As a natural extension of previously defined graph labelings, we introduce in this paper a new magic labeling whose evaluation is based on the neighbourhood of a vertex. We define a 1-vertex-magic vertex labeling of a graph with v vertices as a bijection f taking the vertices to the integers 1, 2, . . . , v with the property that there is a constant k such that at any vertex x, ∑ y∈N(x) f(y) = ...

A group distance magic labeling or aG-distance magic labeling of a graph G = (V, E) with |V | = n is a bijection f from V to an Abelian group G of order n such that the weight w(x) = ∑y∈NG (x) f (y) of every vertex x ∈ V is equal to the same element μ ∈ G, called the magic constant. In this paper we will show that if G is a graph of order n = 2p(2k + 1) for some natural numbers p, k such that d...

A 1-vertex-magic vertex labeling of a graph G(V,E) with p vertices is a bijection f from the vertex set V (G) to the integers 1, 2, . . . , p with the property that there is a constant k such that at any vertex x the sum ∑ f(x) taken over all neighbors of x is k. In this paper, we study the 1-vertex-magic vertex labelings of two families of disconnected graphs, namely a disjoint union of m copi...

In six-dimensional F-theory/heterotic string theory, half-hypermultiplets arise only when they correspond to particular quaternionic K\"ahler symmetric spaces, which are mostly associated with the Freudenthal-Tits magic square. Motivated by intriguing singularity structure previously found in such F-theory models a gauge group $SU(6)$,$SO(12)$ or $E_7$, we investigate, as final magical example,...

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− ...

A graph G = (V (G), E(G)) admits an H-covering if every edge in E(G) belongs to a subgraph of G isomorphic to H. The graph G is H-magic if there exists a bijection f : V (G)∪E(G)→ {1, 2, 3, . . . , |V (G)∪E(G)|} such that for every subgraph H ′ of G isomorphic to H, ∑ v∈V (H′) f(v) + ∑ e∈E(H′) f(e) is constant. Then G is H-supermagic if f(V (G)) = {1, 2, 3, . . . , |V (G)|}. Let {Gi}i=1 be a fi...

Let A be an abelian group. An A-magic of a graph G = (V, E) is a labeling f : E(G) → A\{0} such that the sum of the labels of the edges incident with u ∈ V is a constant, where 0 is the identity element of the group A. In this paper we prove Z3-magic labeling for the class of even cycles, Bistar, ladder, biregular graphs and for a certain class of Cayley digraphs. Mathematics Subject Classifica...

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...