Given a graph $G$, the number of its vertices is represented by $n(G)$, while the number of its edges is denoted as $m(G)$. An independent set in a graph is a set of vertices where no two vertices are adjacent to each other and the size of the maximum independent set is denoted by $\alpha(G)$. A matching in a graph refers to a set of edges where no two edges share a common vertex and the maximum matching size is denoted by $\mu(G)$. If $\alpha(G) + \mu(G) = n(G)$, then the graph $G$ is called a K\"{o}nig-Egerv\'{a}ry graph. Considering a graph $G$ with a degree sequence $d_1 \leq d_2 \leq \cdots \leq d_n$, the annihilation number $a(G)$ is defined as the largest integer $k$ such that the sum of the first $k$ degrees in the sequence is less than or equal to $m(G)$ (Pepper, 2004). It is a known fact that $\alpha(G)$ is less than or equal to $a(G)$ for any graph $G$. Our goal is to estimate the difference between these two parameters. Specifically, we prove a series of inequalities, including $a(G) - \alpha(G) \leq \frac{\mu(G) - 1}{2}$ for trees, $a(G) - \alpha(G) \leq 2 + \mu(G) - 2\sqrt{1 + \mu(G)}$ for bipartite graphs and $a(G) - \alpha(G) \leq \mu(G) - 2$ for K\"{o}nig-Egerv\'{a}ry graphs. Furthermore, we demonstrate that these inequalities serve as tight upper bounds for the difference between the annihilation and independence numbers, regardless of the assigned value for $\mu(G)$.

翻译：给定图$G$，其顶点数记为$n(G)$，边数记为$m(G)$。图中的独立集是指顶点之间互不相邻的顶点集合，最大独立集的大小记为$\alpha(G)$。图中的匹配是指边之间无公共顶点的边集合，最大匹配的大小记为$\mu(G)$。若$\alpha(G) + \mu(G) = n(G)$，则图$G$称为König-Egerváry图。考虑度序列为$d_1 \leq d_2 \leq \cdots \leq d_n$的图$G$，湮灭数$a(G)$定义为满足序列中前$k$个度之和小于等于$m(G)$的最大整数$k$（Pepper, 2004）。已知对任意图$G$，有$\alpha(G) \leq a(G)$。我们的目标是估计这两个参数之间的差值。具体地，我们证明了一系列不等式，包括：对于树，$a(G) - \alpha(G) \leq \frac{\mu(G) - 1}{2}$；对于二分图，$a(G) - \alpha(G) \leq 2 + \mu(G) - 2\sqrt{1 + \mu(G)}$；对于König-Egerváry图，$a(G) - \alpha(G) \leq \mu(G) - 2$。此外，我们证明无论$\mu(G)$取何值，这些不等式都是湮灭数与独立数之差的紧上界。