Scattering Number And Wiener Polarity Index Of The Two Graphs

The structures of several kinds of interconnection networks can be modeled in termsof the corona of two predetermined networks. One example is an electric networks dis-tributed in a big city where each transformer must guarantee the energy supply of itscatchment area. To optimize resources, the transformers in the city are distributed intocatchment areas of the same entity. The structure consists of a network transformerscan be modeled by a graph G and the catchment area to which each transformer is con-nected can be modeled by a graph H. The topological structure of the network is thecorona G H of G and H.To measure the vulnerability of networks more properly, somevulnerability parameters have been introduced and studied. Among them are toughness,scattering number, integrity and rupture degree,and others, each of which measures notonly the difculty of breaking down the network but also the efect of the damage. Ingeneral, for most of the aforementioned parameters, the corresponding computing prob-lem is NP-hard. So it is of interest to give formulae or algorithms for computing theseparameters for special classes of graphs. In this paper we mainly discuss about the scat-tering number of the corona of two graphs and the Wiener polarity index of the corona oftwo graphs.The scattering number S(G) of a non-complete connected graph G=(V, E)is defined as S(G)=max {ω(G S)|S|: ω(G S)>1, S V (G)}, where the max-imum is taken over all cutsets S of vertices of G and ω(G S) denotes the number ofcomponents of the resultant graph G S by deletion of S, and the scattering number ofa complete graph is defined as∞. Moreover we present the Wiener polarity index ofthe corona and edge corona of two graphs. The edge corona of two graphs, G H is thegraph obtained by one copy of G and E(G) copy of H such that the two ends of an edge e,are joined to all vertices to He. The Wiener polarity index Wp(G) of a graph G=(V, E)is the number of unordered pairs of vertices {u, v} of G such that the distance dG(u, v)between u and v is3. The corona of two graphs G and H, written as G H, is the graphobtained by taking one copy of G and|V (G)|copies of H, and then joining the ith vertexof G to every vertex in the ith copy of H. In this paper we present a formula for thescattering number of the corona of two graphs.