For a simple graph G, a vertex labeling f : V (G) → {1, 2,. .. , k} is called a k-labeling. The weight of a vertex v, denoted by wt f (v) is the sum of all vertex labels of vertices in the closed neighborhood of the vertex v. A vertex k-labeling is defined to be an inclusive distance vertex irregular distance k-labeling of G if for every two different vertices u and v there is wt f (u) = wt f (v). The minimum k for which the graph G has a vertex irregular distance k-labeling is called the inclusive distance vertex irregularity strength of G. In this paper we establish a lower bound of the inclusive distance vertex irregularity strength for any graph and determine the exact value of this parameter for several families of graphs.
Let G = (V , E) be a finite (non-empty) graph, where V and E are the sets of vertices and edges of G. An edge magic total labeling is a bijection α from V ∪ E to the integers 1, 2, . . . , n+ e, with the property that for every xy ∈ E, α(x)+ α(y)+ α(xy) = k, for some constant k. Such a labeling is called an a-vertex consecutive edge magic total labeling if α(V ) = {a + 1, . . . , a + n} and a b-edge consecutive edge magic total if α(E) = {b + 1, b + 2, . . . , b + e}. In this paper we study the properties of a-vertex consecutive edge magic and b-edge consecutive edge magic graphs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.