Communications of the
Korean Mathematical Society CKMS

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) \cup A(D)\longrightarrow\{1, 2, \ldots$, $p+q\} $ with the property that for every $ v\in V(D)$, $f(v)+ \sum_{u\in O(v)}f((v,u))=k$, for some constant $ k $. Such a labeling is called a $ V $-super vertex out- magic total labeling ($ V $-SVOMT labeling) if $ f(V(D))=\{1, 2, 3, \ldots, 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.

Keywords: vertex out-magic total labeling, V-super vertex out-magic total labeling, vertex out-magic digraphs, V-super vertex out-magic digraphs

MSC numbers: Primary 05C78