A note on skew spectrum of graphs

We give some properties of skew spectrum of a graph, especially, we answer negatively a problem concerning the skew characteristic polynomial and matching polynomial in [M. Cavers et al., Skew-adjacency matrices of graphs, Linear Algebra Appl. 436 (2012) 4512--4529]

A survey on the skew energy of oriented graphs

Let $G$ be a simple undirected graph with adjacency matrix $A(G)$. The energy of $G$ is defined as the sum of absolute values of all eigenvalues of $A(G)$, which was introduced by Gutman in 1970s. Since graph energy has important chemical applications, it causes great concern and has many generalizations. The skew energy and skew energy-like are the generalizations in oriented graphs. Let $G^\sigma$ be an oriented graph of $G$ with skew adjacency matrix $S(G^\sigma)$. The skew energy of $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of the norms of all eigenvalues of $S(G^\sigma)$, which was introduced by Adiga, Balakrishnan and So in 2010. In this paper, we summarize main results on the skew energy of oriented graphs. Some open problems are proposed for further study. Besides, results on the skew energy-like: the skew Laplacian energy and skew Randi\'{c} energy are also surveyed at the end.Comment: This will appear as a chapter in Mathematical Chemistry Monograph No.17: Energies of Graphs -- Theory and Applications, edited by I. Gutman and X. L