Inclusive vertex irregular 1-distance labelings on triangular ladder graphs

For a simple and connected graph G, the distance between two vertices u, v ϵ V (G), denoted by d(u, v), is the length of the shortest path connecting them. A vertex labeling λ: V (G) → {1, ⋯, k} is called an inclusive vertex irregular d-distance labeling if the weights of the vertices are distinct, where the weight of a vertex v is defined as the sum of the vertex label v and all vertex labels u such that d(u, v) ≤ d. The minimal value of the largest label k of all such labeling of G is called the inclusive d-distance irregularity strength of G and is denoted by disd0(G). Lower and upper bounds for disd0(G) have already been investigated for any graph G with d = 1. The value of dis10(G) for some classes of graphs, such as paths, cycles, complete graphs, and other related types of graphs, have also been determined in the literature. In this paper, we find an upper bound for dis10(G) for triangular ladder graphs L_n, with n ≡ 4 mod 5, n > 4, and the exact value for the rest of the cases with n ≥ 3.