On Edge Irregularity Strength of Products of Certain Families of Graphs with Path \(P_2\)

Abstract

For a simple graph \(G = (V, E)\), a vertex labeling \(\alpha: V \rightarrow \{1, 2, \ldots, k\}\) is called a \(k\)-labeling. The weight of an edge \(xy\) in \(G\), denoted by \(w_\phi(xy)\), 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 \(f\) there is \(w_\phi(e) \neq w_\phi(f)\). The minimum \(k\) for which the graph \(G\) has an edge irregular \(k\)-labeling is called the edge irregularity strength of \(G\), denoted by \(\mathrm{es}(G)\). In this paper, we determine the exact value for certain families of graphs with path \(P_2\).