk in is called non-backtracking if for any and for any (when ). The following result goes back to Singleton [38].

Proposition 2 (Singleton [38]).

Let be a connected -regular graph with adjacency matrix . For any vertices and of and any non-negative integer , the entry of the matrix equals the number of non-backtracking walks of length between and .

The eigenvalues of are real and we denote them by , where . Sometimes, to highlight the dependence of the eigenvalues on a particular graph, we will use for . When is -regular and connected, it is known that and that . It is also known that and that is an eigenvalue if and only if the graph is bipartite. The smallest eigenvalue of a regular graph has been used to determine the independence number of various interesting graphs (see Godsil and Meagher [17]).

The properties of the eigenvalues of a regular graph were essential in the proofs of Hoffman and Singleton [20] as well as Damerell [11] and Bannai and Ito [2].

The spectral gap is an important parameter in spectral graph theory and is closely related to the connectivity [15] and expansion properties of the graph [19]. Informally, expanders are sparse graphs with large spectral gap. More precisely, a family of graphs is called a family of expanders if

  1. there exists such that each is a connected -regular graph for and the number of vertices of goes to infinity as goes to infinity,

  2. there is a positive constant such that for any .

The first condition above explains the denomination of sparse used at the beginning of this paragraph. This condition implies that the number of edges in is linear in its number of vertices for every . This is best possible in order of magnitude for connected graphs.

The second condition is algebraic and is equivalent to a combinatorial condition that each is highly connected (meaning their expansion constants are bounded away from

) and also equivalent to the probability condition that a random walk on

converges quickly to its stationary distribution. We refer to [19] for the precise descriptions of these conditions.

A natural question arising from the previous considerations is how large can the spectral gap be for an -regular connected graph ? Since we are interested in situations where is fixed, this is equivalent to asking how small can be for a connected -regular graph . This was answered by the Alon-Boppana theorem.

Theorem 3.

Let be a natural number.

  1. Alon-Boppana 1986. If is a connected -regular graph with vertices, then

    (4)

    where is a constant and is a quantity that goes to as goes to infinity.

  2. Asymptotic Alon-Boppana Theorem. If is a sequence of connected -regular graphs such that as . Then

    (5)

To my knowledge, there is no paper written by Alon and Boppana which contains the theorem above. The first appearance of this result that I am aware of, is in 1986 in Alon’s paper [1] where it is stated that

R. Boppana and the present author showed that for every -regular graph on vertices .

Note that the in [1] is the smallest positive eigenvalue of the Laplacian of and it equals . Therefore the statement above is equivalent to

(6)

There is a similar result to the Alon-Boppana theorem that is due to Serre [37]. The meaning of this theorem below is that large -regular graphs tend to have a positive proportion of eigenvalues trying to be greater than .

Theorem 4 (Serre [37]).

For any , there exists such that any -regular graph on vertices has at least eigenvalues that are at least .

These results motivated the definition of Ramanujan graphs that was introduced by Lubotzky, Phillips and Sarnak [27]. A connected -regular graph is called Ramanujan if all its eigenvalues (with the exception of and perhaps , if is bipartite) have absolute value at most . Lubotzky, Phillips and Sarnak [27] and independently Margulis [30] constructed infinite families of -regular Ramanujan graphs when is a prime. These constructions used results from algebra and number theory closely related to a conjecture of Ramanujan regarding the number of ways of writing a natural number as a sum of four squares of a certain kind (see [27, 30] and also, [12] for a more detailed description of these results). For the longest time, it was not known whether infinite families of -regular Ramanujan graphs exist for any . Marcus, Spielman and Srivastava [29] obtained a breakthrough result by showing that there exist infinite families of bipartite -regular Ramanujan graphs for any . Their method of interlacing polynomials has been fundamental to this proof and has found applications in other areas of mathematics as well.

§2 Spectral Moore theorems for general graphs

Throughout the years, several proofs of the Alon-Boppana theorem have appeared (see Lubotzky, Phillips and Sarnak [27], Nilli [33], Kahale [25], Friedman [16], Feng and Li [14], Li and Solé [26], Nilli [34] and Mohar [32]). Theorem 4 was proved by Serre [37] with a non-elementary proof (see also [12]). The first elementary proofs appeared around the same time by Cioabă [6] and Nilli [34]. Richey, Stover and Shutty [36] worked to turn Serre’s proof into a quantitative theorem and asked the following natural question.

Problem 5.

Given an integer and , what is the maximum order of a -regular graph with ?

These authors obtained several results involving . In this section, we describe our recent results related to the problem above and its bipartite and hypergraph versions. See [9, 8, 7] and the references therein for more details and other related problems. The method that is fundamental to all these results is due to Nozaki [35] who proved the linear programming bound for graphs.

Theorem 6 (Nozaki [35]).

Let be a connected -regular graph with vertices and distinct eigenvalues . If there exists a polynomial such that , for any , , and for any , then

Nozaki used this result to study the following problem.

Problem 7.

Given integers , what is the -regular graph on vertices that has the smallest among all -regular graphs on vertices ?

While similar to it, this problem is quite different from Problem 5.

In [8], the authors used Nozaki’s LP bound for graphs to obtain the following general upper bound for .

Theorem 8 (Cioabă, Koolen, Nozaki and Vermette [8]).

Given integers and a non-negative real number , let be the tridiagonal matrix:

If equals the second largest eigenvalue of the matrix , then

We sketch below the ideas of the proof of this theorem. For , denote

(7)

where the s are the orthogonal polynomials defined in equations (2) and (3). The polynomials also form a family of orthogonal polynomials. They satisfy the following properties:

(8)
(9)

for .

The eigenvalues of the matrix are the roots of and are distinct (see [8, Theorem 2.3]). If we denote them by , then the polynomial satisfies for . It is a bit more involved to check the other conditions from Theorem 6 and we refer the reader to [8] for the details to see how one can apply Nozaki’s LP bound to and obtain that

To make things more clear, note the following result.

Proposition 9.

Let be an integer. For any , there exists an integer and a positive number such that is the second largest eigenvalue of the matrix .

Let denote the largest root of and denote the largest root of . Note that and

(10)

Bannai and Ito [3, Section III.3] showed that , where . Because , one can show that for any (see [8, Prop 2.6] for other properties of these eigenvalues). From the remarks following Theorem 8, note that the second largest eigenvalue of equals the largest root of the polynomial . Because the roots of and interlace, one obtains that is a decreasing function in and takes values between and . Taking into the account what happens when , namely that , we obtain the following result.

Proposition 10.

For , takes any value in the interval .

Putting these things together, one deduces that can take any possible value between and . There are several infinite families for which the precise values have been determined in [8], but there are several open problems for relatively small values of and . For example, with an example of a -regular graph with on vertices being the 2nd subconstituent of the Hoffman-Singleton graph. Theorem 8 can give (see [7]). Also, we know that (Heawood graph), but we don’t know the exact value of for any . Lastly, has been determined for (equals with Pappus graph as an example attaining it) and (it is

with the odd graph

meeting it), but we don’t know it for .

§3 Alon-Boppana and Serre theorems

We point out the relevance of these results in the context of Alon-Boppana and Serre theorems. A typical Alon-Boppana result is of the form: if is a connected -regular graph with diameter , then

(11)

See Friedman [16, Corollary 3.6] for the inequality above or Nilli [34, Theorem 1] for a slightly weaker bound. The equivalent contrapositive formulation of inequality (11) is the following: if is a connected -regular graph of diameter with , then . By Moore bound (1), this implies that

(12)

Obviously, the best bound one can achieve here is obtained for that with . When applying Theorem 8, let be a real number such that . Given the properties of the largest roots of the polynomials (see Proposition 10 and the paragraph containing it), we must have that either or . If , then there exists such that is the largest eigenvalue of the polynomial . If is a connected -regular graph with , then using Theorem 8 we get that

(13)

which is clearly better than (12). If , then there exists such that is the largest eigenvalue of the polynomial . As above, if is a connected -regular graph with , then Theorem 8 implies that

(14)

which is again better than (11).

§4 Spectral Moore theorems for bipartite graphs

Building on this work, the author with Koolen and Nozaki extended and refined these results to bipartite regular graphs [9]. Let be an integer and be any real number between and . Define as the maximum number of vertices of a bipartite -regular graph whose second largest eigenvalue is at most . Clearly, and a natural question is whether or not these parameters are actually the same or not. One can show that attained by the Petersen graph and that attained by the -dimensional cube. For the bipartite graphs, a linear programming bound similar to Nozaki’s Theorem 6 from [35] was obtained with the use of the following polynomials:

for any , where were defined earlier in (2) and (3). Let be a bipartite connected regular graph. Its adjacency matrix has the form and we call the biadjacency matrix of . Note that for any . The following is called the LP bound for bipartite regular graphs.

Theorem 11 (Cioabă, Koolen and Nozaki [9]).

Let be a connected bipartite -regular graph with vertices and denote by its set of distinct eigenvalues, where . If there exists a polynomial such that , for each , , and for each , then

(15)

Equality holds if and only if for each , and for each , , and , where is the biadjacency matrix of . If equality holds and for each , then the girth of is at least .

For any integers and any positive , let be the tridiagonal matrix with lower diagonal , upper diagonal , and constant row sum . Using Theorem 11, the following general upper bound for was obtained in [9].

Theorem 12 (Cioabă, Koolen and Nozaki [9]).

If is the second largest eigenvalue of , then

(16)

Equality holds if and only if there exists a bipartite distance-regular graph whose quotient matrix with respect to the distance-partition from a vertex is for or for .

Define for . These are orthogonal polynomials and one can show that for as well as that . The first step in proving the above result is showing that the characteristic polynomial of equals . The proof proceeds in similar steps to Theorem 8, but is more technical and we refer the reader to [9] for the details. Similar to the situation for general graphs, one can show the following.

Proposition 13.

Let be an integer. For any , there exists and such that is the second largest eigenvalue of .

In [9], the authors also proved that for given and , the upper bound obtained in Theorem 12 is better than the one in Theorem 8. Theorem 12 has applications to various areas and it improves results obtained in the context of coding theory by Høholdt and Janwa [21] and Høholdt and Justensen [22] and design theory by Teranishi and Yasuno [39].

As in the case of Theorem 8, Theorem 12 has applications for the Alon-Boppana theorems for bipartite regular graphs. Corollary 4.11 in [9] is a consequence of Theorem 12 and states that if is a bipartite -regular graph of order greater than (the right hand-side in Theorem 12), then , where is the second largest eigenvalue of . Li and Solé [26, Theorems 3 and 5] proved that if is a bipartite -regular graph of girth , then . This result follows from Corollary 4.11 in [9] as is the second largest eigenvalue of and having girth implies that has at least vertices.

§5 Classical Moore problem

Since the fundamental work of Singleton [38], Hoffman and Singleton [20], Bannai and Ito [2] and Damerell [11] in the 1970s, the families of orthogonal polynomials and have been important in the study of the Moore problem (1). It has been observed by several authors (see [13] or [31] for example) that if is connected -regular of diameter , eigenvalues and , then

(17)

where is the upper bound from the Moore bound (1). Recall that denotes the largest root of and satisfies

(18)

Inequality (17) will improve the classical Moore bound (1) when . This will happen when . When , from (10) we know that . Note that [13, Theorem 2] contains a typo in the numerator of the right hand-side of the inequality (the in the numerator should be a ). The informal description of the result above is that when is large, then the order of the graph will be smaller than the Moore bound. Note however that if is small, then may be negative and inequality (17) may be worse than the classical Moore bound (1). Our results from [8] may be used to handle some cases when is small (actually when is small). More precisely, Theorem 8 gives an upper bound for the order of an -regular graph with small second largest eigenvalue regardless of its diameter actually.

Known Defect Lower Moore Upper
(8,2) 57 8 2.09503 2.19258 3.40512
(9,2) 74 8 2.29956 2.37228 3.53113
(10,2) 91 10 2.46923 2.54138 3.88473
(4,3) 41 12 2.11232 2.25342 2.88396
(5,3) 72 34 2.42905 2.62620 3.77862
(4,4) 98 63 2.53756 2.69963 3.44307
(5,4) 212 214 2.91829 3.12941 4.41922
(3,5) 70 24 2.32340 2.39309 2.64401
(4,5) 364 121 2.89153 2.93996 3.42069
(3,6) 132 58 2.45777 2.51283 2.75001
(4,6) 740 717 3.00233 3.08314 3.73149
Table 1: Numerical results for small

We explain this argument and give some numerical examples in Table 1 where we listed some values of (these values are from [31, page 4]) where the maximum orders of -regular graphs of diameter are not known. For each such pair, the column labeled Known gives the largest known order of an -regular graph of diameter . The column Defect equals the difference between the Moore bound and the entry in the Known column. The column Moore contains the value of rounded below to decimal points. The column Upper contains the lower bound for that guarantees that inequality (17) will give a lower bound than the value from the Known column. For example, for and , if is an -regular graph with diameter having , then . The column Lower contains an upper bound for that guarantees that the order of such -regular graph would be small. For example, for and , our Theorem 8 implies that if is an -regular graph with , then . Another way to interpret these results in Table 1 is that if one wants to look for a -regular graph of diameter with more than vertices, then the second largest eigenvalue of such putative graph has to be between and .

§6 Acknowledgments

This article is based on a talk I gave at the RIMS Conference Research on algebraic combinatorics, related groups and algebras. I thank the participants of the conference and I am grateful to Hiroshi Nozaki for his help with the computations in Table 1 and for his amazing support.

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