Skew Spectra of Oriented Graphs An oriented graph Gsigma is a simple undirected graph G with an orientation sigma, which assigns to each edge a direction so that Gsigma becomes a directed graph. G is called the underlying graph of Gsigma, and we denote by Sp(G) the adjacency spectrum of G. Skewadjacency matrix S( Gsigma ) of Gsigma is introduced, and its spectrum Sp_S( Gsigma ) is called the skewspectrum of Gsigma. The relationship between Sp_S( Gsigma ) and Sp(G) is studied.  In particular, we prove that (i) Sp_S( Gsigma )  bf i Sp(G) for  some  orientation sigma if and only if G is bipartite, (ii) Sp_S(Gsigma)  bf i Sp(G) for  any  orientation sigma if and only if G is a forest, where bf isqrt1.

Constructing Fifteen Infinite Classes of Nonregular Bipartite Integral Graphs A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs S_1(t)K_1,t, S_2(n,t), S_3(m,n,t), S_4(m,n,p,q), S_5(m,n), S_6(m,n,t), S_8(m,n), S_9(m,n,p,q),  S_10(n),  S_13(m,n),  S_17(m, n, p, q),  S_18(n,p,q,t), S_19(m,n,p,t), S_20(n,p,q) and S_21(m,t) are defined. We construct the fifteen classes of larger graphs from the known 15 smaller integral graphs S_1S_6, S_8S_10, S_13, S_17S_21 (see also Figures 4 and 5, Balinska and Simic, Discrete Math.  236 (2001) 1324). These classes consist of nonregular and bipartite graphs. Their spectra and characteristic polynomials are obtained from matrix theory. We obtain their integral property by using number theory and computer search. All these classes are infinite. They are different from those in the literature. It is proved that the problem of finding such integral graphs is equivalent to solving Diophantine equations. We believe that it is useful for constructing other integral graphs. The discovery of these integral graphs is a new contribution to the search of integral graphs. Finally, we propose several open problems for further study.