讲座题目：Inverses of bicyclic graphs with a unique perfect matching

主讲人：魏二玲

讲座时间：2021.11.12（周五）14：00-15：00

讲座地点：腾讯会议690 515 624

主讲人简介：

魏二玲，中国人民大学副教授，硕士生导师，民进中国人民大学专家委员会委员。研究方向：图论与组合优化、图神经网络。毕业于北京交通大学，主持和参加国家自然科学基金多项。

主讲内容：

Let $G$ be a graph. The inverse of $G$ is a weighted graph $(G^{-1}, w)$ such that $V(G^{-1})=V(G)$ and $[w(ij)]_{n\times n}=A(G)^{-1}$ for each pair of vertices $i$ and $j$ of $G$, where $A(G)$ is the adjacency matrix of $G$. A weighted graph $(G,w)$ with weight function $w: E(G)\to \{-1,0,1\}$ is called a signed graph or a mixed graph. A perfect matching of a graph $G$ is a set of disjoint edges which cover all vertices of $G$. A simple connected graph $G$ of order $n$ is bicyclic if it has $n+1$ edges.Let $\mathcal B$ be the family of all bicyclic graphs with a unique perfect matching. In this paper, we show that every graph $G$ in $\mathcal{B}$ is invertible and its the inverse is bipartite if and only if $G$ is bipartite. Further, we characterize all graphs in $\mathcal{B}$ which have a signed graph as its inverse.