On the A α -spectral radius of a graph

By Jie Xue, Huiqiu Lin, Shuting Liu, Jinlong Shu. Linear Algebra and its Applications, 2018

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Abstract Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G. For any real α ∈ [ 0 , 1 ] , Nikiforov [3] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . The largest eigenvalue of A α ( G ) is called the A α -spectral radius of G. In this paper, we give three edge graft transformations on A α -spectral radius. As applications, we determine the unique graph with maximum A α -spectral radius among all connected graphs with diameter d, and determine the unique graph with minimum A α -spectral radius among all connected graphs with given clique number. In addition, some bounds on the A α -spectral radius are obtained.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2c7ec16a977508ada8da71ac1b90085f5896a224