Computing the Edge Irregularity Strengths of Chain Graphs and the Join of Two Graphs

In computer science, graphs are used in variety of applications directly or indirectly. Especially quantitative labeled graphs have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V,E) with the vertex set V and the edge set E, a vertex k-labeling $\phi: V \rightarrow \{1,2,\dots, k\}$ is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f their $w_\phi(e) \ne w_\phi(f)$, where the weight of an edge $e=xy \in E(G)$ is $w_\phi(xy)=\phi(x)+\phi(y)$. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine the edge irregularity strengths of some chain graphs and the join of two graphs. We introduce a conjecture and open problems for researchers for further research

On The Edge Irregularity Strength of Firecracker Graphs F2,m

Let  be a graph and k be a positive integer. A vertex k-labeling  is called an edge irregular labeling if there are no two edges with the same weight, where the weight of an edge uv is . The edge irregularity strength of G, denoted by es(G), is the minimum k such that  has an edge irregular k-labeling. This labeling was introduced by Ahmad, Al-Mushayt, and Bacˇa in 2014.  An (n,k)-firecracker is a graph obtained by the concatenation of n k-stars by linking one leaf from each. In this paper, we determine the edge irregularity strength of fireworks graphs F2,m