Some Graph With The Skew Energy And Laplacian Eigenvalue

In Graph Theory, the research of energy is active. Gutman [16] studied the energy of graphs in 1970s, which has strong applications in Chemistry. After that, many researchers have studied the skew energy of graphs. Hou et al. investigated the minimal energy of unicycle graphs. Li and Zhou [25] studied the energy of graphs with given diameter. In recent years, people are concerned with the skew energy of digraphs. In 2010, Adiga et al. [1] introduced and studied skew energy of digraphs. Hou et al. in [9] studied the skew energy of unicycle digraphs, and they obtained the relevant conclusions of the skew energy of unicycle digraphs. There are a lot of results on the bounds for skew energy; see the recent survey paper by Li and Lian [23].In this paper, we first study the skew energy of unicyclic digraphs, and obtain the first five digraphs with small skew energy. Next, we investigate the third and fourth minimal Laplacian eigenvalue of graphs.