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 $m=\mathrm{\mid }E\mathrm{\mid }$. $d_i$ will denote the degree of vertex $v_i$ of $G$, and $\mathrm{\Delta }={\max }_{1\le i\le n}{d}_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij}{)}_{n\times n}$, where $m_{ij}=\sqrt{(d_i+d_j-2)\mathrm{/}(d_i{d}_j)}$ 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 $\rho_{ABC}(G) \leq \sqrt{\Delta+(2m-n+1)/\Delta -2}$\mathrm{.}Asanapplication,theuniquetreewith$n \geq 4$ vertices having second largest ABC spectral radius is determined.