Euclidean graph distance matrices of generalizations of the star graph

Abstract

In this paper a relation between graph distance matrices of the star graph and its generalizations and Euclidean distance matrices is considered. It is proven that distance matrices of certain families of graphs are circum Euclidean. Their spectrum and generating points are given in a closed form.

Introduction

A matrix $D\in {\mathbb{R}}^{n\times n}$ is Euclidean distance matrix (EDM), if there exist points ${\mathit{x}}_i\in {\mathbb{R}}^r\text{,}i=1\text{,}2\text{,}\dots \text{,}n$, such that $d_{\mathit{ij}}=\Vert {\mathit{x}}_i-{\mathit{x}}_j{\Vert }_2^2$. The minimal possible r is called the embedding dimension (see [5], e.g.). Euclidean distance matrices were introduced by Menger in 1928, later they were studied by Schoenberg [22], Gower [9], and other authors. In recent years many new results were obtained (see [13], [15] and the references therein). EDMs have many interesting properties and are used in various applications in linear algebra, graph theory, bioinformatics, e.g., where frequently a question arises, what can be said about a configuration of points ${\mathit{x}}_i$, if only distances between them are known.

Schoenberg obtained the following characterization of EDMs.

Theorem 1 [22]

A symmetric hollow (i.e., with zeros on the diagonal) matrix $D\in {\mathbb{R}}^{n\times n}$ is EDM if and only if ${\mathit{x}}^{T}D\mathit{x}\le 0$ for all $\mathit{x}\in {\mathbb{R}}^n$ such that ${\mathit{x}}^T\mathit{e}=0$, where $\mathit{e}≔{[1\text{,}1\text{,}\dots \text{,}1]}^T\in {\mathbb{R}}^n$.

In the paper we will use the notation $\mathit{e}$ and E for the vector and the matrix of ones, respectively. The vectors ${\mathit{e}}_i$ will denote the standard basis.

Based on Schoenberg’s results, Hayden, Reams and Wells gave the following characterization, which will be frequently used in the rest of the paper.

Theorem 2 [13, Thm. 2.2]

Let $D\in {\mathbb{R}}^{n\times n}$ be a nonzero hollow matrix. Then D is EDM if and only if it has exactly one positive eigenvalue and there exists $\mathit{w}\in {\mathbb{R}}^n$ such that $D\mathit{w}=\mathit{e}$ and ${\mathit{w}}^T\mathit{e}⩾0$.

A nonzero EDM has exactly one positive eigenvalue and the sum of its eigenvalues is zero. It is conjectured that there always exists a solution of the inverse eigenvalue problem, i.e., to prove that any set of numbers that meet these conditions is a spectrum of an EDM (see [15], [17], e.g).

There is another very useful characterization of EDMs.

Lemma 1 [13, Lem. 5.3]

Let a symmetric hollow matrix $D\in {\mathbb{R}}^{n\times n}$ have only one positive eigenvalue with the corresponding eigenvector $\mathit{e}\in {\mathbb{R}}^n$. Then D is EDM.

An EDM matrix D is circum-Euclidean (CEDM) (also spherical) if its generating points ${\mathit{x}}_i$ lie on the surface of some hypersphere (see [24], e.g.). Circum-Euclidean distance matrices are important because every EDM is a limit of CEDMs. They can be characterized as follows.

Theorem 3 [24, Thm. 3.4]

An Euclidean distance matrix $D\in {\mathbb{R}}^{n\times n}$ is CEDM if and only if there exist $\mathit{s}\in {\mathbb{R}}^n$ and $\beta \in \mathbb{R}$, such that $D\mathit{s}=\beta \mathit{e}$ and ${\mathit{s}}^T\mathit{e}=1$.

We will often use the following result that is an immediate corollary of , .

Corollary 1

Let D satisfy the assumptions of Theorem 2. If there exists $\mathit{w}\in {\mathbb{R}}^n$ such that $D\mathit{w}=\mathit{e}$ and ${\mathit{w}}^T\mathit{e}>0$, the matrix D is CEDM.

Proof

By Theorem 2, the matrix D is EDM. Since ${\mathit{w}}^T\mathit{e}>0$, by taking $\beta ≔1/({\mathit{w}}^T\mathit{e})$ and $\mathit{s}=\beta \mathit{w}$, Theorem 3 implies that D is CEDM.  □

For a given EDM $D\in {\mathbb{R}}^{n\times n}$ we can compute its generating points ${\mathit{x}}_i$, i.e., a set of points, such that $d_{\mathit{ij}}=\Vert {\mathit{x}}_i-{\mathit{x}}_j{\Vert }_2^2$. First we need to construct a Gower matrix

$$G_D≔-\frac{1}{2}(I-{\mathit{es}}^T)D(I-{\mathit{se}}^T)\text{,}$$

where $\mathit{s}\in {\mathbb{R}}^n$ satisfies the relation ${\mathit{s}}^T\mathit{e}=1$. If D is CEDM and we choose $\mathit{s}=1/n·\mathit{e}$, the obtained points lie on the hypersphere with the center $\mathit{0}$ and the radius $\sqrt{\beta /2}$.

The matrix $G_D$ is positive semidefinite, thus it can be written as $G_D=X^{T}X$, with $X=\mathrm{diag}\left(\sqrt{{\sigma }_i}\right)U^T$. Here $G_D=U\mathrm{\Sigma }{U}^T$ is the singular value decomposition of $G_D$ and $\mathrm{\Sigma }=\mathrm{diag}({\sigma }_i)$. The points ${\mathit{x}}_i$ are then obtained as columns of X.

The embedding dimension of D is therefore equal to the rank of the matrix $G_D$. Since an arbitrary translation, rotation or a mirror map, applied to the points ${\mathit{x}}_i$, preserves the distance matrix, the generating points are not unique.

Let $\mathcal{G}$ be a simple connected graph with a vertex set $\mathcal{V}(\mathcal{G})$. Let us define the distance $d(u\text{,}v)$ between vertices $u\text{,}v\in \mathcal{V}(\mathcal{G})$ as their graph distance, i.e., the length (number of edges) of the shortest path between them. Let $G≔{[d(u\text{,}v)]}_{u\text{,}v\in \mathcal{V}(\mathcal{G})}$ be the distance matrix of $\mathcal{G}$. An example is shown in Fig. 1.

In this paper we will study graph distance matrices of some families of graphs, derived from the star graph. Our goal is to prove that they are CEDMs and to obtain their distance spectrum and generating points in a closed form.

In algebraic graph theory, a lot is known on the adjacency matrix and the Laplacian matrix of a graph. Many results on their spectra exist, particularly on properties of their largest and smallest eigenvalues (see [4], e.g.). Not much is known on the graph distance matrix. It could be efficiently computed by Dykstra algorithm, e.g. Some results on its structure and its spectrum, the so-called D-spectrum for particular graphs were given in [12], [14].

Similar problems were studied in several papers. Line distance matrices (corresponding to paths) were considered in [19], [16], cell matrices (for weighted star graphs) were introduced in [16], distance matrices of weighted trees were studied in [1], [7], and distance matrices of weighted paths and cycles were analysed in [18].

This paper generalizes and extends some of these results to a broader family of graphs. In this way, a deeper insight into the relation between general graphs (and networks) and EDM structure is obtained. Hopefully, this will enable a more thorough study of the problem considered.

The structure of the paper is as follows. In the second section we analyse the star graph, k-star graphs and trees in general and prove that their graph distance matrices are CEDMs. The distance spectrum (D-spectrum) of the star graph is given in a closed form as well as its generating points. In the third section we study properties of circulant matrices that will be an important tool throughout the paper. The generalizations of the star graph (the wheel, the gear and the helm graph) are analysed in the fourth, the fifth and the sixth section. We prove that their distance matrices are CEDMs. We give the distance spectra of the wheel and the gear graph in a closed form. For the wheel graph we also compute the generating points of the corresponding distance matrix. The paper is concluded with an example.