On Friendly Index Set of Graphs

Accepted 18/Aug/2018, Online 30/Aug/2018 Abstract—A function f from V(G) to {0,1} where for each edge xy ,f*(xy) = (f(x) +f(y))(mod2), let vi(f) is the number of vertices v with f(v) = i and ei(f) is the number of edges e with f*(e) = i is called friendly if |v0(f) v1(f)| ≤ 1. The friendly index set of a graph G is FI(G) = {|e0(f) e1(f)|, where f runs over all friendly labelings f of G}. In this paper we find the friendly index set of the umbrella graph, Spl(K1,n), Globe graph ,P2+mk1 and union of a path and a star sharing a vertex in common. Keywords— Friendly labeling, Friendly index set, Umbrella graph, Spl(K1,n), Globe graph


INTRODUCTION
For all terminology and notations in graph theory we follow harary [3]. Unless mentioned or otherwise a graph in this paper shall mean a simple finite graph without isolated vertices. Let G be a graph with vertex set V(G) and edge set E(G). A labeling f:V(G)→Z 2 induces an edge labeling f*: E(G) → Z 2 defined by f*(xy) = f(x) + f(y) for each edge xy∈E(G). For i∈Z 2 , let v f (i) = card{v∈V(G):f(v)=i}and e f (i) = card {e∈E(G) : f*(e) = i}. A labeling f of a graph G is said to be friendly if |v f (0) -v f (1)| ≤ 1. The friendly index set of the graph G, FI(G) is defined as {|e f (0) -e f (1)| : the vertex labeling f is friendly}. Lee and Ng [3] define the friendly index set of graphs. This is a generalization of graph cordiality. This paper consists of four sections. Section I is the introduction part of friendly index set. Section II contains preliminaries and notations.Section III contains main results and illustrations and section IV contains conclusion part .

II. PRELIMINARIES AND NOTATIONS
Definition 2.1 [4] : A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions.
Definition 2.2 [10] : A fan graph f n obtained by joining all vertices of a path P n to further vertex, called the centre. Thus f n contains n+1 vertices c,v 1 ,v 2 , ..., v n and (2n-1) edges say cv i , 1 ≤ i ≤ n and v i v i+1 , 1 ≤ i ≤ n-1 Definition 2.3 [2] : Star K 1,n is the graph with one vertex of degree n called apex and n vertices of degree one.
Definition 2.4 [7] : A Umbrella graph U(m,n) is the graph obtained by joining a path P n with the central vertex of a fan f m .
Definition 2.5 [5] : For each vertex v of a graph G take a new vertex v. Join v 1 to all the vertices of G adjacent to v. The graph spl(G) thus obtained is called splitting graph of G.
Continuously proceeding like this ie, label x 1 by 0 and next continuously label up to vertices by 1 and vertices by 0 and then alternatively 0's and 1's starting with 1.
At the last step we get the vertex is labeled friendly and | e(0) -e(1)| = 0. Then keep x 1 , x 2 , x 3 , x 4 remain unchanged and then label all the other vertices ie, x 5 , x 6 ,...,x m by its complement. Again the vertex labeling is friendly and |e(0) -e(1)| = m-4.
Continue this process to get the friendly index set {2,4,...,m} Atlast change the labels x m and y 1 to its complement and we get the vertex is friendly and | e(0) -e(1)| = 0.

Case (i)
When n is even

Proof :
Consider a path P 2 with two vertices v 1 ,v 2 . Let y 1 ,y 2 ,...,y m be the m isolated vertices. Join v 1 ,v 2 with y i , 1 ≤ i ≤ m. The graph obtained is P 2 +mk 1 . The vertex set of G is V(G) = {v 1 ,v 2 ,y 1 ,y 2 ,...,y m }. The edge set of G is Union of a path and a star sharing a vertex in common has the friendly index set { 1,3,5,…..2n-5}

Proof
Let the n vertices of path P n be u 1 , u 2 , ….., u n. Let the n spokes of star S n be v 1 , v 2 , …. v n and u 1 be the centre vertex of the star. Identify u 1 and v 1. Let the graph so obtained is G.
Then | e(0)e(1) | = 2n-3 Now keep all the vertex remains unchanged and change the vertex u 3 and v 3 to its complement. This labeling is vertex friendly. Then | e(0)e(1) | = 2n-1 Continue this process upto changing the vertex and to its complement.

IV. CONCLUSION
Labeling of graphs in graph theory is one of the interesting research topic in the recent days. Here we have find new results of friendly index sets of five graphs related to labeling of graphs. Similar work can be carried out for other graphs also.