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

On the Weighing Matrices of Order 4n and Weight 4n-2 and 2n-1

We give algorithms and constructions for mathematicaland computer searches which allow us to establish the existence of W (4n; 4n \Gamma 2) and W (4n; 2n \Gamma 1) for many orders 4n less than 4000. We compare these results with the orders for which W (4n; 4n) and W (4n; 2n) are known. We use new algorithms based on the theory of cyclotomy to obtain new T -matrices of order 43 and JM-matrices which yield W (4n; 4n \Gamma 2) for n = 5; 7; 9; 11; 13; 17; 19; 25;31; 37; 41; 43; 61; 71; 73; 157. Key words and phrases: Weighing matrices, Hadamard matrices, conference matrices, cyclotomy. AMS Subject Classification: Primary 05B20. 1 Introduction Definition 1 An orthogonal design A, of order n, and type (s 1 ; s 2 ; : : : ; s u ), denoted OD(n; s 1 ; s 2 ; : : : ; s u ) on the commuting variables (\Sigmax 1 ; \Sigmax 2 ; : : : ; \Sigmax u ; 0) is a square matrix of order n with entries \Sigmax k where each x k occurs s k times in each row and column such that the distinct rows are pairw..