Balanced Rank Distribution Labeling of Crown and Wheel Graphs
P. K. Hemalatha, S. Gokilamani · Global Journal of Pure and Applied Mathematics · 2023
A balanced rank distribution of a simple graph G of order n is defined with the following constraints; (i) For a given k ≥ n, there exist an injective function f : V (G) → {1, 2, ..., k} that gives the vertex labeling of G. (ii) There exists an ontoon the minimum bounded set B = er(G) of edge labelings, called the edge ranks.Then G is said to have a balanced rank distribution labeling if (i) the cardinality of er(G) is less than or equal to the minimum degree of G and (ii) a weakly balanced rank distribution labeling if er(G) lies between minimum and maximum degrees of G. Further, the balanced rank distribution number of G denoted by brd(G) is the minimum k ≥ n for which the defined rank distribution labelings exist.In this paper, we proved that the crown graph C n ⊙ K 1 admits a weakly balanced rank distribution labeling for n ≥ 3 and the wheel graph W n admits a balanced rank distribution labeling for 4 ≤ n ≤ 10 and weakly balanced rank distribution labeling for n ≥ 11.Further the balanced rank distribution number of the graphs C n ⊙ K 1 and W n have also been obtained for the given positive integer n.