A Study on C -Exponential Mean Labeling of Graphs

Abstract

A function $h$ is mentioned as a $C$ -exponential mean labeling of a graph $G\left(V,E\right)$ that has $s$ vertices and $r$ edges if $h:V\left(G\right)⟶\left\{1,2,3,\cdots ,r+1\right\}$ is injective and the generated function $h^{\ast }:E\left(G\right)⟶\left\{2,3,4,\cdots ,r+1\right\}$ defined by $h^{\ast }\left(ab\right)=⌈1/e{\left(\left(h{\left(b\right)}^{h\left(b\right)}\right)/\left(h{\left(a\right)}^{h\left(a\right)}\right)\right)}^{1/\left(h\left(b\right)-h\left(a\right)\right)}⌉,$ for all $ab\in E\left(G\right)$ , is bijective. A graph which recognizes a $C$ -exponential mean labeling is defined as $C$ -exponential mean graph. In the following study, we have studied the exponential meanness of the path, the graph triangular tree of $T_n$ , $C_m^{P_n}$ , cartesian product of two paths $P_m▫P_n$ , one-sided step graph of $S{T}_n$ , double-sided step graph of $2S{T}_{2n}$ , one-sided arrow graph of $A_r^s$ , double-sided arrow graph of $D{A}_r^s$ , and subdivision of ladder graph $S\left(L_t\right)$ .