Distance signless Laplacian eigenvalues, diameter, and clique number

Let G be a connected graph of order n . Let D iag ( Tr ) be the diagonal matrix of vertex transmissions and let D ( G ) be the distance matrix of G . The distance signless Laplacian matrix of G is deﬁned as D Q ( G ) = D iag ( Tr ) + D ( G ) and the eigenvalues of D Q ( G ) are called the distance signless Laplacian eigenvalues of G . Let ∂ Q 1 ( G ) ≥ ∂ Q 2 ( G ) ≥ · · · ≥ ∂ Q n ( G ) be the distance signless Laplacian eigenvalues of G . The largest eigenvalue ∂ Q 1 ( G ) is called the distance signless Laplacian spectral radius. We obtain a lower bound for ∂ Q 1 ( G ) in terms of the diameter and order of G . With a given interval I , denote by m D Q ( G ) I the number of distance signless Laplacian eigenvalues of G which lie in I . For a given interval I , we also obtain several bounds on m D Q ( G ) I in terms of various structural parameters of the graph G , including diameter and clique number.


Introduction
Let G = (V (G), E(G)) be a simple connected graph with the vertex set V (G) = {v 1 , v 2 , . . . , v n } and edge set E(G). The order and size of G are |V (G)| = n and |E(G)| = m, respectively. The degree of a vertex v, denoted by d G (v) is the number of edges incident to the vertex v. In G, N G (v) is the set of all vertices which are adjacent to v. Further, K n denotes the complete graph on n vertices. In a graph G, the subset M ⊆ V (G) is called an independent set if no two vertices of M are adjacent. A clique is a complete subgraph of a given graph G. The cardinality of the maximum clique is called the clique number of G and is denoted by ω. A vertex u ∈ V (G) is called a pendant vertex if d G (u) = 1. For other standard definitions, we refer the reader to [6,11].
For v i , v j ∈ V (G), the distance between v i and v j , denoted by d ij or d G (v i , v j ), is the length of a shortest path between v i and v j . The diameter d (or d(G)) of a graph G is the maximum distance between any two vertices of G. The distance matrix of G, denoted by D(G), is defined as D(G) = (d ij ) vi,vj ∈V (G) . The transmission T r G (v i ) (we will write T r(v i ) if the graph G is understood) of a vertex v i is defined as the sum of the distances from v i to all other vertices in G, that is, , . . . , T r(v n )) be the diagonal matrix of vertex transmissions of G. For a connected graph G, Aouchiche and Hansen [4] defined the distance Laplacian matrix of G as D L (G) = Diag(T r) − D(G) (or simply D L ) and the distance signless Laplacian matrix as D Q (G) = T r(G) + D(G) (or simply D Q ). The eigenvalues of D Q (G) are called the distance signless Laplacian eigenvalues of G. Clearly, D Q (G) is a real symmetric matrix. We denote its eigenvalues by ∂ Q i (G)'s and order them as The largest eigenvalue ∂ Q 1 (G) is called the distance signless Laplacian spectral radius. Recent work on distance Laplacian matrix can be seen in [13,14]. For more work done on distance signless Laplacian matrix of a graph G, we refer the reader to [1][2][3][7][8][9]12,[15][16][17][18][19]. If the graph G is understood, we may write ∂ Q i in place of ∂ Q i (G) and refer the distance signless Laplacian eigenvalues as D Q − eigenvalues. Let m D Q (G) I be the number of distance signless Laplacian eigenvalues of G that lie in the interval I. Also, let m D Q (G) (∂ Q i (G)) be the multiplicity of the distance signless Laplacian eigenvalue ∂ Q i (G). In this paper, we obtain a lower bound for the distance signless Laplacian spectral radius of the graph G in terms of diameter d and order n. We show that the number of distance signless Laplacian eigenvalues in the interval [n − 2, dn] is at least d + 1, where d is the diameter of the graph G. We also obtain a lower bound for the number of distance signless Laplacian eigenvalues which fall in the interval (n − 2, 2n − 2), in terms of the order n and the number of vertices having degree n−1. Moreover, we show that the number of distance signless Laplacian eigenvalues in the interval [n−2, 2n−ω −2) is at most n − ω + 2, where n is the order and ω is the clique number of the graph G.

Distribution of distance signless Laplacian eigenvalues
We require the following lemmas to prove our main results.
A particular case of the well known min − max theorem is the following result.
Lemma 2.2. [20] If N is a symmetric n × n matrix with eigenvalues µ 1 ≥ µ 2 ≥ · · · ≥ µ n , then for any x ∈ R n (x = 0), we have where the equality holds if and only if x is an eigenvector of N corresponding to the largest eigenvalue µ 1 .

Lemma 2.3. [10]
Let M = (m ij ) be a n × n complex matrix having l 1 , l 2 , . . . , l p as its distinct eigenvalues. Then, If we apply Lemma 2.3 for the distance signless Laplacian matrix of a graph G with n vertices, we get

Theorem 2.1 (Cauchy Interlacing Theorem).
Let M be a real symmetric matrix of order n, and let A be a principal submatrix of M with order s ≤ n. Then In the following theorem, we give the lower bound for the distance signless Laplacian spectral radius of the graph G in terms of diameter d and order n.
Consider the n-vector y = (y 1 , y 2 , . . . , y d−1 , y d , y d+1 , . . . , y n ) T defined by Now, we have On substituting the above inequality in Inequality (2), we get The next result shows that the number of distance signless Laplacian eigenvalues in the interval [n − 2, dn] is at least d + 1, where d is the diameter of the graph G. Proof. We consider the principal submatrix, say M , corresponding to the vertices v 1 , v 2 , . . . , v d+1 which belong to the induced path P d+1 in the distance signless Laplacian matrix of G. Clearly, for all i = 1, 2, . . . , d + 1. Also, the sum of the off diagonal elements of any row of M is less than or equal to d(d + 1)/2. Using Lemma 2.3, we conclude that the maximum eigenvalue of M is at most dn. Using Lemma 2.1 and Theorem 2.1, we see there are at least d + 1 distance signless Laplacian eigenvalues of G which are greater than or equal to n − 2 and less than or equal to dn, that is An immediate consequence of Theorem 2.3 is the following result.
Proof. Since dn < 2T r max , by Lemma 2.1 and Inequality (1), we have Thus, using Theorem 2.3, we get For proving the next result, we need the following lemma which can be found in [5]. Now, we obtain a lower bound for the number of distance signless Laplacian eigenvalues which fall in the interval (n − 2, 2n − 2), in terms of the order n and the number of vertices having degree n − 1.

Equality holds when
Proof. We consider the following two cases.

Case 1.
Let m d = n, that is, G ∼ = K n . By Lemma 2.1, we see that the equality holds. By Lemma 2.4, we observe that n − 2 is a distance signless Laplacian eigenvalue of G with multiplicity at least m d − 1.
Also, we know that the distance signless Laplacian matrix corresponding to any connected graph H is symmetric, positive and irreducible. Therefore, by the Perron-Frobenius Theorem, ∂ Q 1 (H − uv) > ∂ Q 1 (H) whenever uv ∈ E(H) and H − uv is connected. As m d ≤ n − 1, therefore, G K n . Thus, from the above information ∂ Q 1 (G) > ∂ Q 1 (K n ) = 2n − 2. Hence, The following lemma is used in proving Theorem 2.5.

Lemma 2.5. [5]
Let G be a graph with n vertices.
The next result shows that the number of distance signless Laplacian eigenvalues in the interval [n − 2, 2n − 4) is at most n − p + 1, where n ≥ 3 is the order of G and p is the number of pendant vertices adjacent to common neighbour.
Proof. Clearly all the vertices in S form an independent set. Since all the vertices in S are adjacent to same vertex, therefore, all the vertices of S have the same transmission . Now, for any v i (i = 1, 2, . . . , p) of S, we have Next, we show that the number of distance signless Laplacian eigenvalues in the interval [n − 2, 2n ω − 2) is at most n − ω + 2, where n is the order and ω is the clique number of the graph G. Theorem 2.6. Let G be a connected graph of order n having clique number ω ≤ n − 1. If only one vertex of the corresponding maximum clique is adjacent to the vertices outside of the clique, then Proof. Let S = {v 1 , v 2 , . . . , v ω } be the set of vertices of the maximum clique such that v ω is the only vertex having neighbours outside of S. Clearly, the set of vertices N = {v 1 , v 2 , . . . , v ω−1 } also form a clique such that every vertex of N is adjacent to v ω only outside of N . It is easy to see that all the vertices belonging to N have the same transmission. For any v i ∈ N , i = 1, 2, . . . , ω − 1, we have T = T r(v i ) ≥ ω − 1 + 2(n − ω) = 2n − ω − 1.