Let G = (V,E) be a graph of order n. Let f : V → {1, 2, . . . , n} be a bijection. For any vertex v ∈ V , the neighbor sum ∑u∈N(v) f(u) is called the weight of the vertex v and is denoted by w(v). If w(v) = k, (a constant) for all v ∈ V , then f is called a distance magic labeling with magic constant k. If the set of vertex weights forms an arithmetic progression {a, a+ d, a+2d, . . . , a+ (n− ...

An (a, d)-edge-antimagic total labeling of G is a one-to-one mapping g taking the vertices and edges onto 1, 2, . . . , |V (G)| + |E(G)| so that the edgeweights w(uv) = g(u) + g(v) + g(uv), uv ∈ E(G), form an arithmetic progression with initial term a and common difference d. Such a labeling is called super if the smallest labels appear on the vertices. In this paper, we investigate the existen...

Let G=(V,E,F) be an embedding of a graph and a,b,c belong to {0,1}. A labeling type (a,b,c) is assignment labels from the set {1,2,...,a|V|+b|E|+c|F|} so that each vertex receives labels, edge b face c no label used more than once. If weight every s-sided face, defined as sum with vertices edges border equal some fixed constant w(s), then said face-magic. We investigate face-magic labelings all...

Let G = (V,E) be a graph of order n. A closed distance magic labeling of G is a bijection l : V (G) → {1, . . . , n} for which there exists a positive integer k such that ∑ x∈N [v] l(x) = k for all v ∈ V , where N [v] is the closed neighborhood of v. We consider the closed distance magic graphs in the algebraic context. In particular we analyze the relations between the closed distance magic la...

Let G = (V, E) be a graph with v vertices and e edges. An (a, d)-vertex-antimagic total labeling is a bijection λ from V (G) ∪ E(G) to the set of consecutive integers 1, 2, . . . , v + e, such that the weights of the vertices form an arithmetic progression with the initial term a and common difference d. If λ(V (G)) = {1, 2, . . . , v} then we call the labeling a super (a, d)-vertex-antimagic t...

Let Γ=(V,E) be a graph of order n. A closed distance magic labeling Γ is bijection ℓ:V→{1,2,…,n} for which there exists positive integer r such that ∑x∈N[u]ℓ(x)=r all vertices u∈V, where N[u] the neighborhood u. said to if it admits labeling. In this paper, we classify connected circulants with valency at most 5, is, Cayley graphs Cay(Zn;S) |S|≤5, S=−S, 0∉S and S generates Zn.