A note on the ABC spectral radius of graphs

The ABC matrix of a graph G, recently introduced by Estrada, is the square matrix of order vertical bar G vertical bar whose (0)-entry is root(d(i) + d(j) - 2)/(d(i)d(j)) if the ith vertex and the jth vertex of G are adjacent, and 0 otherwise, where d(i) is the degree of the ith 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.