Let $m_G$ denote the number of perfect matchings graph $G$. We introduce a combinatorial tools for determining parity and giving lower bound on power 2 dividing $m_G$. In particular, we certain vertex sets called channels, which correspond to elements in kernel adjacency matrix $G$ modulo $2$. A result Lov\'asz states that existence nontrivial channel is equivalent being even. give new proof th...

In this paper, we obtain the upper and lower bounds on the eccen- tricity connectivity index of unicyclic graphs with perfect matchings. Also we give some lower bounds on the eccentric connectivity index of unicyclic graphs with given matching numbers.

Matching preclusion is a measure of robustness in the event of edge failure in interconnection networks. The matching preclusion number of a graph G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching, and the conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resultin...