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Open AccessArticle

The Generalized Distance Spectrum of the Join of Graphs

1
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood P.O. Box 316-3619995161, Iran
2
Department of Mathematics, University of Kashmir, Srinagar 190001, India
3
Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK
*
Authors to whom correspondence should be addressed.
Symmetry 2020, 12(1), 169; https://doi.org/10.3390/sym12010169
Received: 16 December 2019 / Revised: 10 January 2020 / Accepted: 11 January 2020 / Published: 15 January 2020
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.
Keywords: generalized distance matrix (spectrum); distance signless Laplacian matrix; joined union; lexicographic product; complete split graph; graph operation generalized distance matrix (spectrum); distance signless Laplacian matrix; joined union; lexicographic product; complete split graph; graph operation
MDPI and ACS Style

Alhevaz, A.; Baghipur, M.; A. Ganie, H.; Shang, Y. The Generalized Distance Spectrum of the Join of Graphs. Symmetry 2020, 12, 169.

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