Maximum first Zagreb index of orientations of unicyclic graphs with given matching number

Abstract

Let D=(V,A) be a digraphs without isolated vertices. The first Zagreb index of a digraph D is defined as a summation over all arcs, M1(D)=12∑uv∈A(du++dv−), where du+(resp. du−) denotes the out-degree (resp. in-degree) of the vertex u. In this paper, we give the maximal values and maximal digraphs of first Zagreb index over the set of all orientations of unicyclic graphs with n vertices and matching number m(2≤m≤⌊n2⌋).