On edge irregularity strength of line graph and line cut-vertex graph of comb graph

Abstract

For a simple graph $G$, a vertex labeling $\phi:V(G) \rightarrow \{1, 2,\ldots,k\}$ is called $k$-labeling. The weight of an edge $xy$ in $G$, written $w_{\phi}(xy)$, is the sum of the labels of end vertices $x$ and $y$, i.e., $w_{\phi}(xy)=\phi(x)+\phi(y)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, $w_{\phi}(e) \neq w_{\phi}(f)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this paper, we find the exact value of edge irregularity strength of line graph of comb graph $P_n \bigodot K_1$ for $n=2,3,4$; and determine the bounds for $n \geq 5$. Also, the edge irregularity strength of line cut-vertex graph of $P_n \bigodot K_1$ for $n=2$; and determine the bounds for $n \geq 3$.