Proper Lucky Labeling of Graph

Abstract

The labeling is said to be lucky labeling of the graph if the vertices of the graph are labeled by natural number with satisfying the condition that sum of labels over the adjacent of the vertices in the graph are not equal, and if vertices are isolated vertex, then the sum of the vertex is zero. The least natural number which labeled the graph is the lucky number. The lucky number of graph \(G\) is denoted by \(\eta \left( G \right)\). The labeling is defined as proper labeling if the vertices of the graph are labeled by natural number with fulfilling the condition that label of adjacent vertices is not the same. The labeling is defined as proper lucky labeling if labeling is proper and also lucky. The proper lucky number of graph \(G\) is denoted by \(\eta_{p} \left( G \right)\). Here, we obtain a proper lucky number for complete bipartite graph \(K_{m,n}\), friendship graph \(F_{n}\) and certain book graph such as triangular book graph \(B_{3}^{t}\) and rectangular book graph \(B_{4}^{t}\).