A note on the ABC spectral radius of graphs. Issue 5 (24th March 2022) - Chen, Xiaodan
ABSTRACT: The ABC matrix of a graph G, recently introduced by Estrada, is the square matrix of order | G | whose ( i, j ) -entry is ( d i + d j − 2 ) / ( d i d j ) if the i th vertex and the j th vertex of G are adjacent, and 0 otherwise, where d i is the degree of the i th vertex of G . The ABC spectral radius of G is the largest eigenvalue of the ABC matrix of G . In this paper, we completely characterize the (connected) graphs and (connected) triangle-free graphs which have the maximum, the minimum and second-minimum ABC spectral radius. These results answer a general question posed in [X. Chen, On ABC eigenvalues and ABC energy, Linear Algebra Appl. 544 (2018) 141–157] for (connected) graphs and (connected) triangle-free graphs.