On total edge irregularity strength of arithmetic ladder graphs

Let $G$ be a simple, connected, and undirected graph. An edge irregular total $\kappa$-labeling of $G$ is a mapping that assigns each vertex and each edge an integer from $\{1,2,\dots ,\kappa \}$ such that distinct edges receive distinct weights, where the weight of an edge is defined as the sum of its label and the labels of its end vertices. The smallest such $\kappa$ is called the total edge irregularity strength of $G$, denoted by $tes(G)$. In this paper, we determine the exact value of $tes(G)$ for the class of arithmetic ladder graphs with $l$ levels, $k$ columns, and difference $d$, where $l\ge 2$, $k\ge 2$, and $d\ge 1$. The results are obtained via explicit constructions of optimal total labelings.