On the first outdegree Zagreb index of a digraph

Abstract

Let D be a digraph of order n and having a arcs. Let d1+,d2+,…,dn+ be the vertex outdegrees of D. The first Zagreb index of D is denoted by Zg+(D) and is defined as Zg+(D)=∑i=1ndi+. We obtain several upper and lower bounds for the first outdegree Zagreb index Zg+(D) of a digraph D in terms of the number of vertices n and the number of arcs a. We characterize the extremal digraphs attaining these bounds. Further, we determine the digraphs which attain the maximum, the second maximum and the third maximum value for Zg+(D) among all digraphs of order n and among all bipartite digraphs of order n. We determine the digraphs which attain the maximum value for Zg+(D) among all digraphs with its underlying graph having m≤n+2 edges. Also, we discuss the orientations of a path Pn and show that the path with n−1 vertices of outdegree 1 attains the minimum value for ZG+(D).