完全图删边后的Zagreb指数研究

The Zagreb Indices of Complete Graphs Deleting Some Edges

摘要: 在分子图论中，第一与第二Zagreb指数是Gutman与Trinajstić于1972年提出的重要拓扑指数。第一Zagreb指数$M_1\left(G\right)$ 定义为所有顶点度数的平方和；第二Zagreb指数$M_2\left(G\right)$ 定义为所有相邻顶点对的度数乘积之和。这些指数是表征分子结构的重要工具。设${\mathcal{K}}_n^p$ 表示从完全图$K_n$ 中删去$p$ 条边所得到的所有图的集合，其中$n,p$ 为正整数，且$n\ge 2p,p\ge 2$ 。本文为属于${\mathcal{K}}_n^p$ 的图的Zagreb指数给出了精确的上界与下界，并刻画了达到这些界的极值图。由此，我们确定了在${\mathcal{K}}_n^p$ 中与Zagreb指数相对应的极值图。

Abstract: In molecular graph theory, the first and second Zagreb indices, $M_1\left(G\right)$ and $M_2\left(G\right)$ , are prominent topological descriptors introduced by Gutman and Trinajstić in 1972. The first Zagreb index $M_1\left(G\right)$ is calculated by summing the squares of each vertex’s degree, while the second Zagreb index $M_2\left(G\right)$ involves summing the products of the degrees of pairs of adjacent vertices. These indices serve as important tools in characterizing molecular structures. Let ${\mathcal{K}}_n^p$ be the set of graphs obtained from the complete graph $K_n$ by deleting $p$ edges, where $n,p$ are positive integers with $n\ge 2p$ and $p\ge 2$ . This paper establishes precise upper and lower bounds for the Zagreb indices of graphs from ${\mathcal{K}}_n^p$ , and identifies the extremal graphs where these bounds are achieved. As a result, we identify the extremal graph from ${\mathcal{K}}_n^p$ that corresponds to the Zagreb indices.