Further results on the distance signless Laplacian spectrum of graphs

Abstract

The distance signless Laplacian matrix [Formula: see text] of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main entries are the vertex transmissions of [Formula: see text], and the spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of [Formula: see text]. In this paper, first we obtain the [Formula: see text]-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signless Laplacian spectral radius of a graph [Formula: see text] in terms of graph parameters and characterize the extremal graphs. Utilizing these results we present some upper and lower bounds on the distance signless Laplacian energy of a graph [Formula: see text].

Bounds for the Generalized Distance Eigenvalues of a Graph

“…with equality if and only if αTr 2 i + (1 − α)T i is same for i. Similar to Remark 2, it can be seen that the lower bound given by Theorem 4 is better than the lower bound given by (8) for all graphs G with T i ≥ Tr 2 i , for all i. Again, if in particular we take the parameter β in Theorem 4 equal to the vertex covering number τ, the edge covering number, the clique number ω, the independence number, the domination number, the generalized distance rank, minimum transmission degree, maximum transmission degree, etc, then Theorem 4 gives a lower bound for ∂(G), in terms of the vertex covering number τ, the edge covering number, the clique number ω, the independence number, the domination number, the generalized distance rank, minimum transmission degree, maximum transmission degree, etc.…”

“…Moreover, the transmission degree sequence and the second transmission degree sequence of C 4 are {4, 4, 4, 4} and {16, 16, 16, 16}, respectively. Now, putting β = Tr max = 4 in the given bound of Theorem 3, we can see that the equality holds: , 5 2 , 3 2 [2] , and then the distance signless Laplacian spectral radius is…”

“…(a) Let C 4 be the cycle of order 4. One can easily see that C 4 is a 4-transmission regular graph and the generalized distance spectrum of C 4 is {4, 4α, 6α − 2 [2] }. Hence, ∂(C 4 ) = 4.…”

“…Hence the generalized distance spectrum of C 4 is {4, 4α, 6α − 2 [2] }. Moreover, the signless Laplacian spectrum of C 4 is {2 [2] , 0 [2] }. Since the diameter of C 4 is 2, hence, applying Theorem 5, for k = 1, we have,…”

“…In [26], chromatic number is used to derive a lower bound for distance signless Laplacian spectral radius. The distance signless Laplacian spectrum has varies connections with other interesting graph topics such as chromatic number [10]; domination and independence numbers [21], Estrada indices [4,5,22,23,[34][35][36]38], cospectrality [11,42], multiplicity of the distance (signless) Laplacian eigenvalues [25,29,30] and many more, see e.g., [1][2][3]27,28,32].…”