AI 中文总结
该研究确定了顶点数为$n$的连通双环图中最大匹配平均大小的上界，刻画了达到上界的图类，完善了图的最大匹配平均大小的极值研究。
AI 中文摘要
对于图$G$，令avm($G$)表示其最大匹配的平均大小。Engbers与Erey开启了该参数的极值研究，并提出需将研究对象从树和单环图扩展到$k$-环图。本文确定了顶点数为$n$、边数为$n+1$的所有连通双环图中avm($G$)的最大值：当$n\boldsymbol{\text{≥5}}$且为奇数时，$\text{avm}(G)\boldsymbol{\text{≤}\frac{n-1}{2}}$，并刻画了所有达到等号的图；当$n=6$时，最大值为$\frac{13}{5}$，仅由$\boldsymbol{\text{Θ(1,3,3)}}$达到；当$n\boldsymbol{\text{≥8}}$且为偶数时，$\text{avm}(G)\boldsymbol{\text{≤}\frac{n}{2}-1+\frac{2}{n-4}}$，等号恰好由两种图达到：第一种是将两个$C_4$副本通过一条边连接，再在该连接边的一个端点附加$(n-8)/2$个悬挂2-路径得到的图；第二种是当$n\boldsymbol{\text{≥10}}$时，将两个$C_4$副本通过一条长度为2的路径连接，再在该连接路径的内部顶点附加1个叶节点和$(n-10)/2$个悬挂2-路径得到的图。证明过程结合了奇数阶极值图的结构刻画，以及基于完美匹配的计数与转换论证。
英文摘要
For a graph $G$, let avm($G$) denote the average size of its maximal matchings. Engbers and Erey initiated the extremal study of this parameter and asked for extensions from trees and unicyclic graphs to $k$-cyclic graphs. In this paper, we determine the maximum value of avm($G$) over all connected bicyclic graphs with $n$ vertices and $n+1$ edges. If $n\ge 5$ is odd, then \[ \text{avm}(G)\le \frac{n-1}{2}, \] and we characterize all graphs attaining equality. For $n=6$, the maximum value is 13/5, attained uniquely by $Θ(1,3,3)$. If $n\ge 8$ is even, then \[ \text{avm}(G)\le \frac{n}{2}-1+\frac{2}{n-4}. \] Equality holds precisely for the graph obtained from two copies of $C_4$ joined by an edge by attaching $(n-8)/2$ pendant 2-paths to one endpoint of the joining edge, and, when $n\ge 10$, for the graph obtained from two copies of $C_4$ joined by a path of length 2 by attaching one leaf and $(n-10)/2$ pendant 2-paths to the internal vertex of the joining path. The proofs combine structural characterizations of odd-order extremal graphs with counting and switching arguments based on perfect matchings.
Comments21 pages, 4 figures
