Radio Heronian Mean k-Graceful Labeling on Degree Splitting of Graphs

Abstract

A mapping g:V\left(G\right)\rightarrow{k,k+1,\ldots,k+N-1} is a radio heronian mean k-labeling such that if for any two distinct vertices s and t of G, d\left(s,t\right)+\left\lceil\frac{g\left(s\right)+g\left(t\right)+\sqrt{g\left(s\right)g\left(t\right)}}{3}\right\rceil\geq1+D,for every s,t\in\ V(G), where D is the diameter of G. The radio heronian mean k-number of g, {rrhmn}_k(g), is the maximum number assigned to any vertex of G. The radio heronian mean number of G, {rhmn}_k(g), is the minimum value of {rhmn}_k(g) taken overall radio heronian mean labelings g of G. If {rhmn}_k(g)=\left|V\left(G\right)\right|+k-1, we call such graphs as radio heronian mean k-graceful graphs. In this paper, we investigate the radio heronian mean k-graceful labeling on degree splitting of graphs such as comb graph P_n\bigodot K_1, rooted tree graph {RT}_{n,n} hurdle graph {Hd}_n and twig graph\ {TW}_n.

A  mapping    is a radio heronian mean k-labeling such that  if for any two distinct vertices  and  of , ,for every V(G), where  is the diameter of . The   radio heronian mean k-number of g, , is the maximum number assigned to any vertex of . The   radio heronian mean number of , , is the minimum value of  taken overall radio heronian mean labelings  of . If , we call such graphs as  radio heronian mean k-graceful graphs. In this paper, we investigate the  radio heronian mean k-graceful labeling  on degree splitting of graphs  such as comb graph ,  rooted tree graph   hurdle  graph     and  twig graph.