total edge irregularity strength

In the context of a finite undirected graph $\zeta$, an edge irregular labelling is defined as a labelling of its vertices with some labels in such a way that each edge has a unique weight, which is determined by the sum of the labels of its endpoints. The main objective of this study is to determine the smallest positive integer $n$ for which it is possible to assign a total edge irregular labelling to $\zeta$ with $n$ as the biggest label. This investigation focuses particularly on cases where $\zeta$ represents the generalized arithmetic and generalized geometric staircase graphs. Within this paper, the definition of generalized geometric staircase graph is proposed. Moreover, we not only establish the edge irregularity strength of these two kind of graphs but also present a method for creating the corresponding edge irregular labelling.

Let G=(V(G),E(G)) be a connected simple undirected graph with non empty vertex set V(G) and edge set E(G). For a positive integer k, by an edge irregular total k-labeling we mean a function f : V(G)UE(G) --> {1,2,...,k} such that for each two edges ab and cd, it follows that f(a)+f(ab)+f(b) is different from f(c)+f(cd)+f(d), i.e. every two edges have distinct weights. The minimum k for which G has an edge irregular total k-labeling is called the total edge irregularity strength of graph G and denoted by tes(G). In this paper, we determine the exact value of total edge irregularity strength for staircase graphs, double staircase graphs and mirror-staircase graphs.