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 $Gin Omega(n,n+1)$ and $nge 6$. As a byproduct, we give the locally optimal graphs in $Omega(n, n + 1)$, for $n ge 8$.
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机译：对于具有完全可靠的边和不可靠的顶点的图$ G $，我们考虑了顶点相互独立地以恒定概率$ p $失效的$ G $的可靠性。由$ P_n（G，p）$表示的图$ G $的可靠性被定义为诱导的存活顶点子图连接的概率。用$ Omega（n，m）$表示具有$ n $个顶点和$ m $个边的连通图族。在本文中，我们确定$ R_n（G，p）$的每个系数的最优值以及$ Gin Omega（n，n + 1）$和$ nge 6 $的对应图。作为副产品，对于$ n ge 8 $，我们给出了$ Omega（n，n + 1）$的局部最优图。
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