ON THE GENERALIZED DISTANCE EIGENVALUES OF GRAPHS

Authors

  • A. Alhevaz Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran Author
  • M. Baghipur Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran Author
  • H. A. Ganie Department of School Education, JK Govt. Kashmir, India Author
  • K. C. Das Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea Author

DOI:

https://doi.org/10.57016/MV-vJ5201ZF

Keywords:

Generalized distance matrix (spectrum), spectral radius, hypercube, unicyclic graph

Subjects:

05C50, 05C12, 15A18

Abstract

For a simple connected graph G, the generalized distance matrix Dα(G) is defined as Dα(G)=αTr(G)+(1α)D(G),  0α1. The largest eigenvalue of Dα(G) is called the generalized distance spectral radius or Dα-spectral radius of G. In this paper, we obtain some upper bounds for the generalized distance spectral radius in terms of various graph parameters associated with the structure of graph G, and characterize the extremal graphs attaining these bounds. We determine the graphs with minimal generalized distance spectral radius among the trees with given diameter d and among all unicyclic graphs with given girth. We also obtain the generalized distance spectrum of the square of the cycle and the square of the hypercube of dimension n. We show that the square of the hypercube of dimension n has three distinct generalized distance eigenvalues.

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Published

2023-06-20

Issue

Section

Articles