Sharp Bounds on (Generalized) Distance Energy of Graphs

Abdollah Alhevaz,Maryam Baghipur,Yilun Shang,Kinkar Ch Das

doi:10.3390/math8030426

Abdollah Alhevaz, Maryam Baghipur + Show 2 more

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https://doi.org/10.3390/math8030426

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Abstract

Given a simple connected graph G, let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian matrix, D Q ( G ) be the distance signless Laplacian matrix, and T r ( G ) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . Noting that D 0 ( G ) = D ( G ) , 2 D 1 2 ( G ) = D Q ( G ) , D 1 ( G ) = T r ( G ) and D α ( G ) − D β ( G ) = ( α − β ) D L ( G ) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.

Highlights

We study in this paper simple connected graphs G = (V, E) with V ( G ) = {v1, v2, . . . , vn } being the vertex set and E( G ) being the edge set
The notion of generalized distance energy of a graph G was first motivated in Alhevaz et al [17]
The distance and the distance signless Laplacian play a pivotal role in mathematics as they offer more information than the classical binary adjacency matrix

Summary

Introduction

The distance signless Laplacian energy of a graph G is defined as follows: EQ (G) = The following lemma characterizes the graphs with exactly two distinct generalized distance eigenvalues. = Kn or G is a n k-transmission regular graph with three different generalized distance eigenvalues represented as k, αk + 1 and αk − 1.

Results

Conclusion