GRAPHS WITH MOST NUMBER OF 3-POINT INDUCED CONNECTED SUBGRAPHS

Let G be a simple graph with e perfectly reliable edges and n nodes which fail independently and with the same probability p. The residual connectedness reliability R(G,rho) of G is the probability that the graph induced by the surviving nodes is connected. If Gamma(n,e) is the collection of all n node e edge simple graphs, then G is an element of Gamma(n,e) is uniformly most reliable if R(G,rho) greater than or equal to R(G',rho) for all G' is an element of Gamma(n,e) and all 0 < rho < 1. If S-3(G) is the number of three point induced connected subgraphs of G, then G is an element of Gamma(n,e) is S-3-maximum if S-3(G) greater than or equal to S-3(G') for all G' is an element of Gamma(n,e). It is known that if G is an element of Gamma(n,e) is S-3-maximum and rho is sufficiently large then R(G,rho) > R(G,rho') for all non-S-3-maximum graphs G' is an element of Gamma(n,e). This paper characterizes the S-3-maximum graphs in Gamma(n,e) for the range e less than or equal to (n(2)/4) + (2n - 3)/4.