New bounds on the distance Laplacian and distance signless Laplacian spectral radii

Let G be a simple undirected connected graph. In this paper, new upper bounds on the distance Laplacian spectral radius of G are obtained. Moreover, new lower and upper bounds for the distance signless Laplacian spectral radius of G are derived. Some of the above mentioned bounds are sharp and the graphs attaining the corresponding bound are characterized. Several illustrative examples are included.


Introduction
Let G = (V(G), E(G)) be a simple undirected graph on n vertices with vertex set V(G) = {v 1 , . . . , v n } and edge set E(G).
The distance between u, v ∈ V(G) for a connected graph G, denoted by d (u, v), is the length of the shortest path connecting u and v. The Wiener index W(G) of a connected graph G is d (u, v) and 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, The graph G is said to be k− transmission regular if Tr(v) = k for each vertex v ∈ V(G).
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 = d(v i , v j ). Two of the oldest works on the distance matrix are [19] (1935) and [26] (1938). In [10] (1964), the authors ask for the conditions under which a given a square symmetric matrix with real and nonnegative entries is realized as the distance matrix of a graph. Different realizability problems of distance matrices are investigated in [20,21,22,23], [7], [24] and [6], among many others papers. A remarkable result on the distance matrix, due to Graham and Pollak [12] (1971), is : If T is a tree of order n ≥ 2 with distance matrix D(T), then detD(T) = (−1) n−1 (n − 1) 2 n−2 .
Thus, the determinant of the distance matrix of a tree depends only on its order. The result established by Graham and Pollak attracted much interest among algebraic graph theory researchers.
The eigenvalues of D(G) are called the distance eigenvalues of G and denoted by Some properties about the eigenvalues of the distance matrix are already known. For example, in [28], the author characterizes the graphs with minimal spectral radius of the distance matrix in three classes of simple connected graphs with n vertices: with fixed vertex connectivity, matching number and chromatic number, respectively. In [14], the authors characterize all connected graphs with ∂ n (G) = −2. Furthermore, they characterize all connected graphs of diameter 2 with exactly three distance eigenvalues when ∂ 1 (G) is not an integer. They also conjecture that the complete k-partite graph is determined by its distance spectrum. Later, in [25], the author determines all graphs which satisfy ∂ n (G) ∈ [−2.383 , 0]. A very complete survey of the state of the art on distance matrices up to 2014 appears in [3] (see also [18]).
In [2] Aouchiche and Hansen introduce, for a connected graph G, the distance Laplacian matrix L(G) and the distance signless Laplacian matrix Q(G) as follows where is the diagonal matrix of the vertex transmissions in G.
In [2], among other results, the above mentioned authors prove the equivalence between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. 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. L(G) and Q(G) are both real symmetric matrices. From this fact and Geršgorin's Theorem, it follows that their eigenvalues are nonnegative real numbers. Let 1 be the all ones vector. Clearly each row sum of L(G) is 0. Then (0, 1) is an eigenpair of L(G) and, since G is connected graph, 0 is a simple eigenvalue.
Clearly, if G is a k− transmission regular graph then where I n is the identity matrix of order n; and, for i = 1, . . . , n, From the Perron-Frobenius Theory for nonnegative matrices, we have Theorem 1 If A is a nonnegative matrix then its spectral radius ρ(A) is an eigenvalue of A, and it has an associated nonnegative eigenvector. Furthermore, if A is irreducible then ρ(A) is a simple eigenvalue of A with an associated positive eigenvector.
In particular, since Q(G) is a positive matrix, ρ(Q(G)) is a simple eigenvalue of Q(G). The Frobenius norm of an n × n matrix M = (m i,j ) is . . , λ n (M) are the eigenvalues of M. In particular, Throughout this paper, we assume that G is a connected graph of order n and K n denotes the complete graph on n vertices. Some lower and upper bounds on ∂ L 1 (G) and ∂ Q 1 (G) are already known. The purpose of this paper is to search for new bounds on these spectral radii. Section 2, is dedicated to upper bounds on ∂ L 1 (G), we recall some known upper bounds as well as some previous results that allow us to derive new upper bounds. Finally, in Section 3, we recall some known results and we obtain new lower and upper bounds on ∂ Q 1 (G). In Section 2 as well as in Section 3, illustrative examples are given.

Bounds on
Theorem 2 Let G be a connected graph of order n ≥ 3. Then A basic result, due to Aouchiche and Hansen, concerning a graph with only two distinct distance Laplacian eigenvalues is Moreover, if the equality in (1) holds, then Tr(v i ) + (n − 1) for any i, j ∈ {1, . . . , n}.
Theorem 5 ([29], Theorem 4.3) Let G be a connected graph on n ≥ 4 vertices. Then Theorem 6 ([29], Theorem 4.4) Let G be a connected graph on n vertices. Then In the following propositions, we derive some new upper bounds on ∂ L 1 (G). We begin recalling a useful result due to Brauer [9] : Theorem 7 Let A be an n × n arbitrary matrix with eigenvalues be an eigenvector of A corresponding to the eigenvalue λ k and let q be any n-dimensional column vector. Then the matrix A + xq T has eigenvalues We next derive our first new upper bound on ∂ L 1 (G).
Theorem 8 Let G be a connected graph of order n. For i = 1, . . . , n, let Then If the equality in (5) holds then B(G) is a reducible matrix. This necessary condition for the equality in (5), it is not a sufficient condition. If B(G) is a irreducible matrix the inequality in (5) is strict.
Proof Since (0, 1) is an eigenpair for the distance Laplacian matrix L(G), using Theorem 7, we obtain that the eigenvalues of B(G) are is a repeated eigenvalue of the nonnegative matrix B(G) and then, from Theorem 1, it is a reducible matrix. The converse is not true (see Example 1 below).
Suppose that B(G) is a irreducible matrix. From Theorem 1, ρ(B(G)) is a simple eigenvalue and then ∂ L To four decimal places, the distance Laplacian eigenvalues of G are 0, 5, 5.5858, 7 and 8.4142. We There are also some results on upper bounds for the second largest modulus ξ (B) of the eigenvalues of a nonnegative matrix B. We recall the result [8] : From this inequality, (6) is immediate. ✷ Our next upper bound for ∂ L 1 (G) is given in the following theorem.
Theorem 10 If G is a connected graph of order n then and this upper bound does not exceed the upper bound given in (5).
Proof From the proof of Theorem 8, we have that 1 is an eigenvector corresponding to ρ(B(G)) with B(G) = L(G) + 1p T as defined in (4) . Applying Theorem 9 to B(G), we obtain We have Replacing in (8), (7) is obtained. From Corollary 1, the right hand side of (7) does not exceed We now recall the following lemma that will play an important role in getting another upper bound on ∂ L 1 (G).

Theorem 11
If G is a connected graph of order n then The equality holds if and only if G = K n or G is a connected graph with three distinct distance Laplacian and 0.
Proof The distance Laplacian eigenvalues of G are Applying Lemma 1, we get Since the upper bounds (9) and (10) are equivalent. Moreover, from Lemma 1, the equality in (10) holds if and only if Therefore, the equality in (9) holds if and only if ∂ L then G is a graph in which the multiplicity of ∂ L 1 (G) is n − 1 and thus, from Theorem 3, and 0. The proof is complete. ✷ In the following example, we apply the above upper bounds to the transmission regular graph but not degree regular graph of the smallest order [2].

Proposition 12
Let G be a connected k− transmission regular graph of order n. Then 3. Let Hence c 2 ≤ c 1 . Thus (11) is immediate from (3).
2. Moreover, for a such graph G, we have Hence (12) is immediate from (9).
3. For brevity, let ∆ = D(G) F . We will prove that c 1 − c 2 ≥ 0. We have Since the last inequality is clearly true, it follows that c 1 − c 2 ≥ 0.

✷
In the next example, we apply the above upper bounds to a graph which is not a transmission regular graph.

Bounds on ∂ Q 1 (G)
We begin observing that for a k− transmission regular graph its distance signless Laplacian spectral radius is 2k. In fact, Q(G)1 = 2k1 and thus, by Theorem 1, ∂ Q 1 (G) = 2k. For instance, the graph in Example 2 is 14− transmission regular graph and then ∂ Q 1 (G) = 28.
From now on, for i = 1, . . . , n, R i (M) denotes the sum of the i− row of a matrix M of order n.
We recall another important result on nonnegative matrix.
Theorem 13 [13] Let A = (a i,j ) be an n × n nonnegative matrix with spectral radius ρ(A). Then Moreover, if A is an irreducible matrix, then equality holds on either side (and hence both sides) of (13) if and only if all row sums of A are equal.
Moreover, equality holds on either side (and hence both sides) of (14) if and only if G is a transmission regular graph.
Proof Since Q(G) is a positive matrix and 2Tr(v i ) = R i (Q(G)) for i = 1, . . . , n, the corollary is an immediate consequence of Theorem 13. ✷ We mention below some already known bounds on ∂ Q 1 . For a simple connected graph G with n vertices such that Tr(v 1 ) ≥ Tr(v 2 ) ≥ · · · ≥ Tr(v n ), the authors in [11] defined the second distance degree of a vertex v i ∈ V(G), denoted by T i , as Theorem 14 ([11], Theorem 3.8) Let G be a simple connected graph with n vertices such that Tr(v 1 ) ≥ Tr(v 2 ) ≥ · · · ≥ Tr(v n ). Then, Moreover, any equality holds if and only if G has the same value Tr(v i ) + T i Tr(v i ) for all i.
The following lemma is proved in [15] and [16].

Lemma 2
Let G be a connected graph and let p(x) be a polynomial on x. Let q 1 (G) the largest signless Laplacian eigenvalue of the matrix Q(G) = D(G) + A(G). Then Moreover, if the row sums of p(Q(G)) are not all equal, then both inequalities in (20) are strict.
The following two results are neccesary for the proof of Theorem 18 (below). In the next lemma, we extend Lemma 2 to the distance signless Laplacian matrix.

Lemma 3 Let G be a connected graph and let p(x) be a polynomial on x. Then
{R i (p(Q(G)))}. (21) Moreover, if the row sums of p(Q(G)) are not all equal, then both inequalities in (21) are strict.
Proof Q(G) is a positive matrix. Then there exists a positive vector x = [x 1 , . . . , x i R i (p(Q(G))).
Since the entries of x are positive and their is equal to 1, we have and thus the result follows. ✷

Theorem 17
Let G be a connected graph on n vertices. Let T and t be the maximum and the minimum transmissions of G, respectively. Then, for any u ∈ V(G), Using (22), we obtain Also, using (22), we get This completes the proof. ✷ Let e i be the n− dimensional vector of zeros except for the i− entry equal to 1.

Theorem 18
Let G be a connected graph on n vertices. Let T and t the maximum and the minimum transmissions of G, respectively. Then and Proof Since Q(G) = Tr(G) + D(G), we have Q 2 (G) = Tr 2 (G) + Tr(G)D(G) + D(G)Tr(G) + D 2 (G).
Therefore, the i-th row sum of Q 2 (G) is From Theorem 17 and (25), we have and Let From (26) and (28), we obtain From (29), for i = 1, . . . , n, we have From this inequality and Lemma 3, we get This inequality allows to conclude the bound in (23). Similarly, using the polynomial p(x) = x 2 − (T − 1)x and (27), the upper bound in (24) can be obtained. ✷
We now recall the following result.
Lemma 4 [5] A connected graph G has only two distinct distance signless Laplacian eigenvalues if and only if G is a complete graph.
Finally, we derive another new upper bound on the largest distance singless Laplacian eigenvalue.

Theorem 19
If G is a connected graph of order n then The equality holds if and only if G = K n .
Applying Lemma 1, we get with equality if and only if the upper bound (31) is equivalent to with the necessary and sufficient condition for the equality given in (32). We claim that the equality in (33) holds if and only if G = K n . Suppose the equality in (33). Then the eigenvalues ∂ Q 2 (G), . . . , ∂ Q n (G) satisfy (32) and thus Moreover, from the fact that Q(G) is a positive matrix, ∂ Q 1 (G) is a simple eigenvalue. Hence the equality in (33) implies that G has only two distinct distance signless Laplacian eigenvalues. Hence, from Lemma 4, G = K n . Conversely, using the fact that the distance signless Laplacian eigenvalues of K n are ∂ Q 1 (K n ) = 2n − 2 and ∂ Q i (K n ) = n − 2, for i = 2, . . . , n, one can easily see that the equality in (33) holds. The proof is complete. ✷ Finally, we apply some bounds presented in this section to the graphs given in the following example, in each vertex is indicated its corresponding transmission. These graphs are taken from [1] and they are the all regular but not transmission regular graphs on 12 vertices. As before, the results are given to four decimal places. Consider now the graph G 2 displayed below. For G 2 , we have ∂ Q 1 (G 2 ) = 47.5268 and Lower bounds: An immediate conclusion from the above example is that the lower bounds (17) and (23) as well as the upper bounds (18) and (24) are not comparable. In fact, (i) (23) gives a better lower bound for ∂ Q 1 (G 3 ) than (17) does, but this is not the case for the graphs G 1 and G 2 in which (17) gives better lower bounds and (ii) (24) gives a better upper bound for ∂ Q 1 (G 1 ) than (18) does, but this is not the case for the graphs G 2 and G 3 in which (18) gives better upper bounds. Finally, we recall if G is k− transmission regular graph, that is, if t = T = k then ∂ Q 1 (G) = 2k. Then, we can expect for tight bounds in (23) and (24) if 0 < T − t ≤ 2.