On the integral weighted oriented unicyclic graphs with minimum skew energy

Let G^σ be a weighted oriented graph, which is obtained from a simple weighted undirected graph by assigning an orientation to each of its edges. For a (weighted) oriented graph G^σ, the undirected graph obtained from G^σ by removing the orientation and the weight of each of its arcs is called the underlying graph of G ^σ, denoted by Ĝ. By U(n,m)(m≥n) we denote the set of all connected integral weighted oriented unicyclic graphs with order n in which each arc is assigned a positive integral weight and the sum of the weights of all arcs is equal to the specified integer m. In this paper, we investigate the minimal skew energies of integral weighted unicyclic oriented graphs, showing that the underlying graph of the oriented graph with minimum skew energy among all graphs over U(n,m)(n≥6) is S_n,3, the graph obtained from a triangle by attaching n-3 pendent edges in exactly one of its vertices. Moreover, we show that its weight sequence has form(w₁,a, ^a,.,aï̧·k,a+ ^{1,a+1,.,a+1ï̧·n-3-k},w₂,w₃)in which the arc lying on the cycle ^C3 and incident to no pendent arcs has weight w₁, and the two largest weights correspond the other two arcs of ^C3, where positive integer numbers w₁,w ₂,w₃,a and k satisfy 1≤w₁≤a,a+1≤w ₂≤w₃≤a+w₁ and m=w₁+ka+(n-k-3) (a+1)+w₂+w₃. In addition, we determine the weight sequence above for n≤m≤3n-1.