Inertia and spectral symmetry of the eccentricity matrices of a class of bi-block graphs

Let $G$ be a simple connected graph on $n$ vertices with the vertex set $V_{G}=\{v_{1},v_{2},\ldots,v_{n}\}$. In literature, many properties and structural characteristics of $G$ have been studied by associating several matrices corresponding to $G$, see [2]. Let us recall some of them which are pertinent to our discussion here. The adjacency matrix of $G$, denoted by $A(G)$, is an $n\times n$ matrix whose $(i,j)$-th entry is one if $v_{i}$ and $v_{j}$ are adjacent and zero elsewhere. Let $d(v_{i},v_{j})$ denote the length of a shortest path between the vertices $v_{i}$ and $v_{j}$. The distance matrix $D(G)\mathrel{\mathop{\mathchar 58\relax}}=(d_{ij})$ of $G$, is an $n\times n$ matrix such that $d_{ij}=d(v_{i},v_{j})$ for all $i$ and $j$. Clearly, the adjacency matrix can be obtained from the distance matrix by retaining the smallest non-zero distance (which is equal to one) in each row and column and by setting the rest of the entries equal to zero. Inspired by this, Randić [12] associated a new matrix corresponding to $G$, namely $D_{\text{MAX}}$, which is derived from the distance matrix $D(G)$ by keeping the largest non-zero distance in each row and each column of $D(G)$ and defining the remaining entries equal to zero. Later, Wang et al. [15] obtained an equivalent definition of $D_{\text{MAX}}$ using the notion of eccentricity of a vertex and called it the eccentricity matrix. Recall that the eccentricity of a vertex $v_{i}$ is defined by $e(v_{i})$= max $\{d(v_{i},v_{j})\mathrel{\mathop{\mathchar 58\relax}}v_{j}\in V_{G}\}$. The eccentricity matrix $E(G)$ of $G$ is an $n\times n$ matrix whose $(i,j)$-th entry is given by

This matrix has been well studied in the literature (see [7, 10, 8, 9, 11, 12, 15, 14, 16]) and has applications in Chemistry, see [12, 15, 14].

It is significant to note that the adjacency and distance matrices of connected graphs are always irreducible, but the eccentricity matrix fails to satisfy this property for a general connected graph. For example, the eccentricity matrix of a complete bipartite graph $K_{l,m}$ is reducible for all $l,m\geq 2$, see [15]. However, for some classes of graphs, the associated eccentricity matrices are irreducible. It is proved in [15] that the eccentricity matrix of a tree with at least two vertices is irreducible, and an alternative proof of this result is given in [8]. This result has been extended to some larger classes of graphs in [7] and [16]. It is shown in [11] that the eccentricity matrices of the coalescence of complete graphs are irreducible. Characterizing the graphs whose eccentricity matrices are irreducible, posed by Wang et al. [15], remains an open problem. In this paper, we provide a class of bi-block graphs $\mathscr{B}$, defined below, whose eccentricity matrices are irreducible.

To introduce the graph class $\mathscr{B}$, let us recall the following definitions. A block of a graph $G$ is a maximal connected subgraph of $G$ which has no cut-vertex. A connected graph $G$ is called a bi-block graph (block graph) if all its blocks are complete bipartite graphs (respectively, complete graphs) of possibly varying orders. Note that the complete bipartite graphs are bi-block graphs with exactly one block. Since the spectral properties of the eccentricity matrices of complete bipartite graphs are already explored in [8], it is sufficient to consider the bi-block graphs with at least two blocks. The main focus of this article is to analyze the spectral symmetry of the eccentricity matrices of bi-block graphs. In Section 4, it is shown that the spectral symmetry result is not true for a bi-block graph having more than two cut-vertices in a block. In view of these, we define the class $\mathscr{B}$ which is the collection of all bi-block graphs with at least two blocks and at most two cut-vertices in each block. Clearly, $\mathscr{B}$ contains all trees with at least three vertices.

In what follows, we present a brief survey of the literature on bi-block graphs. In [4], the authors studied the spectral radius of the adjacency matrix of a class of bi-block graphs with a given independence number. The permanent, determinant, and the rank of the adjacency matrices of bi-block graphs were computed in [13]. The inverse formula given for the distance matrix of a tree was extended to a subclass of bi-block graphs in [6]. Inspired by these, in this article, we study the eccentricity matrices of the subclass $\mathscr{B}$ of bi-block graphs.

A primary motivation for studying this paper comes from the following spectral symmetry results. In related to this, we first recall a well-known characterization for bipartite graphs. A graph $G$ is a bipartite graph if and only if the eigenvalues of the adjacency matrix $A(G)$ are symmetric with respect to the origin (i.e., if $\mu$ is an eigenvalue of $A(G)$ with multiplicity $l$ then -$\mu$ is also an eigenvalue of $A(G)$ with multiplicity $l$). In [10], the authors studied a similar equivalence for the eccentricity matrices of trees and gave a characterization for trees of odd diameters. Specifically, the eigenvalues of $E(T)$ of a tree $T$ are symmetric with respect to the origin if and only if $\operatorname{diam}(T)$ is odd. An analogous result has been proved for a subclass of block graphs (namely, clique trees) in [7]. The problem of characterizing the graphs whose eigenvalues of the eccentricity matrices are symmetric with respect to the origin, posed in [10], remains open. Motivated by these, in this article, we consider the spectral symmetry equivalence for the eccentricity matrices of graphs in $\mathscr{B}$. Precisely, we prove that the eigenvalues of $E(G)$ are symmetric with respect to the origin if and only if $\operatorname{diam}(G)$ is odd, whenever $G\in\mathscr{B}$ with $\operatorname{diam}(G)\geq 4$. By means of examples, we show that this equivalence does not hold for a general bi-block graph and for $G\in\mathscr{B}$ with $\operatorname{diam}(G)=3$.

To present another motivation for this paper, we now turn our attention to the inertias of graph matrices (i.e., matrices that arise from graphs). Let us recall the notion of inertia. Let $A$ be an $n\times n$ real symmetric matrix and let $i_{+}(A)$, $i_{-}(A)$ and $i_{0}(A)$ denote the number of positive, negative and zero eigenvalues of $A$, respectively, including the multiplicities. The inertia of $A$ is the ordered triple $(i_{+}(A),i_{-}(A),i_{0}(A))$, and is denoted by $\operatorname{In}(A)$. It is known that $i_{+}(A)+i_{-}(A)=\operatorname{rank}(A)$. An interesting and challenging problem in spectral graph theory is finding the eigenvalues and inertias of graph matrices, see [2, 3, 8, 9, 11, 15, 14, 16] and the references therein. Among other results, it has been proved that the inertia of the distance matrix of a tree on $n(\geq 2)$ vertices is $(1,n-1,0)$, see [2]. The inertias of the eccentricity matrices of lollipop graphs, coalescence of complete graphs, coalescence of two cycles, trees, and clique trees have been computed in [7, 10, 8, 11]. One of the objectives of this article is to find the inertias of the eccentricity matrices of the graphs in $\mathscr{B}$. Some of the results obtained and the ideas used in this paper are similar to those in [7, 10] but the proofs are different in many cases.

The outline of this article is as follows. In the next section, we recall some results and notation used in this article. In Section 3, we associate a tree $T_{G}$ for each $G\in\mathscr{B}$ and study their relationships where $\mathscr{B}$ is a subclass of bi-block graphs. Using these results, we derive the centers of graphs in $\mathscr{B}$. Section 4 deals with the inertia and the spectral symmetry of eccentricity matrix of $G\in\mathscr{B}$. Among other things, we give an equivalent condition for the spectrum of $E(G)$ to be symmetric with respect to the origin where $G\in\mathscr{B}$. Finally, we prove the irreducibility of the eccentricity matrices of graphs in $\mathscr{B}$.

In this section, we recall basic definitions and notation that will be used in the sequel.

We denote the vertex set and the edge set of a graph $G$ by $V_{G}$ and $E_{G}$, respectively. A graph $G^{\prime}=(V_{G^{\prime}},E_{G^{\prime}})$ is said to be a subgraph of $G$ if $V_{G^{\prime}}\subseteq V_{G}$ and $E_{G^{\prime}}\subseteq E_{G}$. A subgraph $G^{\prime}$ of $G$ is called an induced subgraph of $G$ if the edges of $G^{\prime}$ are precisely the edges of $G$ whose ends are in $V_{G^{\prime}}$. We denote the induced subgraph $G^{\prime}$ by $G[V_{G^{\prime}}]$ and call it as the subgraph induced by $V_{G^{\prime}}$. A vertex $v\in V_{G}$ is said to be a cut-vertex of $G$ if $G\setminus\{v\}$ is a disconnected graph. The notation $C_{G}$ stands for the collection of all cut-vertices of $G$. A maximal connected subgraph of a graph $G$ is known as a component of $G$. The radius and the diameter of $G$ are, respectively, denoted by $\operatorname{rad}(G)$ and $\operatorname{diam}(G)$, and are defined by $\operatorname{rad}(G)\mathrel{\mathop{\mathchar 58\relax}}=\min\{e(v)\mathrel{\mathop{\mathchar 58\relax}}v\in V_{G}\}$ and $\operatorname{diam}(G)\mathrel{\mathop{\mathchar 58\relax}}=\max\{e(v)\mathrel{\mathop{\mathchar 58\relax}}v\in V_{G}\}$. A path $P$ in $G$ is a subgraph of $G$ whose vertices are arranged in a sequence such that two vertices are adjacent in $P$ if and only if they are consecutive in the sequence. We denote a path of length $k$ between two vertices $a$ and $b$ in $G$ by $P_{G}(a,b)=au_{1}u_{2}\ldots u_{k-1}b$, where $u_{i}\in V_{G}$ for all $i$. We write the length of the path $P_{G}(a,b)$ by $l\left(P_{G}(a,b)\right)$. A diametrical path in $G$ is a shortest path between two vertices $u$ and $v$ such that $d(u,v)=\operatorname{diam}(G)$. A vertex $v\in V_{G}$ is said to be a central vertex if $e(v)=\operatorname{rad}(G)$. The center of $G$, denoted by $C(G)$, is the collection of all central vertices of $G$.

A graph $G$ is said to be bipartite if $V_{G}$ can be partitioned into two non-empty subsets $V_{1}$ and $V_{2}$ such that each edge of $G$ has one end in $V_{1}$ and the other end in $V_{2}$. The pair $(V_{1},V_{2})$ is called a bipartition of the bipartite graph $G$ and the sets $V_{1}$ and $V_{2}$ are referred to as the partite sets of $G$. A bipartite graph with bipartition $(V_{1},V_{2})$ is said to be a complete bipartite graph if every vertex of $V_{1}$ is adjacent to all the vertices of $V_{2}$, and is denoted by $K_{\lvert V_{1}\rvert,\lvert V_{2}\rvert}$ where $\lvert V_{1}\rvert$ stands for the cardinality of $V_{1}$. For more details on graph-theoretic notions and terminologies, we refer to the book [1].

Let $G\in\mathscr{B}$ and let $B$ be a block of $G$. Then $B$ is a complete bipartite graph $K_{l,m}$. Throughout this article, we assume that $\left(V_{1}(B),V_{2}(B)\right)$ is the bipartition of $B$. The block $B$ of $G$ is said to be a bridge block if $\lvert V_{B}\cap C_{G}\rvert=2$, and a leaf block if $\lvert V_{B}\cap C_{G}\rvert=1$.

Let $A$ be an $m\times n$ matrix. We write the transpose of $A$, $i$-th row of $A$, $i$-th column of $A$ and the rank of $A$ by $A^{\prime}$, $A_{i*}$, $A_{*i}$ and $\operatorname{rank}(A)$, respectively. We denote the principal submatrix of $A$ whose rows and columns are indexed, respectively, by the sets $U\subseteq\{1,2,\ldots,m\}$ and $V\subseteq\{1,2,\ldots,n\}$ by $A\left([U\mid V]\right)$. The notations $J$ and $O$ are used to denote the matrices with all elements equal to $1$ and $0$, respectively, and the orders of the matrices are clear from the context. The determinant of a square matrix $A$ is written by $\det(A)$.

In the following, we collect some known results which are needed in this paper.

In fact, the above result holds for any non-singular principal submatrix $A$ of $M$.

In this section, we associate a tree $T_{G}$ for each graph $G\in\mathscr{B}$ and obtain some interconnections between $G$ and $T_{G}$ by employing the properties of trees. In particular, we show that $\operatorname{diam}(G)=\operatorname{diam}(T_{G})$ and $C(T_{G})\subseteq C(G)$. Making use of these relations, we explicitly find the centers of graphs in $\mathscr{B}$. These results will be used in the next section to study the inertia and the spectral symmetry of the eccentricity matrix of a graph in $\mathscr{B}$.

The idea of analyzing a graph $G\in\mathscr{B}$ through the associated tree $T_{G}$ is motivated by [7, 10]. In [7], the authors studied the eccentricity matrices of clique trees by constructing trees of particular types. While defining trees, they considered non-cut-vertices only from the leaf blocks of clique trees. In our case, we have to include non-cut-vertices from both leaf blocks and bridge blocks of $G\in\mathscr{B}$ in order to obtain a tree satisfying specific properties and thus the construction is different from [7].

Let us begin this section by associating a subgraph $T_{G}$ for each $G\in\mathscr{B}$. It will be shown later that $T_{G}$ is a tree. We collect some selected vertices from each block $B$ of $G$ to define $T_{G}$. If $B=K_{1,1}$, then we collect both the vertices of $B$. Suppose that $B\neq K_{1,1}$ and $B$ is a bridge block with $V_{B}\cap C_{G}=\{z_{1},z_{2}\}$. If both the cut-vertices $z_{1}$ and $z_{2}$ lie on the same partite set, say $V_{1}(B)$, then we choose three vertices from $V_{B}$, which are $z_{1}$, $z_{2}$ and a non-cut-vertex in $V_{2}(B)$ with the minimum vertex label; otherwise, we collect exactly two vertices $z_{1}$ and $z_{2}$ from $V_{B}$. In the case of leaf block $B\neq K_{1,1}$, we collect exactly three vertices, which are the cut-vertex in $V_{B}$, and non-cut-vertices with the minimum vertex label in each partite sets $V_{1}(B)$ and $V_{2}(B)$. The graph $T_{G}$ is defined as the subgraph induced by the vertex set which is the union of all selected vertices in each block of $G$. The precise construction of $T_{G}$ is given below.

In the following, we mention a few properties of $G\in\mathscr{B}$ that are needed in the sequel. For more details, we refer to [1, 5].

• $(P_{1})$

  Suppose that $B$ is a block of $G$ with bipartition $(V_{1}(B),V_{2}(B))$. Then $\lvert V_{1}(B)\rvert=1$ if and only if $\lvert V_{2}(B)\rvert=1$.

• $(P_{2})$

  Any shortest path in $G$ can contain at most three consecutive vertices from a single block $B$ of $G$.

• $(P_{3})$

  Cycles of $G$ are exactly cycles of its blocks.

• $(P_{4})$

  Let $v\in V_{G}$. Then $v$ is a cut-vertex of $G$ if and only if it lies in at least two blocks of $G$.

Using the property $(P_{4})$ and the fact that each edge of $G$ lies on exactly one block of $G$ (P. $59$ in [1]), we obtain the following remark.

Many results of this section deals the relationship between $G\in\mathscr{B}$ and its associated graph $T_{G}$. We first show that the graph $T_{G}$ is a tree. In order to distinguish, we use the notations $d_{G}(a,b)$ and $d_{T_{G}}(a,b)$, respectively, to denote the distance between the vertices $a$ and $b$ with respect to $G$, and with respect to $T_{G}$.

The following lemma is useful in establishing the result $\operatorname{diam}(G)=\operatorname{diam}(T_{G})$. For notational simplicity, we also use the notation $d(a,b)$ to denote the distance between two vertices $a$ and $b$ in $G$ instead of $d_{G}(a,b)$.

The next two lemmas are the consequences of the above result.

Let $H$ be a connected graph and let $u\in V_{H}$. We denote the eccentricity of a vertex $u$ with respect to $H$ by $e_{H}(u)$, and the subscript is omitted if it is clear from the context.

Since most of the results of this paper hold for graphs in $\mathscr{B}$ with diameters at least four (for instance, see Example 4.2), we deal only with those graphs hereafter. In the following lemma, the eccentricity of a non-cut-vertex (when it exists) of $G\in\mathscr{B}$ is presented in terms of the eccentricities of the cut-vertices of $G$. In Section 4, we see the usefulness of this lemma in proving the spectral symmetry of $E(G)$.