The Generalized Distance Spectrum of the Join of Graphs

Abdollah Alhevaz,Yilun Shang,Maryam Baghipur,Hilal A Ganie

doi:10.3390/sym12010169

Abdollah Alhevaz, Yilun Shang + Show 2 more

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https://doi.org/10.3390/sym12010169

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Abstract

Let G be a simple connected graph. In this paper, we study the spectral properties of the generalized distance matrix of graphs, the convex combination of the symmetric distance matrix D ( G ) and diagonal matrix of the vertex transmissions T r ( G ) . We determine the spectrum of the join of two graphs and of the join of a regular graph with another graph, which is the union of two different regular graphs. Moreover, thanks to the symmetry of the matrices involved, we study the generalized distance spectrum of the graphs obtained by generalization of the join graph operation through their eigenvalues of adjacency matrices and some auxiliary matrices.

Highlights

Complicated graph structures can often be built from relatively simple graphs via graph-theoretic binary operations such as products
Graph spectrum provides a unique way of characterizing graph structures, sometimes even identifying the entire graph classes
A regular graph has the same degree for all vertices

Summary

Introduction

Complicated graph structures can often be built from relatively simple graphs via graph-theoretic binary operations such as products. A regular graph has the same degree for all vertices. The sum of the distances from a vertex v to all other vertices, TrG (v) = ∑ duv , is called the transmission degree u ∈V ( G ). A k-transmission regular graph admits TrG (v) = k for any vertex v. We will show that the generalized distance spectrum of join of two regular graphs can be obtained from their adjacency spectrum. We determine the generalized distance spectrum of join of a regular graph with the union of two different regular graphs.

On the Generalized Distance Spectrum of Join of Graphs

The Generalized Distance Spectrum of the Joined Union