Modular irregularity strength of vertex amalgamation and comb product path with cycle related graphs

Abstract

Consider a graph G = (V(G), E(G)), where V(G) is a nonempty set of vertices and E(G) is a set of edges. Let Z_n be the group of integers modulo n, and let k be a positive integer. A modular irregular labeling of a graph G of order n is a k-edge labeling ϕ : E(G) → {1, 2, … , k}, such that an induced weight function wt_ϕ : V(G) → Z_n is bijective. The weight function is defined as follows: wt_ϕ(u) = Σ_{u ∈ N(v)} ϕ(uv) (mod n) for all vertices v in V(G). The minimum value of k is called the modular irregularity strength of G, denoted as ms(G). Suppose G and H are two connected graphs, with G has order n. Vertex amalgamation of graphs G and H is a graph obtained by identifying one vertex from each graph. Suppose that o is a given vertex of H. The comb product of G ▷ H is the graph obtained by taking one copy of G and n copies of H and then attaching the vertex o of the i-th copy of H to the i-th vertex of G. In this paper, we discuss on the exact values of the modular irregularity strength for several graphs such as: vertex amalgamation of cycles; comb product path (or cycle) and cycle and comb product path (or cycle) and regular graphs.