On the Maximum ABC Spectral Radius of Connected Graphs and Trees

Let \(G=(V,E)\) be a connected graph, where \(V=\{v_1, v_2, \cdots, v_n\}\) and \(m=|E|\). \(d_i\) will denote the degree of vertex \(v_i\) of \(G\), and \(\Delta=\max_{1\leq i \leq n} d_i\). The ABC matrix of \(G\) is defined as \(M(G)=(m_{ij})_{n \times n}\), where \(m_{ij}=\sqrt{(d_i + d_j -2)/(d_i d_j)}\) if \(v_i v_j \in E\), and 0 otherwise. The largest eigenvalue of \(M(G)\) is called the ABC spectral radius of \(G\), denoted by \(\rho_{ABC}(G)\). Recently, this graph invariant has attracted some attentions. We prove that \(\rho_{ABC}(G) \leq \sqrt{\Delta+(2m-n+1)/\Delta -2}\). As an application, the unique tree with \(n \geq 4\) vertices having second largest ABC spectral radius is determined.