论文标题
在连接图和树的最大ABC光谱半径上
On the Maximum ABC Spectral Radius of Connected Graphs and Trees
论文作者
论文摘要
令$ g =（v，e）$为连接的图，其中$ v = \ {v_1，v_2，\ cdots，v_n \} $和$ m = | e | $。 $ d_i $将表示$ g $的顶点$ v_i $的程度，$δ= \ max_ {1 \ leq i \ leq n} d_i $。 $ g $的ABC矩阵定义为$ m（g）=（m_ {ij}）_ {n \ times n} $，其中$ m_ {ij} = \ sqrt {（d_i + d_i + d_j -2）/（d_i + d_i d_i d_i d_j）} $ if $ v_i v_i v_j v_j \ $否则， $ m（g）$的最大特征值称为$ g $的ABC光谱半径，由$ρ_{abc}（g）$表示。最近，该图不变引起了一些注意。我们证明$ρ_{abc}（g）\ leq \ sqrt {δ+（2m-n+1）/δ-2} $。作为应用程序，确定了具有第二大ABC光谱半径的$ n \ geq 4 $顶点的唯一树。
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.
