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

TL;DR

This paper determines the maximum first Zagreb index for orientations of unicyclic graphs with a fixed number of vertices and matching number, identifying the extremal digraphs.

Contribution

It provides the maximal values and characterizes the extremal orientations of unicyclic graphs with given matching number for the first Zagreb index.

Findings

Maximal first Zagreb index values are established.

Extremal digraphs achieving these maxima are characterized.

Results apply to all orientations of unicyclic graphs with specified parameters.

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, $M_1(D)=\frac{1}{2}\sum\limits_{uv\in A}(d^{+}_{u}+d^{-}_v)$, 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 $m$ $(2\leq m\leq \lfloor \frac{n}{2}\rfloor)$.