Distance matrices on the H − join of graphs: a general result and applications

Given a graph H with vertices 1, . . . , s and a set of pairwise vertex disjoint graphs G 1 , . . . , G s , the vertex i of H is assigned to G i . Let G be the graph obtained from the graphs G 1 , . . . , G s and the edges connecting each vertex of G i with all the vertices of G j for all edge ij of H . The graph G is called the H − join of G 1 , . . . , G s . Let M ( G ) be a matrix on a graph G . A general result on the eigenvalues of M ( G ) , when the all ones vector is an eigenvector of M ( G i ) for i = 1, 2, . . . , s , is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of G when G 1 , . . . , G s are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.


Introduction
The distance matrix of a graph G of order n is the n × n matrix D(G) = d i,j , indexed by the vertices of G, where d i,j is the distance (number of edges of a shortest path) between vertices v i and v j . The very beginning of the distance matrix research goes back to the thirties of the twentieth century [22] and [33]. The main motivation from graph theory point of view was the problem of realizability of distance matrices, first presented in [8] and investigated in [23,24,25,29], [5], [30] and [4], among many other papers. However, it was proven in [1] and [31] that the problem of determining the minimum weight graph realization of a distance matrix with integer entries is NP-complete. The distance matrices have also deserved the attention of the spectral graph theory community since the paper published by Graham and Pollak in 1971 [14], where a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems is established. In the same paper the authors also proved the following very impressive result, which drew the attention of many researchers, transforming the spectral properties of distance matrices in a hot topic of spectral graph theory. Assuming that T is a tree of order n ≥ 2 with distance matrix D(T), then detD(T) = (−1) n−1 (n − 1) 2 n−2 . (1) Thus, the determinant of the distance matrix of a tree depends only from its number of vertices. The concepts of distance Laplacian and distance signless Laplacian were introduced in [3], where the authors proven the equivalence between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. A very complete survey of the state of the art on distance matrices up to 2014 appears in [2] (see also [27]). More recently, extremal graphs were characterized in terms of the eigenvalues of distance signed Laplacian and distance Laplacian matrices. Namely, the graphs of order n with least distance signed Laplacian eigenvalue equal to n − 2 and the graphs of order n ≥ 11 with second least distance signed Laplacian eigenvalue in the interval [n − 2, n] [19]; the graphs of order n with largest distance Laplacian eigenvalue equal to n − 2 [18] (in this paper the authors proven the conjecture proposed in [2] that the multiplicity of the largest eigenvalue of the distance Laplacian matrix of a non complete graph is not greater than n − 2 with equality if and only if either the graph is isomorphic to a star or to a regular complete bipartite graph).
This paper is devoted to the determination of distance, distance Laplacian and distance signless Laplacian eigenvalues of graphs obtained by the H-joint operation [7] over a family of regular graphs. Furthermore, sharp lower bounds on distance incidence energy and distance Laplacian-energy like of a graph are deduced for graphs of prescribed order in terms of the vertex connectivity and the graphs for which these bounds are attained are characterized.
In the remaining part of this section, we introduce the notation and basic definitions of the concepts used throughout the paper.
Let G = (V (G) , E (G)) be a simple undirected graph of order n, that is, on n vertices, with vertex set V (G) and edge set E (G) . The cardinality of V (G) is called the order of G. If e ∈ E (G) has end vertices u and v, then we say that u and v are adjacent and this edge is denoted by uv.
The cardinality of N G (u) is said to be the degree of u. A graph G is called d-degree regular when every vertex has the same degree equal to d. The vertex connectivity (or just connectivity) of a graph G, denoted by κ (G) , is the minimum number of vertices of G whose deletion disconnects G. It is conventional to define κ (K n ) = n − 1. The adjacency matrix of a graph G of order n, A(G), is a 0 − 1-matrix of order n with entries a ij such that a ij = 1 if ij ∈ E(G) and a ij = 0 otherwise. Other matrices on G are the Laplacian matrix L (G) = D (G) − A (G) and the signless Laplacian matrix Q (G) = D (G) + L (G) , where D (G) is the diagonal matrix of vertex degrees. It is well known that L (G) and Q (G) are positive semidefinite matrices and that (0, 1) is an eigenpair of L (G) where 1 is the all ones vector. Let us denote by σ(M) the spectrum (the multiset of eigenvalues) of a square matrix M. An eigenvalue of a matrix M will be denoted by λ(M) and throughout the paper, assuming that M has order n, the eigenvalues M are indexed in non increasing order, that is, λ 1 (M) ≥ · · · ≥ λ n (M). Given an eigenvalue λ i (M) its eigenspace will be denoted by Λ λ i (M). If M is a nonnegative matrix then, by the Perron-Frobenius Theorem, M has an eigenvalue equal to its spectral radius, called the Perron root of M. In addition, if M is irreducible then the Perron root of M is a simple eigenvalue with a corresponding positive eigenvector, called the Perron vector of M.
Given two vertex disjoint graphs G 1 and G 2 , the join of G 1 and G 2 is the graph This join operation can be generalized as follows [6,7]: Let H be a graph of order s. Let V(H) = {1, . . . , s}. Let {G 1 , . . . , G s } be a set of pairwise vertex disjoint graphs. For 1 ≤ i ≤ s, the vertex i ∈ V(H) is assigned to the graph G i . Let G be the graph obtained from the graphs G 1 , . . . , G s and the edges connecting each vertex of G i with all the vertices of G j if and only if This graph operation introduced in [6] under the designation of H − join of the graphs G 1 , . . . , G s it is denoted by The same graph operation was introduced in [11] as a generalization of the lexicographic product [9,10]. In [11] this graph operation was designated generalized composition and denoted by H[G 1 , G 2 , . . . , G s ]. Clearly if each G i is a graph of order n i , then the H − join of G 1 , . . . , G s (generalized composition H[G 1 , G 2 , . . . , G s ]) is a graph of order n = n 1 + n 2 + . . . + n s . The same operation appear in [26] under the designation of joined union. In [26] the distance spectrum of the joined union of regular graphs is determined and this technique is applied to the construction of distance equienergetic graphs with diameter greater than two. As usual, let P n , C n , S n and K n be the path, cycle, star and the complete graph on n vertices, respectively.
, that is, the length of the shortest path connecting u and v. The transmission Tr(v) of a vertex v ∈ V(G) is the sum of the distances from v to all other vertices of G, that is, A graph H is said to be r− transmission regular if Tr(v) = r for each vertex v ∈ V(H). The eigenvalues of the distance matrix D(G) of the graph G are called the distance eigenvalues of G and they are denoted by In [3] Aouchiche and Hansen introduce, for a connected graph G, the distance Laplacian matrix L(G) and the signless Laplacian matrix Q(G), respectively, as follows is the diagonal matrix of the vertex transmissions in G. The eigenvalues of L(G) and Q(G) are called the distance Laplacian eigenvalues and the distance signless Laplacian eigenvalues of G and they are denoted by respectively. Notice that L(G) and Q(G) are both real symmetric matrices and then, from Geršgorin's Theorem, it follows that their eigenvalues are nonnegative real numbers. Let 1 be the all one vector. Clearly each row sum of L(G) is 0. Then (0, 1) is an eigenpair of L(G) and, when G is connected graph, 0 is a simple eigenvalue. It is clear that if G is a r− transmission regular graph, then where I n is the identity matrix of order n and, for i = 1, . . . , n, A basic result on ∂ L 1 (G) is the following: it is assumed that G i is a graph order n i such that the all one vector 1 n i is an eigenvector for the eigenvalue μ i of G i . Moreover, I and 0 are the identity and the zero matrices of the appropriate order, respectively.
Let M(G) be a matrix on a graph G. In this paper, a general result on the eigenvalues of M (G), when the all one vector is an eigenvector of M (G i ) for i = 1, 2, . . . , s, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of G when G 1 , . . . , G s are regular graphs.
Finally, we introduce the notions of the distance incidence energy and distance Laplacianenergy like of a graph and derive sharp lower bounds on these two distance energies among all the connected graphs G of order n, with m ≥ n edges and κ(G) ≤ k. The graphs for which those bounds are attained are characterized.

A general result on the H-join of graphs
Consider the vertices of G with the labels 1, . . . , ∑ s i=1 n i starting with the vertices of G 1 , continuing with the vertices of G 2 , G 3 , . . . , G s−1 and finally with the vertices of G s . With the above mentioned labeling, we get

Example 2 For the graph in Example 1, our labeling is
where the diagonal blocks M i are symmetric matrices such that for i = 1, . . . , s and δ i,j are scalars for 1 ≤ i < j ≤ s. Using a strategy similar to the one used in [6], the following lemmas are proven and the spectrum of the matrix M(G) is deduced as it is stated by Theorem 2. (2) and the eigenvalues μ i in (3), both for i = 1, . . . , s. Then

Lemma 1 Consider the block matrices M i which appear in the expression of M(G) in
and thus λ 1 ∈ σ(M (G)) with multiplicity m 1 . Similarly, and thus λ 2 , . . . , λ s ∈ σ(M (G)), with multiplicity m 2 , . . . , m s , respectively. Furthermore, for each i ∈ {1, . . . , s}, if μ i has multiplicity p i > 1, we can consider p i − 1 linear independent vectors from Λ μ i (M i ) \ {0} orthogonal to 1 i and using each of them in a similar way as above, we obtain p i − 1 linear independent eigenvectors of M(G) associated to μ i . Therefore, (2) and the s × s symmetric matrix

Lemma 2 Considering the matrix M(G) in
it follows that We claim that λ ∈ σ(M(G)) with an associated eigenvector The last equality is a consequence of (5). Hence M (G) y = λy and then σ(F s ) ⊆ σ(M(G)). 2 From Lemma 1 and Lemma 2, we get the following more strong result on the spectrum of Theorem 2 Consider the matrix M(G) (2) and the matrix F s (4). Then the spectrum of M (G) is

Determination of the eigenvalues of H-join distance matrices
In this section, we assume that H is a connected graph of order s and d i,j denotes the distance between i, j ∈ V(H). Here, we apply Theorem 2 to determine the eigenvalues of the distance matrix, distance Laplacian matrix and distance signless Laplacian matrix of the H-join G = . . , G s are regular graphs. Therefore, throughout this section we deal with the H-join of a family of regular graphs G 1 , . . . , G s .

Distance eigenvalues
Taking into account (2), we get that the distance matrix of the H-join G = H {G i : 1 ≤ i ≤ s}, where G i is a d i -degree regular graph of order n i , for each i = 1, . . . , s, it follows that and from (4) (6)), where J n i is the all one square matrix of order n i , I n i is the identity matrix of order n i and A(G i ) is the adjacency matrix of G i , for each i = 1, . . . , n. Further, as G i is a d i -degree regular graph it follows that M i 1 (7)) are given by
Taking into account the Remark 1, applying Theorem 2, we obtain the following corollary.

Corollary 1 Let H a connected graph of order s. If for each i ∈ {1, . . . , s}, G i is a d i -degree regular graph, then the spectrum of D(G), where G = H {G
where F s is the s × s matrix given in (7).
Notice that the matrices M i in the Theorem 2 are not necessarily the distance matrices D(G i ). This can be observed with the following example.

Distance Laplacian eigenvalues
for each i = 1, . . . , s and therefore, we can write for each i = 1, . . . , s. Taking into account (2), we get that the distance Laplacian matrix of the where D(G) is as in (6). Notice also that L(G)1 n = 0 n . On the other hand, from (4) we obtain where for each i = 1, . . . , s, λ n i (L i ) is as in (9). Therefore, applying Theorem 2, we obtain the following corollary. (11) and λ n i (L i ) is as in (9), for each i = 1, . . . , s.

Distance signless Laplacian eigenvalues
Finally, we consider the distance signless Laplacian matrix of the H-join of regular graphs. For i = 1, . . . , s, let us consider the matrices Q i = k i I n i + M i , where k i is again as in (10) and M i , λ 1 (M i ) are as in the Remark 1, for each i = 1, . . . , s. Moreover, notice that Q i 1 n i = λ 1 (Q i )1 n i , where for each i = 1, . . . , s. Taking into account (2), we get that the distance signless Laplacian matrix where D(G) is as in (6). Also, from (4) we obtain where for each i ∈ {1, . . . , s}, λ 1 (Q i ) is as in (12). Therefore, applying Theorem 2, we obtain the following corollary.

Corollary 3
Let H be a connected graph of order s. If for each i ∈ {1, . . . , s}, G i is a regular graph, the spectrum of where F s is the s × s matrix in (13) and λ 1 (Q i ) is as in (12) [4] , 13 [2] , 12, 10.0957, 9}.

Sharp lower bounds on the distance incidence energy and distance Laplacian-energy like
In this section, we define the distance incidence energy and the distance Laplacian-energy like of connected graphs. Then we derive sharp lower bounds on these two energies among the graphs G on n vertices in terms of connectivity. These lower bounds are attained if and only if Three well known energies on a graph are the (adjacency) energy, introduced by Gutman in 1978 as the sum of the absolute values of the eigenvalues of the adjacency matrix, the Laplacian energy introduced by Gutman and Zhou in [15] and the signless Laplacian energy which was defined in analogy with the Laplacian energy. The distance energy [28], the distance Laplacian energy [32] and the distance signless Laplacian energy [13] have been introduced as follows respectively. Other energies on a graph are the incidence energy [17] and the Laplacian-energy like [34]. For a graph G of order n, its incidence energy, denoted by IE(G), becomes where q 1 (G) ≥ q 2 (G) ≥ . . . ≥ q n (G), are the signless Laplacian eigenvalues of G; and its Laplacian-energy like energy, denoted by LEL(G), is From now on, V (n, k) is the family of connected graphs G of order n such that κ(G) ≤ k.
In [21] and in [34], sharp upper bounds on the incidence energy and on the Laplacian-energy like, respectively, for the graphs in V (n, k) are obtained. These upper bounds are attained if and only if G = K k ∨ (K 1 ∪ K n−k−1 ).
In this section we extend the above concepts of incidence energy and Laplacian-energy like to the distance matrix context, introducing the notions of distance Laplacian energy like, denoted by DLEL(G), and distance incidence energy, denoted by DIE(G), as follows In this study the following theorem plays a crucial role.

Theorem 3 ([3], Theorem 3.5) Let G be a connected graph on n vertices and m ≥ n edges. Consider the connected graph G obtained from G by the deletion of an edge.
• Let ∂ L 1 (G), . . . , ∂ L n (G) and ∂ L 1 ( G), . . . , ∂ L n ( G) be the distance Laplacian eigenvalues of G and G, respectively. Then ∂ L i ( G) ≥ ∂ L i (G) for i = 1, . . . , n.

Corollary 4 If G and G are connected graphs such that G is obtained from G by the deletion of an edge, then DLEL( G) > DLEL(G) and DIE( G) > DIE(G).
Corollary 5 Among the all connected graphs on n vertices and m ≥ n edges, the complete graph K n has the smallest distance Laplacian-energy like and the smallest distance incidence energy.
For i = 1, 2, 3, let G i be a d i −regular graph of order n i . Then G = G 1 ∨ (G 2 ∪ G 3 ) is a graph of order n = n 1 + n 2 + n 3 . Observe that G = G 1 ∨ (G 2 ∪ G 3 ) is a P 3 − join graph in which the central vertex of P 3 is assigned to G 1 , one pendent vertex of P 3 is assigned to G 2 and the other to G 3 .
Labelling the vertices of G = G 1 ∨ (G 2 ∪ G 3 ) starting with the vertices of G 1 , continuing with the vertices of G 2 and finishing with the vertices of G 3 , and using the results obtained in the Subsection 3.3, the distance signless Laplacian matrix Q (G) becomes where, for i = 1, 2, 3, and Clearly, the largest eigenvalues of Q 1 , Q 2 and Q 3 are with eigenvectors 1 n 1 , 1 n 2 and 1 n 3 , respectively. Applying Corollary 3, we get the following result.
and, for i = 1, 2, 3, G i is a d i −regular graph then where Q 1 , Q 2 , Q 3 are as in (15), (17) and Let n and k be positive integers, with k ≤ n − 1 and consider the graph where, without loss of generality, we assume 1 ≤ i ≤ n−k 2 . Then, for the graph G(i), the matrices Q 1 , Q 2 and Q 3 in (15) are respectively, and the matrix F 3 (G(i)) in (18) becomes Taking into account that the adjacency eigenvalues of K s are s − 1 and −1 with multiplicity s − 1, the spectra of Q 1 , Q 2 (i) and Q 3 (i) in (20) are where λ [t] denotes that λ is an eigenvalue with multiplicity t. One can obtain that the spectrum of where f 1,3 (i) = 2(n − 1) − k 2 ± ( k 2 ) 2 + 4i(n − k − i) and f 2 = n − 2. Now, applying Theorem 4 to Q(G(i)) the next corollary follows. (19). Then

Corollary 6 Let G(i) be the graph defined in
Let |S| be the cardinality of a finite set S and let us define W (n, k) as follows: Then, denoting by K 0 the empty graph (that is, the graph without edges and without vertices) we have the following theorem.
Proof Let G ∈ W(n, k). We first consider k = n − 1. From Corollary 5, DIE (G) ≥ DIE (K n ) with equality if and only if G = K n . Moreover b (n, n − 1) = 2(n − 1) Then the result is true for k = n − 1. Now let 1 ≤ k ≤ n − 2 and let G ∈ W(n, k) such that DIE (G) is a minimum. Let S ⊆ V (G) such that G − S is a disconnected graph. Let C 1 , C 2 , . . . , C r be the connected components of G − S. We claim that r = 2. If r > 2 then we can construct a new graph H = G ∪ {e} where e is an edge connecting a vertex in C 1 with a vertex in C 2 . Clearly, H ∈ W(n, k) and G = H − e. By Corollary 4, DIE (G) > DIE (H) , which is a contradiction. Therefore r = 2, that is, G − S = C 1 ∪ C 2 . By hypothesis |S| ≤ k. Now, we claim that |S| = k. Suppose |S| < k. Since G − S = C 1 ∪ C 2 , we may construct a graph H = G + e where e is an edge joining a vertex u ∈ V (C 1 ) with a vertex v ∈ V (C 2 ) . We see that H − S is a connected graph and the deletion of the vertex u disconnected it. This tell us that H ∈ W(n, k). By Corollary 4, DIE (G) > DIE (H) , which is also a contradiction.
We have proved DIE (G) ≥ DIE (G (i)) for all G ∈ W k n . We now search for the value of i for which DIE (G (i)) is minimum. From Corollary 6, Defining the function and observing that f (x) = f (n − k − x), for x ∈ [0, n − k], after some algebraic manipulation, we may conclude that f is a strictly increasing function in the interval [0, n−k 2 ]. Hence DIE (G) ≥ DIE (G (1)) for all G ∈ W(n, k). Moreover, since G (1) = K k ∨ (K 1 ∪ K n−k−1 ) and DIE (G (1)) = b (n, k), the inequality (21) holds as equality if and only if G = K k ∨ (K 1 ∪ K n−k−1 ) . 2 To find a sharp lower bound on the distance Laplacian-energy like among the graphs in W (n, k) is easier. Using the above mentioned labelling for the vertices of G = G 1 ∨ (G 2 ∪ G 3 ) and the results obtained in the subsection 3.2, we obtain where, for i = 1, 2, 3, and k i are as in (16). The smallest eigenvalues of L 1 , L 2 and L 3 are λ n 1 (L 1 ) = n 2 + n 3 , λ n 2 (L 2 ) = n 1 + 2n 3 and λ n 3 (L 3 ) = n 1 + 2n 2 , with eigenvectors 1 n 1 , 1 n 2 and 1 n 3 , respectively. Applying Corollary 2, the following theorem is obtained.
Applying Theorem 6 to L(G(i)), we obtain the next corollary.
By similar arguments to the ones used in the proof of Theorem 5, we get the following result.