A total k-labeling is a function fe from the edge set to {1, 2, . , ke} and fv vertex {0, 4, 2kv}, where k = max{ke, 2kv}. distance irregular reflexive of graph G k-labeling, if for every two different vertices u 0 G, w(u) 6= w(u ), Σui∈N(u)fv(ui) + Σuv∈E(G)fe(uv). The minimum which has k-labelling called strength denoted by Dref (G). In this paper, we determine exact value some connected graph...

The study of graph labellings has focused on finding classes of graphs which admit a particular type of labelling. Here we consider variations of the well-known edge-magic and vertex-magic total labellings for which all graphs admit such a labelling. In particular, we consider two types of injections of the vertices and edges of a graph with positive integers: (1) for every edge the sum of its ...

Given a simple graph G (V, E) and a positive number d, an Ld(2, 1)-labelling of G is a function f V(G) [0, oc) such that whenever x, y E V are adjacent, If(x)f(Y)l >2d, and whenever the distance between x and y is two, If(x) f(Y)l >d. The Ld(2, 1)-labelling number A(G, d) is the smallest number m such that G has an Ld(2, 1)-labelling f with max{f(v) v E V} m. It is shown that to determine A(G, ...

The context of this paper is the use of formal methods for topology-based geometric modelling. Topology-based geometric modelling deals with objects of various dimensions and shapes. Usually, objects are defined by a graph-based topological data structure and by an embedding that associates each topological element (vertex, edge, face, etc.) with relevant data as their geometric shape (position...

A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. We characterize magic line graphs of general graphs and describe some class of supermagic line graphs of bipartite graphs.

By strengthening an edge-decomposition technique for gracefully labelling a generalised Petersen graph, we provide graceful labellings for a new infinite family of such graphs. The method seems flexible enough to provide graceful labellings for many other classes of graphs in the future.

Roman domination in graphs is concerned with the problem of finding a vertex labelling, with minimum weight, satisfaying certain conditions. In this work, the authors initiate the study of a generalization to labellings of both vertices and edges in a graph.

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In the paper we establish some bounds for the number of edges in supermagic graphs.

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. Some constructions of supermagic labellings of regular graphs are described. Supermagic regular complete multipartite graphs and supermagic cubes are characterized.

A graph is called degree-magic if it admits a labelling of the edges by integers 1, 2, . . . , |E(G)| such that the sum of the labels of the edges incident with any vertex v is equal to 1+|E(G)| 2 deg(v). Degree-magic graphs extend supermagic regular graphs. In this paper we characterize complete tripartite degree-magic graphs.