Graphs with most number of three point induced connected subgraphs Original Research Article

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, p) of G is the probability that the graph induced by the surviving nodes is connected. If Γ(n, e) is the collection of all n node e edge simple graphs, then G ϵ Γ(n, e) is uniformly most reliable if R(G, p) ⩾ R(G′, p) for all G′ ϵ Γ(n, e) and all 0 < p < 1. If S3(G) is the number of three point induced connected subgraphs of G, then G ϵ Γ(n, e) is S3-maximum if S3(G) ⩾ S3(G′) for all G′ ϵ Γ (n, e). It is known that if G ϵ Γ(n, e) is S3-maximum and p is sufficiently large then R(G, p) > R(G, p′) for all non-S3-maximum graphs G′ ϵ Γ (n, e). This paper characterizes the S3-maximum graphs in Γ(n, e) for the range e ⩽ (n22/4) + (2n− 3)/4.