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Let $S(G^sigma)$ be the skew-adjacency matrix of an oriented graph $G^sigma$. The skew energy of $G^sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^sigma)$. In this paper, we first deduce an integral formula for the skew energy of an oriented graph. Then we determine all oriented graphs with minimal skew energy among all connected oriented graphs on $n$ vertices with $m (nle m < 2(n-2))$ arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi {it et al.} [G. Caporossi, D. Cvetkovi$acute{c}$, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.]
Let $D$ be an oriented graph. The inversion of a set $X$ of vertices in $D$ consists in reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by ${rm inv}(D)$, is the minimum number of
We show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separator
We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson in the case of unoriented
A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of gain graphs
This paper has been withdrawn by the author.