The Study On The Graceful Labelings, The Cordial Labelings And The Friendly Index Sets Of Spider's Web Graph

Graceful graph and cordial graph are study topics in labeling graph, they are very inter-esting and important. They have good applications value and prospect of research. In the early 1960's, graceful graph once suggested that get people's attention. Let G = (V,E)be a graph with V(G) vertices and E(G) edges. Let |E(G)| have q edges, if there ex-ists f:V(G)? {0,1,2, …,q} being injective mapping, and define an induced function f':E(G)? {1,2,3, …,q} by setting f'{u,v) = |f(u) - f(v)| for all(u, v)?E(G). The graph G is called graceful graph.Let f be a function from the vertices of G to {0,1} and for each edge xy(xy ? E) assign the label |f(x) — f(y)|. Let v f (i) denote the number of vertices G(V, E)with i and e f(i)denote the number of edges G(V,E) with i. If |vf(1) — vf(0)| ?1, |ef(1) - ef(0)|?1;the graph is said to be a cordial graph.Let f be a labeling from V(G) to {0,1} and for each edge xy define f*(xy) = |f(x)-f(y)|. For i = and 1, let vf(i) denote the number of vertices with f(v) = i and ef(i) denote the number of edges with f*(e) = i. They call a labeling f friendly if |vf(1) - vf(0)|?1. In this way, we define the friendly index set of a graph G as FI(G)= {|ef(1)-ef(0)|} where f runs over all friendly labelings of G.Graceful graph has important applications in practice, for example in radio astronomy,cryptography, X ray, missile control code design, the whole voltage transmitter design, circuit design has been widely used. Especially in recent years, graceful graph and cordial graph research at home and abroad won a lot of research results. This paper mainly studies a class of graceful, cordial and friendly index set. In this paper, the main content and research results are as follows:In the first part, this paper introduces the study of graceful graph, cordial graph and friend-ly index set as well as the concept of graceful graph, cordial graph and friendly index set. It introduces the basic concepts of graph and a spider's web graph CW(m,n).In the next part, the graceful of a spider's web graph CW(4, n) is proven.In the last part, the cordial labelings of CW(m,n) and the friendly index set of CW(m, 2)(m? 3 is odd) are proven.