Graphs with Most Number of Three 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 ρ. The residual connectedness reliability R(G, ) ρ 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 n e ∈Γ( , ) is uniformly most reliable if R G R G ( , ) ( , ) ρ ρ ≥ ′ for all ′ ∈ G n e Γ( , ) and all 0 1 < < ρ . If S G 3( ) if the number of three point induced connected subgraphs of G, the G n e ∈Γ( , ) is S3-maximum if S G S G 3 3 ( ) ( ) ≥ ′ for all ′ ∈ G n e Γ( , ) . It is known that if G n e ∈Γ( , ) is S3maximum and ρ is sufficiently large then R G R G ( , ) ( , ) ρ ρ > ′ for all non S3-maximum graphs G′ ∈ Γ(n, e). This paper characterizes the S3-maximum graphs in Γ( , ) n e for the range e n n ≤ + − ( / ) ( ) / 2 4 2 3 4. * Supported by NSF Grant ECS−8600782