Abstract:
 1. For a perfect matching $M$ of a graph $G$, a forcing set $S$ of $M$ is a subset of $M$ contained in no other perfect matchings of $G$. The forcing number of the perfect matching $M$ is the cardinality of a forcing set of $M$ with the smallest size. A set $S$ of edges of $G$ is called an antiforcing set of $M$ if $GS$ has a unique perfect matching $M$. The antiforcing number of $M$ is the smallest cardinality of antiforcing sets of $M$. In this talk, we introduce necessary conditions and sufficient conditions for forcing sets and antiforcing sets, and with which calculate forcing numbers and antiforcing numbers for some extremal graphs.
2. Given a simple graph $G$ with $V(G)=\{1,2,\dots,n\}$, let $\mathcal{S}(G)$ be the set of all $n\times n$ real symmetric matrices $A=[a_{i,j}]$ such that for all $i\neq j$, $a_{i,j}\neq 0$ if and only if $ij$ is an edge in $G$. The inertia set of $G$ is the set $\{(p,q):A\in\mathcal{S}(G) \mbox { and } A \mbox{ has } p \mbox{ positive and } q \mbox{ negative eigenvalues}\}.$ The inertia sets of graphs was introduced by Barret, Tracy Hall and Loewy (2009). The inverse inertia problem is a problem between the minimum rank problem and the inverse eigenvalue problem. In this talk, I will introduce some properties about the inertia sets of graphs and the inertia sets of some class of graphs.
