Bounds for the Generalized Distance Eigenvalues of a Graph

Alhevaz, Abdollah, Baghipur, Maryam, Ganie, Hilal Ahmad and Shang, Yilun (2019) Bounds for the Generalized Distance Eigenvalues of a Graph. Symmetry, 11 (12). p. 1529. ISSN 2073-8994

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Official URL: https://doi.org/10.3390/sym11121529

Abstract

Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D(G) be the distance matrix, DL(G) be the distance Laplacian, DQ(G) be the distance signless Laplacian, and Tr(G) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by Dα(G) the generalized distance matrix, i.e., Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . In this paper, we establish some new sharp bounds for the generalized distance spectral radius of G, making use of some graph parameters like the order n, the diameter, the minimum degree, the second minimum degree, the transmission degree, the second transmission degree and the parameter α , improving some bounds recently given in the literature. We also characterize the extremal graphs attaining these bounds. As an special cases of our results, we will be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance eigenvalue.

Item Type: Article
Uncontrolled Keywords: distance matrix (spectrum); distance signlees Laplacian matrix (spectrum); (generalized) distance matrix; spectral radius; transmission regular graph
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Computer and Information Sciences
Depositing User: Elena Carlaw
Date Deposited: 17 Dec 2019 15:08
Last Modified: 17 Dec 2019 15:15
URI: http://nrl.northumbria.ac.uk/id/eprint/41757

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