We investigate the ratio $\avM(G)$ of the average size of a maximal matching to the size of a maximum matching in a graph $G$. If many maximal matchings have a size close to $\maxM(G)$, this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, $\avM(G)$ approaches $\frac{1}{2}$. We propose a general technique to determine the asymptotic behavior of $\avM(G)$ for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of $\avM(G)$ which were typically obtained using generating functions, and we then determine the asymptotic value of $\avM(G)$ for other families of graphs, highlighting the spectrum of possible values of this graph invariant between $\frac{1}{2}$ and $1$.

翻译：本文研究了图$G$中最大匹配的平均大小与最大匹配大小之比$\avM(G)$。若许多最大匹配的大小接近$\maxM(G)$，则该图不变量的值接近1。反之，若许多最大匹配的大小较小，则$\avM(G)$趋近于$\frac{1}{2}$。我们提出了一种通用技术来确定各类图$\avM(G)$的渐近行为。为说明该技术的应用，我们首先展示如何利用它找到$\avM(G)$的已知渐近值（这些值通常通过生成函数获得），随后确定了其他图族$\avM(G)$的渐近值，凸显了该图不变量在$\frac{1}{2}$到$1$之间可能取值的谱系。