Further Results on Radio Number of Wedge sum of Graphs

Abstract

Let G be a simple connected graph with vertex set V and diameter d . An injective function c : V → { 1 , 2 , 3 , … } is called a radio labeling of G if | c ( x ) c ( y ) | + d ( x , y ) ≥ d + 1 for all distinct x , y ∈ V , where d ( x , y ) is the distance between vertices x and y . The largest number in the range of c is called the span of the labeling c . The radio number of G is the minimum span taken over all radio labelings of G . For a fixed vertex z of G , the sequence ( l 1 , l 2 , … , l r ) is called the level tuple of G , where l i is the number of vertices whose distance from z is i . Let J k ( l 1 , l 2 , … , l r ) be the wedge sum (i.e. one vertex union) of k ≥ 2 graphs having same level tuple ( l 1 , l 2 , … , l r ) . Let J ( l 1 l ′ 1 , l 2 l ′ 2 , … , l r l ′ r ) be the wedge sum of two graphs of same order, having level tuples ( l 1 , l 2 , … , l r ) and ( l ′ 1 , l ′ 2 , … , l ′ r ) . In this paper, we compute the radio number for some sub-families of J k ( l 1 , l 2 , … , l r ) and J ( l 1 l ′ 1 , l 2 l ′ 2 , … , l r l ′ r ) .