Generalized Arithmetic Staircase Graphs and Their Total Edge Irregularity Strengths

Abstract

Let Γ=(VΓ,EΓ) be a simple undirected graph with finite vertex set VΓ and edge set EΓ. A total n-labeling α:VΓ∪EΓ→{1,2,…,n} is called a total edge irregular labeling on Γ if for any two different edges xy and x′y′ in EΓ the numbers α(x)+α(xy)+α(y) and α(x′)+α(x′y′)+α(y′) are distinct. The smallest positive integer n such that Γ can be labeled by a total edge irregular labeling is called the total edge irregularity strength of the graph Γ. In this paper, we provide the total edge irregularity strength of some asymmetric graphs and some symmetric graphs, namely generalized arithmetic staircase graphs and generalized double-staircase graphs, as the generalized forms of some existing staircase graphs. Moreover, we give the construction of the corresponding total edge irregular labelings.

Total Edge Irregularity Strength of the Cartesian Product of Bipartite Graphs and Paths

“…In the following years, the exact value of total edge irregularity strength has been proven for hexagonal grid graphs [4]; barbell graph for n ≥ 3 [1]; staircase graphs, double staircase graphs, and mirror-staircase graphs [16]; generalized arithmetic staircase graphs and generalized double-staircase graphs [17].…”

On the 2-Token Graphs of Some Disjoint Union of Graphs

“…Consider a particular graph studied in [15] called a staircase graph (see also [16]). According to [15], the staircase graph SC n is the graph with vertex set,…”