Super Edge-Magic Deficiency of Disjoint Union of Shrub Tree, Star and Path Graphs

Abstract

Let C = (M, N) be a finite, undirected and simple graph with |M(C)| = t and |N(C)| = s. The labeling of a particular graph is a function which maps vertices and edges of graph or both into numbers (generally +ve integers). If the domain of the given graph is the vertex-set then the labeling is described as a vertex labeling and if the domain of the given graph is the edge-set then the labeling is defined as an edge labeling. If the domain of the graph is the set of vertices and edges then the labeling defined as a total labeling. A graph will be termed as magic, if there is an edge labeling, using the positive numbers, in such a way that the sums of the edge labels in the order of a vertex equals a constant (generally called an index of labeling), without considering the choice of the vertex. An edge magic total labeling of a given graph comprising t vertices and s edges is a (1 − 1) function that maps the vertices and edges onto the integers 1, 2, . . . , t+s, with the intention that the sums of the labels on the edges and the labels of their end vertices are always an identical number, consequently they are independent of any specific edge. To a greater extent, we can define a labeling as super if the t least possible labels happen at the vertices. The Super edge-magic deficiency of a graph C, signified as µs(C), is the least non negative integer m0 so that C ∪m0K1 hasa Super edge-magic total labeling or +∞ if such m0 does not exist. In this paper, we will take a look at the Super edge-magic deficiencies of acyclic graphs for instance disjoint union of shrub graph with star, disjoint union of the shrub graph with two stars and disjoint union of the shrub graph with path.