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 $D$ defined as a summation over all arcs, $M_1(D)=\frac{1}{2}\sum _{uv\in A}(d_u^{+}+d_v^{-})$, where $d_u^{+}$(resp. $d_u^{-}$) 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 $m$ $(2\le m\le \lfloor \frac{n}{2}\rfloor )$.