On the Maximum ABC Spectral Radius of Connected Graphs and Trees

Abstract

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 $\Delta ={\max }_{1\le i\le n}{d}_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij}{)}_{n\times n}$, where $m_{ij}=\sqrt{(d_i+d_j-2)/(d_i{d}_j)}$ 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 ${\rho }_{ABC}(G)\le \sqrt{\Delta +(2m-n+1)/\Delta -2}$. As an application, the unique tree with $n\ge 4$ vertices having second largest ABC spectral radius is determined.

10 pages

References in corpus (1)

• Forbidden branches in trees with minimal atom-bond connectivity index

Cited by in corpus (1)

• Ordering Trees by Their ABC Spectral Radii