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Let G =(V, E) be a graph and φ be a total (or an edge) coloring of G by using the color set {1, 2,…, k}.Let f(v) denote the sum of the color of the vertex v and the colors of all incident edges (or just the colors of all incident edges) of v.We say that φ is neighbor sum distinguishing if for each edge uv ∈ E(G), f(u) ≠ f(v).The smallest number k is called the neighbor sum distinguishing chromatic number (or neighbor sum distinguishing index).Such graph coloring problems can be translated into searching for a substitute for each variable of some polynomial such that the polynomial is nonzero.