The Isolated Scattering Number Of Some Graphs

Let G=(V(G),E(G)) be an undirected, simple and connected graph, and let S(?) V(G). When G is a complete graph.S is called a vertex cut of G if G-S is a trivial graph. When G is not a complete graph, G is called a vertex cut of G if G-S is not connected. The vertex-cut set of G is defined as C(G)={S:S is a vertex-cut set of G}. Let i(G) be the number of isolated vertices of G. The isolated scattering number of G, is defined as isc(G)=max{i(G-S)-|S|:S∈C{G)}. Let S*be a vertex cut. of G. S*is called a isolated scattering number set of G if S*satisfies isc(G)=i{G-S*)-|S*|. The complement graph of G, denoted by G. is a graph with the same vertex set of G and u.v∈V(G) is adjacent in G if and only if u, v is not adjacent in G. In this paper, we investigate the isolated scattering number of graph and this paper consists of five chapter.In Chapter1, some definitions and notations used in this thesis are introduced.In Chapter2. the maximum number of edges and minimum number of edges of connected graphs with given numbers of vertices and the isolated scattering numbers are investigated, and the main results are as follows:1) Let v(≥2) be an integer. G be a connected graph of order v with the maximum number of edges and isc(G)=B. Let S*be the isolated scattering number set of G, then two results are obtained:(1)If all of components of G-S*are isolated vertices, then(2)If all of components of G-S*are not isolated vertices, then1°If1≤B<≤v-4. then where a-[2/v-B-d].2°If-(v-4)≤B≤0; then|E(G)|=(v-12)+(1-B).3°If B=-(v-2),then |E(G)|=(v2)2) Let v(≥6) be an integer, G be a connected graph of order v with the minimum number of edges and isc(G)=B. Let S*be the isolated scattering number set of G, thenIn chapter3, the isolated scattering number of the self-complement graph of G is dis-cussed. The upper and lower bounds of the isolated scattering number of self-complement graph of G are given. The result are:-[2/v-3]≤isc(G)≤2.In chapter4, the isolated scattering number of the gear graph of G is discussed. The main results are as follows:(1) Let Gn be an gear graph, then isc(Gn)=1.(2) Let Gn be an complement graph of a gear graph Gn, then isc(Gn)=1-n.(3) Let Gn be an gear graph, then the isolated scattering number of the Cartesian product K2×Gn is isc(K2x Gn)=0.(4) Let n≥3and m≥3be positive integers, then the isolated scattering number of the Cartesian product Gn×Gm is isc(Gn×Gm)=1.(5) Let n≥5be a positive integer, then the isolated scattering number of the join graph with G3, G4,..., Gn is isc((G3∨G4)∪(G4∨G5)∪...∪(Gn-1∨Gn))=-8.In chapter5, the minimum isolated scattering number of a connected graph with m edges and order v are researched. The main results are as follows:Let G be a connected graph of m edges and order v, then where:v-1≤m≤(v2).