On the distance spectra of graphs
Abstract
The distance matrix of a graph is the matrix containing the pairwise distances between vertices. The distance eigenvalues of are the eigenvalues of its distance matrix and they form the distance spectrum of . We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.
Keywords. distance matrix, eigenvalue, distance regular graph, Kneser graph, double odd graph, halved cube, Doob graph, lollipop graph, barbell graph, distance spectrum, strongly regular graph, optimistic graph, determinant, inertia, graph
AMS subject classifications. 05C12, 05C31, 05C50, 15A15, 15A18, 15B48, 15B57
1 Introduction
The distance matrix of a graph is the matrix indexed by the vertices of where is the distance between the vertices and , i.e., the length of a shortest path between and . Distance matrices were introduced in the study of a data communication problem in [16]. This problem involves finding appropriate addresses so that a message can move efficiently through a series of loops from its origin to its destination, choosing the best route at each switching point. Recently there has been renewed interest in the loop switching problem [13]. There has also been extensive work on distance spectra (eigenvalues of distance matrices); see [2] for a recent survey.
In [19], the authors classify the distance regular graphs having only one positive distance eigenvalue. Such graphs are directly related to a metric hierarchy for finite connected graphs (and more generally, for finite distance spaces, see [3, 11, 12, 18]), which makes these graphs particularly interesting. In Section 3 we find the distance spectra of Doob graphs, double odd graphs, and halved cubes, completing the determination of distance spectra of distance regular graphs that have one positive distance eigenvalue. In Section 2 we characterize strongly regular graphs having more positive than negative distance eigenvalues in terms of their parameters, generalizing results in [5], and apply this characterization to show several additional infinite families of strongly regular graphs have this property.
Section 4 contains examples of graphs with specific properties and a small number of distinct distance eigenvalues. Answering a question in [4], we provide a construction for a family of connected graphs with arbitrarily large diameter that have no more than 5 distinct distance eigenvalues but are not distance regular (Example 4.2). We exhibit a family of graphs with arbitrarily many distinct degrees but having exactly 5 distinct distance eigenvalues (Example 4.5). Finally, we give two lower bounds for the number of distinct distance eigenvalues of a graph. The first bound is for a tree, in terms of its diameter, and the second is for any graph in terms of the zero forcing number of its complement.
In Persi Diaconis’ talk on distance spectra at the “Connections in Discrete Mathematics: A celebration of the work of Ron Graham” [13], he suggested it would be worthwhile to study the distance matrix of a clique with a path adjoined (sometimes called a lollipop graph), and in Section 5 we determine determinants and inertias of these graphs, of barbell graphs, and of generalized barbell graphs (a family that includes both lollipops and barbells). The remainder of this introduction contains definitions and notation used throughout.
All graphs are connected, simple, undirected, and finite of order at least two. Let be a graph. The maximum distance between any two vertices in is called the diameter of and is denoted by . Two vertices are adjacent in the complement of , denoted by , if and only if they are nonadjacent in . Let denote the adjacency matrix of , that is, is the matrix indexed by the vertices of where if and is otherwise. The all ones matrix is denoted by and the all ones vector by . A graph is regular if every vertex has the same degree, say ; equivalently, ; observe that is the spectral radius of .
Since is a real symmetric matrix, its eigenvalues, called distance eigenvalues of , are all real. The spectrum of is denoted by where is the spectral radius, and is called the distance spectrum of the graph .
The inertia of a real symmetric matrix is the triple of integers , with the entries indicating the number of positive, zero, and negative eigenvalues, respectively (counting multiplicities). Note the order , while customary in spectral graph theory, is nonstandard in linear algebra, where it is more common to use . The spectrum of a matrix can be written as a multiset (with duplicates as needed), or as a list of distinct values with the exponents () in parentheses indicating multiplicity.
2 Strongly regular graphs
A regular graph of order is strongly regular with parameters if every pair of adjacent vertices has common neighbors and every pair of distinct nonadjacent vertices has common neighbors. For a strongly regular graph with parameters , is equivalent to is copies of , so we assume and thus has diameter at most 2. There is a well known connection between the adjacency matrix of a graph of diameter at most and its distance matrix that was exploited in [5].
Remark 2.1.
A real symmetric matrix commutes with if and only if it has constant row sum. Suppose commutes with and is the constant row sum of , so and have a common eigenvector of . Since eigenvectors of real symmetric matrices corresponding to distinct eigenvalues are orthogonal, every eigenvector of for an eigenvalue other than is an eigenvector of for eigenvalue 0.
Now suppose is a graph that has diameter at most . Then . Now suppose in addition that is regular, so commutes with . Thus .
Let be a strongly regular graph with parameters . It is known that the (adjacency) eigenvalues of are of multiplicity , of multiplicity , and of multiplicity [15, Chapter 10].^{1}^{1}1These formulas do work for , but in that case . Thus the distance eigenvalues of are
For a derivation of these values using quotient matrices, see [4, p. 262].
2.1 Optimistic strongly regular graphs
A graph is optimistic if it has more positive than negative distance eigenvalues. Graham and Lovász raised the question of whether optimistic graphs exist (although they did not use the term). This question was answered positively by Azarija in [5], where the term ‘optimistic’ was introduced. A strongly regular graph is a conference graph if . In [5] it is shown that conference graphs of order at least 13 are optimistic and also that the strongly regular graphs with parameters are optimistic for . Additional examples of optimistic strongly regular graphs, such as the Hall–Janko graph with parameters (100, 36, 14, 12), and examples of optimistic graphs that are not strongly regular are also presented there.
Theorem 2.2.
Let be a strongly regular graph with parameters . The graph is optimistic if and only if and . That is, is optimistic if and only if
Proof.
Observe that . Thus is optimistic if and only if and . Simple algebra shows that is equivalent to :
There are two cases. First assume , so
Now assume , or equivalently, . For any strongly regular graph, . Thus
It is well known [15, p. 222] (and easy to see) that
(1) 
The denominator is always positive, and thus if and only if . ∎
Corollary 2.3.
Whether or not a strongly regular graph is optimistic depends only on its parameters .
There are several additional families of strongly regular graphs for which the conditions in Theorem 2.2 hold.
Corollary 2.4.
A strongly regular graph with parameters is optimistic if and only if .
Proof.
By Theorem 2.2, implies , and is equivalent to
The family of symplectic graphs is defined using subspaces of a vector space over a field with a finite number of elements. Let be the field with elements and consider as vertices of the one dimensional subspaces of for ; let denote the subspace generated by . The alternate matrix of order over is the matrix with copies of . The vertices and are adjacent in if . See [15, Section 8.11] for and [23] for more general . It is known that the symplectic graph is a strongly regular graph with parameters
(see [15, Section 10.12] for or [23, Theorem 2.1] for more general ). The next result is immediate from Corollary 2.4.
Corollary 2.5.
The symplectic graphs are optimistic for every and except and .
There are additional families of optimistic strongly regular graphs with parameters . One example is the family on one type of nonisotropic points, which has parameters
with and [7].
The more common definition of a conference graph is a strongly regular graph with ; this is equivalent to the definition given earlier as [15, Lemma 10.3.2].
Theorem 2.6.
Let be a strongly regular graph with parameters . Both and are optimistic if and only if is a conference graph and .
2.2 Strongly regular graphs with one positive distance eigenvalue
Distance regular graphs (see Section 3 for the definition) having one positive distance eigenvalue were studied in [19]; strongly regular graphs are distance regular. Here we make some elementary observations about strongly regular graphs with one positive distance eigenvalue that will be used in Section 3.
Proposition 2.7.
Let be a strongly regular graph with parameters and (adjacency) eigenvalues , and . Then is a conference graph, , or . Thus has exactly one positive distance eigenvalue if and only if , , or .
Proof.
Assume has exactly one positive distance eigenvalue. Since and , . It is known that if a strongly regular graph is not a conference graph, then and are integers [15, Lemma 10.3.3]. If , then (this follows from [15, p. 219]), implying . Since , if and only if . As shown in [5] (see also Theorem 2.6), conference graphs of order at least 13 are optimistic and so have more than one positive eigenvalue. The conference graph on 5 vertices is , and the conference graph on 9 vertices has . Thus if and only if , , or . ∎
Observation 2.8.
There are several well known families of strongly regular graphs having , and thus having one positive distance eigenvalue. Examples of such graphs and their distance spectra include:

The cocktail party graphs are complete multipartite graphs on partite sets of order 2; is a strongly regular graph with parameters and has distance spectrum .

The line graph with parameters has distance spectrum
. 
The line graph with parameters has distance spectrum
.
3 Distance regular graphs having one positive distance eigenvalue
Let be integers. The graph is called distance regular if for any choice of with , the number of vertices such that and is independent of the choice of and . Distance spectra of several families of distance regular graphs were determined in [4]. In this section we complete the determination of the distance spectra of all distance regular graphs having exactly one positive distance eigenvalue, as listed in [19, Theorem 1]. For individual graphs, it is simply a matter of computation, but for infinite families the determination is more challenging. The infinite families in [19, Theorem 1] are (with numbering from that paper):

cocktail party graphs ,

cycles (called polygons in [19]),

Hamming graphs ,

Doob graphs ,

Johnson graphs ,

double odd graphs , and

halved cubes .
First we summarize the known distance spectra of these infinite families. For a strongly regular graph, it is easy to determine the distance spectrum and we have listed the distance spectra of cocktail party graphs in Observation 2.8. The distance spectra of cycles are determined in [14], and in [4] are presented in the following form: For odd , , and for even the distance eigenvalues are and if is odd. Hamming graphs, whose distance spectra are determined in [17], are discussed in Section 3.1 (since they are used to construct Doob graphs). Johnson graphs, whose spectra are determined in [4], are discussed in Section 3.2 (because they are used to determine distance spectra of double odd graphs). The remaining families are defined and their distance spectra determined in Section 3.1 for Doob graphs, Section 3.2 for double odd graphs, and Section 3.3 for halved cubes. Thus all the infinite families in [19, Theorem 1] now have their distance spectra determined.
For completeness we list the distance spectra (some of which are known) for the individual graphs having exactly one positive distance eigenvalue; these are easily computed and we provide computational files in Sage [21]. Definitions of these graphs can be found in [2].
Proposition 3.1.
The graphs listed in [19, Theorem 1] that have one positive distance eigenvalue and are not in one of the infinite families, together with their distance spectra, are:

the Gosset graph; ,

the Schläfli graph; ,

the three Chang graphs (all the same spectra); ,

the icosahedral graph; ,

the Petersen graph; ,

the dodecahedral graph; .
A graph is transmission regular if (where is the constant row sum of ). Any distance regular graph is transmission regular. Here we present some tools for transmission regular graphs and matrices constructed from distance matrices of transmission regular graphs. We first define the cartesian product of two graphs: For graphs and define the graph to be the graph whose vertex set is the cartesian product and where two vertices and are adjacent if ( and ) or ( and ). The next theorem is stated for distance regular graphs in [17], but as noted in [4] the proof applies to transmission regular graphs.
Theorem 3.2.
[17, Theorem 2.1] Let and be transmission regular graphs with and . Then
Lemma 3.3.
Let be an irreducible nonnegative symmetric matrix that commutes with . Suppose
, and (where is the row sum so ). Then
where the union is a multiset union.
Proof.
Since commutes with , is an eigenvector of for eigenvalue . Thus is an eigenvector for and is an eigenvector for . Let be an eigenvector for (and assume is linearly independent). Since eigenvectors for distinct eigenvalues are orthogonal, , and thus . Define and . Then is an eigenvector of for eigenvalue and is an eigenvector of for eigenvalue . ∎
3.1 Hamming graphs and Doob graphs
A Doob graph is the cartesian product of copies of the Shrikhande graph and the Hamming graph . The Shrikhande graph is the graph where and . The distance spectrum is [21].
For and , the Hamming graph has vertex set consisting of all tuples of elements taken from , with two vertices adjacent if and only if they differ in exactly one coordinate; is equal to with copies of . In [17] it is shown that the distance spectrum of the Hamming graph is
(4) 
Observe that the line graph is the Hamming graph .
Theorem 3.4.
The distance spectrum of the Doob graph is
3.2 Johnson, Kneser, and double odd graphs
Before defining Johnson graphs, we consider a more general family that includes both Johnson and Kneser graphs. For fixed and , let and denote the collection of all subsets of . For fixed , the graph is the graph defined on vertex set such that two vertices and are adjacent if and only if .^{2}^{2}2Note the definition of this family of graphs varies with the source. Here we follow [6], whereas in [15] the graph defined by , and is what is here denoted by . The Johnson graphs are the graphs , and they are distance regular. Observe that the line graph is the Johnson graph . The distance spectrum of Johnson graphs are determined in [4, Theorem 3.6]:
(6) 
where .
Although they do not necessarily have one distance positive eigenvalue and are not all distance regular, Kneser graphs can be used to construct double odd graphs. The Kneser graph is the graph on the vertex set such that two vertices and are adjacent if and only if . Of particular interest are the odd graphs .
A double odd graph is a graph whose vertices are element or element subsets of , where two vertices and are adjacent if and only if or , as subsets. Double odd graphs can also be constructed as tensor products of odd graphs. We first define the tensor product of two graphs: For graphs and define the graph to be the graph whose vertex set is the cartesian product and where two vertices and are adjacent if and . To see that , observe that has as vertices two copies of the vertices of , call them and . Then there are no edges just between the vertices of the form , no edges just between the , and if and only if . Equivalently, if and only if . We will work with the representation of as .
Remark 3.5.
Let be a graph that is not bipartite. Then is a connected bipartite graph and has the form , where and are nonnegative symmetric matrices with the entries of even and those of odd; all of these statements are obvious except the symmetry of . Observe that is the matrix whose entry is the shortest even distance between vertices and in , and is the matrix whose entry is the shortest odd distance between vertices and in .
It is known [24] that the distance between two vertices and in is given by the formula
which for the odd graph is
(7) 
The distance between two vertices and in the Johnson graph is given by the formula
(8) 
where is the symmetric difference. Let .
Proposition 3.6.
In , if and only if . Furthermore, for , .
Proof.
The first statement follows from equations (7) and (8) (it also follows from the definition). To prove the second part, let be a path of minimum even length between and , then is a path of length in the Johnson graph by the first statement. Conversely any path of the length in the Johnson graph between and provides a path of length between and in the odd graph (note that the new vertices are pairwise distinct). This implies the second statement. ∎
Proposition 3.7.
For , .
Proof.
Let and be two vertices of the odd graph . Let be an subset of containing . Thus has the maximum size among all subsets of . Now the minimum odd distance between and is one more than the minimum even distance between and a neighbor of in the Kneser graph . By Proposition 3.6, this is , where is a neighbor of in that maximizes . It suffices to set . This implies that the minimum odd distance between and is
Theorem 3.8.
Let be the distance matrix of the Johnson graph , let be the all ones matrix of order , and let . The distance matrix of the double odd graph is
and its distance spectrum is
Proof.
We now return to arbitrary Kneser graphs and determine their distance spectra. Let be the distance matrix of . Let be the identity matrix of order and the adjacency matrix of for . It follows that
It is known that forms an association scheme called the Johnson scheme and the following properties come from the Corollary to Theorem 2.9 and from Theorem 2.10 in [6]. For more information about association schemes, see, e.g., Section 2.3 in [6].
Theorem 3.9.
[6] The matrices form a commuting family and are simultaneously diagonalizable. There are subspaces such that

the whole space is the direct sum of ;

for each ,
is an eigenvalue with multiplicity whose eigenspace is .
The value