topological indices

Discrete Applied Mathematics

Let D n be the set of digraphs with n non-isolated vertices. Let D ∈ D n and denote by d u + and d u − the outer degree and inner degree, respectively, of the vertex u of D . We define the vertex-degree-based (VDB, for short) topological index φ induced by the real numbers φ i j , as φ D = 1 2 ∑ 1 ≤ i , j ≤ n − 1 a i j φ i j , where a i j is the number of arcs in D of the form u v , where d u + = i and d v − = j . We show in this paper that this is a generalization of the concept of VDB topological indices of graphs. In the case φ i j = 1 i j , we obtain the Randić index of digraphs, which we denote by χ . We find the extremal values of χ over D n . We also find the extremal values of χ over OT ( n ) , the set of all oriented trees with n vertices. On the other hand, given a graph G , we consider the set O G of all orientations of G , and show that when G is a bipartite graph, the sink–source orientations of G uniquely attain the minimal value of χ over O G . We find the extremal values of χ over O P n and O C n , where P n and C n are the path and the cycle with n vertices, respectively. Finally, we find the extremal values of χ over O H d , the set of all orientations of the hypercube H d of dimension d .

Vertex-degree based topological indices of digraphs