Some results on the energy of the minimum dominating distance signless Laplacian matrix assigned to graphs

Subject Areas : Statistics

Abdollah Alhevaz ¹ ( Department of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran. )
Maryam Baghipur ² ( Department of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran. )
Ebrahim Hashemi ³ ( Department of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran. )

Received: 2018-03-13 Accepted : 2018-06-24 Published : 2018-03-01

Keywords: گراف, ماتریس لاپلاسین بدون علامت فاصله, انرژی ماتریس لاپلاسین بدون علامت فاصله, ماتریس فاصله, انرژی ماتریس لاپلاسین بدون علامت فاصله‌ی مینیمم احاطه‌گری,

Abstract :

Let G be a simple connected graph. The transmission of any vertex v of a graph G is defined as the sum of distances of a vertex v from all other vertices in a graph G. Then the distance signless Laplacian matrix of G is defined as D^{Q}(G)=D(G)+Tr(G), where D(G) denotes the distance matrix of graphs and Tr(G) is the diagonal matrix of vertex transmissions of G. For a given minimum dominating set of a graph G, our aim in this paper is to define and study the so called minimum dominating distance signless Laplacian matrix, denoted by MDD^{Q}(G). We study some properties of the matrix MDD^{Q}(G). We also define the minimum dominating distance signless Laplacian energy of a graph G, denoted by EDD^{Q}(G), as the sum of the absolute values of the eigenvalues of MDD^{Q}(G), and give some upper and lower bounds for the energy and spectral radius of MDD^{Q}(G).