Extremal graphs for average size of maximal matchings in bicyclic graphs

For a graph $G$, let $avm(G)$ denote the average size of its maximal matchings. This parameter was introduced by Engbers and Erey in the study of extremal problems for maximal matchings, and they asked for extensions from trees and unicyclic graphs to $k$-cyclic graphs. In this paper, we solve the first non-unicyclic case by determining the minimum value of $avm(G)$ over all connected bicyclic graphs with $n$ vertices and $n+1$ edges. We prove that, for every connected bicyclic graph $G$ of order $n\ge 5$, $$\mathrm{avm}(G)\ge \frac{4n-11}{2n-5}\mathrm{.}$$ Moreover, equality holds uniquely for the graph obtained from two triangles sharing a common edge by attaching all remaining $n-4$ pendant edges to one of the two vertices of degree $3$. The key point is to translate the minimization of $\mathrm{avm}(G)$ into structural restrictions on small maximal matchings, which are then analyzed through the three possible bicyclic core types.