On the distance α-spectral radius of a connected graph

Abstract

Abstract For a connected graph G and α ∈ [ 0 , 1 ) $\alpha \in [0,1)$ , the distance α-spectral radius of G is the spectral radius of the matrix D α ( G ) $D_{\alpha }(G)$ defined as D α ( G ) = α T ( G ) + ( 1 − α ) D ( G ) $D_{\alpha }(G)=\alpha T(G)+(1-\alpha )D(G)$ , where T ( G ) $T(G)$ is a diagonal matrix of vertex transmissions of G and D ( G ) $D(G)$ is the distance matrix of G. We give bounds for the distance α-spectral radius, especially for graphs that are not transmission regular, propose local graft transformations that decrease or increase the distance α-spectral radius, and determine the graphs that minimize and maximize the distance α-spectral radius among several families of graphs.

Keywords

• Distance spectral radius
• Distance signless Laplacian spectral radius
• Local graft transformation
• Extremal graph