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Keywords = distance signlees Laplacian matrix (spectrum)

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17 pages, 330 KiB  
Article
Bounds for the Generalized Distance Eigenvalues of a Graph
by Abdollah Alhevaz, Maryam Baghipur, Hilal Ahmad Ganie and Yilun Shang
Symmetry 2019, 11(12), 1529; https://doi.org/10.3390/sym11121529 - 17 Dec 2019
Cited by 16 | Viewed by 3100
Abstract
Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and [...] Read more.
Let G be a simple undirected graph containing n vertices. Assume G is connected. Let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian, D Q ( G ) be the distance signless Laplacian, and T r ( G ) be the diagonal matrix of the vertex transmissions, respectively. Furthermore, we denote by D α ( G ) the generalized distance matrix, i.e., D α ( G ) = α T r ( 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. Full article
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
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