1. For a perfect matching of a graph , a forcing set of is a subset of contained in no other perfect matchings of . The forcing number of the perfect matching is the cardinality of a forcing set of with the smallest size. A set of edges of is called an anti-forcing set of if has a unique perfect matching . The anti-forcing number of is the smallest cardinality of anti-forcing sets of . In this talk, we introduce necessary conditions and sufficient conditions for forcing sets and anti-forcing sets, and with which calculate forcing numbers and anti-forcing numbers for some extremal graphs.
2. Given a simple graph with , let be the set of all real symmetric matrices such that for all , if and only if is an edge in . The inertia set of is the set 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.|