On the Reliability of Unicyclic Networks with Vertex Failure

For a graph $G$ with perfectly reliable edges and unreliable vertices, we consider the reliability of $G$ for which vertices fail independently of each other with a constant probability $p$. The reliability of graph $G$, denoted by $P_n(G,p)$, is defined to be the probability that the induced sub graphs of surviving vertices connected. Denote by $\Omega(n,m)$ the family of connected graphs with $n$ vertices and $m$ edges. In this paper, we determine the optimal value of each coefficient of $R_n(G, p)$ and the corresponding graphs for $G\in \Omega(n,n+1)$ and $n\ge 6$. As a byproduct, we give the locally optimal graphs in $\Omega(n, n + 1)$, for $n \ge 8$.