基于图的Euler-Sombor指标的一些界

Some Bounds on the Euler-Sombor Index of Graphs

摘要: 设$G$ 为一个简单图，$V\left(G\right)$ 表示图$G$ 的顶点集，$E\left(G\right)$ 表示图$G$ 的边集，$d_v$ 表示顶点$v$ 在图$G$ 中的度数。近年来，Euler-Sombor指标作为一种新型的基于顶点度的拓扑指标被提出，其被定义为$Eu\left(G\right)={\displaystyle \sum }_{uv\in E\left(G\right)}\sqrt{d_u^2+d_v^2+d_u{d}_v}$ 。本文研究了Euler-Sombor指标关于图的度序列和边的界，并给出了该指标关于第一Zagreb指标$M_1\left(G\right)$ 的上界与下界。

Abstract: Let $G$ be a simple graph, with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$ . For each vertex $v$ , let $d_v$ denote its degree in $G$ . Recently, the Euler-Sombor index has been proposed as a novel vertex degree-based topological index, defined as $Eu\left(G\right)={\displaystyle \sum }_{uv\in E\left(G\right)}\sqrt{d_u^2+d_v^2+d_u{d}_v}$ . In this paper, we investigate bounds on the Euler-Sombor with respect to the degree sequence and the edges of a graph, and establish upper and lower bound relating this index to the first Zagreb index $M_1\left(G\right)$ .