Abstract: A proper[k]-edge coloring σ of graph G is a k-proper-edge-coloring of graph G using colors in[k]={1, 2, …, k}. Let w_σ(x) denote the sum of the colors of edges incident with x, i.e., w_σ(x)=∑_e∋x σ(e), and w_σ(x) is called the weight of the vertex x with respect to σ. A neighbor sum distinguishing edge coloring σ of G is a proper[k]-edge coloring of G such that no pair adjacent vertices receive the same weights. The smallest value k for which G has such a coloring is called the neighbor sum distinguishing edge chromatic number of G and denoted by χ'_∑(G). We obtained the exact values of this parameter for the lexicographic product P_n[H] of a path P_n and a connected simple graph H, where H is a Class 1 regular graph, a path, the complement of a complete graph, respectively.

Key words: path, lexicographic product, neighbor sum distinguishing edge coloring, neighbor sum distinguishing edge chromatic number