V-SUPER VERTEX OUT-MAGIC TOTAL LABELINGS OF DIGRAPHS

Abstract

Let D be a directed graph with p vertices and q arcs. A vertex out-magic total labeling is a bijection f from V (D) ∪ A(D) −→ {1, 2, . . ., p + q} with the property that for every v ∈ V (D), f(v) + ∑ u∈O(v) f((v, u)) = k, for some constant k. Such a labeling is called a V super vertex outmagic total labeling (V -SVOMT labeling) if f(V (D)) = {1, 2, 3, . . . , p}. A digraph D is called a V -super vertex out-magic total digraph (V -SVOMT digraph) if D admits a V -SVOMT labeling. In this paper, we provide a method to find the most vital nodes in a network by introducing the above labeling and we study the basic properties of such labelings for digraphs. In particular, we completely solve the problem of finding V -SVOMT labeling of generalized de Bruijn digraphs which are used in the interconnection network topologies. 1. Background A labeling of a graph G is a mapping that carries a set of graph elements, usually the vertices and edges into a set of numbers, usually integers. We deal with digraphs which possibly admit self-loops but not multiple arcs. For standard graph theory terminology we follow [6]. Specifically, let D = (V,A) be a digraph with vertex set V and arc set A. If (u, v) ∈ A, then there is an arc from u to v and u is called a head, v is called a tail. If (u, u) ∈ A, the arc (u, u) is called a self-loop or loop. For a vertex v ∈ V, the sets O(v) = {u | (v, u) ∈ A} and I(v) = {u | (u, v) ∈ A} are called the out-neighborhood and the inneighborhood of the vertex v, respectively. The out-degree and in-degree of v are deg(v) = |O(v)| and deg(v) = |I(v)|, respectively. MacDougall et al. [12, 15] introduced the notion of vertex magic total labeling. If G is a finite simple undirected graph with p vertices and q edges, then a vertex magic total labeling is a bijection f from V (G) ∪ E(G) to the integers 1, 2, . . . , p + q with the property that for every u in V (G), f(u) + Received October 20, 2015. 2010 Mathematics Subject Classification. Primary 05C78.