Generalized Polygonal Sum Labeling of Some Families of Graphs

Abstract

A graph $G$ with "p" vertices and "q" edges is called a Polygonal sum graph($PSG$) if it admits a labeling known as Polygonal sum labeling ($PSL$). $PSL$ is an injective function $h:V(G)\rightarrow N$, where $N$ represents the set of all non-negative integers that induces a bijection $h^+:E(G)\rightarrow\{P_K(1),P_K(2),\dots,P_K(q)\}$ of the edges of G defined by $h^{+}(uv)=h(u)+h(v)$ for every $e=uv\in E(G),$ where $P_K(1),P_K(2),\dots,P_K(q)$ are the first "q" polygonal numbers. In this paper we prove that Olive tress, Caterpillars $S(n_1,n_2,\dots,n_m)$ Shrub $St(n_1,n_2,\dots,n_m)$, Banana tree $Bt(n_1,n_2,\dots,n_m)$ and H-graphs are $PSG's$.