A note on the \(ABC\) spectral radius of graphs. (English) Zbl 1485.05099
Summary: The \(ABC\) matrix of a graph \(G\), recently introduced by E. Estrada [J. Math. Chem. 55, No. 4, 1021–1033 (2017; Zbl 1380.92097)], is the square matrix of order \(|G|\) whose \((i,j)\)-entry is \(\sqrt{(d_i + d_j - 2)/(d_id_j)}\) 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, Linear Algebra Appl. 544, 141–157 (2018; Zbl 1388.05112)] for (connected) graphs and (connected) triangle-free graphs.
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