Peifang Fu | Searcharxiv

arXiv2020

Ordering Trees by Their ABC Spectral Radii

Let $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$. Let $d_i$ denote the degree of vertex $v_i$. 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 ABC spectral radius of $G$ is the largest eigenvalue of $M(G)$. In the present paper, we establish two graph perturbations with respect to ABC spectral radius. By applying these perturbations, the trees with the third, fourth, and fifth largest ABC spectral radii are determined.

math.CO↗

arXiv2020

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 $Δ=\max_{1\leq i \leq n} d_i$. 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 $ρ_{ABC}(G)$. Recently, this graph invariant has attracted some attentions. We prove that $ρ_{ABC}(G) \leq \sqrt{Δ+(2m-n+1)/Δ-2}$. As an application, the unique tree with $n \geq 4$ vertices having second largest ABC spectral radius is determined.

math.SP↗