Related properties of pseudo--inverse graphs

• 1、School of Mathematics and Information Science, Yantai University, Yantai 264005

Abstract：The inverse of a graph is an important research topic in graph theory. The pseudo-inverse of a graph was proposed by D. Cvetkovi\'{c}, I. Gutman and S. Simic in 1978.A perfect matching of $G$ is a set of disjoint edges in $G$, and these edges contain all vertices of $G$.The pseudo--inverse $PI(G)$ of a graph $G$ is defined as a graph with the same vertex set as $G$, and the two vertices $x$ and $y$ in $PI(G)$ are adjacent if and only if $G-x-y$ has a perfect matching. If a graph is isomorphic to its selfpseudo--inverse graph(i.e. $G=PI(G)$), it is called a selfpseudo--inverse graph.In this paper, we mainly study several properties of the pseudo--inverses of graphs.First, we show that a connected graph $G$ has perfect matching if and only $PI(G)$ is connected. Second, we give the sufficient and necessary conditions for two vertices $u$ and $v$ being adjacent in $PI(G)$.Thirdly, we characterized graphs with perfect matchings such that $PI(G)=G^+$, where $G^+$ is the positive inverse of $G$.Finally, we first characterize selfpseudo--invertible unicyclic graphs with perfect matchings. Then, using this result, we further characterize selfpseudo--invertible bicyclic graphs with perfect matchings.

Keywords： Pseudo-inverse perfect matching bicyclic graph