Further results on edge irregularity strength of graphs

Muhammad Imran (Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates)
Adnan Aslam (University of Engineering and Technology Lahore, Pakistan (RCET))
Sohail Zafar (Department of Mathematics University of Management and Technology, Lahore, Pakistan)
Waqas Nazeer (Division of Science and Technology, University of Education, Lahore, Pakistan)

Indonesian Journal of Combinatorics
Vol 1, No 2 (2017)

A vertex $k$-labelling $\phi:V(G)\longrightarrow \{1,2,\ldots,k\}$ is called irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, there is $w_{\phi}(e)\neq w_{\phi}(f)$; where the weight of an edge is given by $e=xy\in E(G)$ is $w_{\phi (xy)=\phi(x)+\phi(y)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labelling is called \emph{edge irregularity strength} of $G$, denoted by $es(G)$. In the paper, we determine the exact value of the edge irregularity strength of caterpillars, $n$-star graphs, $(n,t)$-kite graphs, cycle chains and friendship graphs.