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A distance antimagic graph is a graph G admitting a bijection f:V(G)→{1,2,…,|V(G)|} such that for two distinct vertices x and y, ω(x)≠ω(y), where ω(x)=∑y∈N(x)f(y), for N(x) the open neighborhood of x. It was conjectured that a graph G is distance antimagic if and only if G contains no two vertices with the same open neighborhood. In this paper, we study several distance antimagic product graphs. The products under consideration are the three fundamental graph products (Cartesian, strong, direct), the lexicographic product, and the corona product. We investigate the consequence of the non-commutative (or sometimes called non-symmetric) property of the last two products to the antimagicness of the product graphs.

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In computer science, graphs are used in variety of applications directly or indirectly. Especially quantitative labeled graphs have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V, E) with the vertex set V and the edge set E, a vertex k-labeling φ : V → {1, 2, . . . , k} is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f their w φ (e) = w φ (f ), where the weight of an edge e = xy ∈ E(G) is w φ (xy) = φ(x) + φ(y). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine the edge irregularity strengths of some chain graphs and the join of two graphs. We introduce a conjecture and open problems for researchers for further research.