A note on edge irregularity strength of firefly graph

Abstract

Let $G$ be a simple graph. A vertex labeling $\psi:V(G) \rightarrow \{1, 2,\ldots,\alpha\}$ is called $\alpha$-labeling. For an edge $uv \in G$, the weight of $uv$, written $z_{\psi}(uv)$, is the sum of the labels of $u$ and $v$, i.e., $z_{\psi}(uv)=\psi(u)+\psi(v)$. A vertex $\alpha$-labeling is said to be an edge irregular $\alpha$-labeling of $G$ if for every two distinct edges $a$ and $b$, $z_{\psi}(a) \neq z_{\psi}(b)$. The minimum $\alpha$ for which the graph $G$ contains an edge irregular $\alpha$-labeling is known as the edge irregularity strength of $G$ and is denoted by $\es(G)$. In this paper, we find the exact value of edge irregularity strength of different cases of firefly graph $F_{s,t,n-2s-2t-1}$ for any $s \geq 1, t \geq 1, n-2s-2t-1 \geq 1 $.