Ordering Of The Oriented Unicyclic Graphs With Skew Energies

The graph theory is a widely used mathematical disciplines. It has im-portant applications in the natural science, social science and so on. Research about the theory of graph spectra is an important field and a pretty active re-search direction in the graph theory. It has extremely widespread applications in the research of quantum chemistry, physics, computer science and so on. The spectral graph theory mainly studies the eigenvalue and eigenspaces of graphs of the correlation matrix (such as adjacency matrix and Laplacian matrix etc).The energy of a graph is a part of the spectral graph theory. An impor-tant application of graph energy in structural chemistry is as follows. Espe-cially, there exists a close correspondence between eigenvalues of graphs and the molecular orbital energy levels of ?-electrons in conjugated hydrocarbons. For a simple undirected graph G, the energy E(G) is defined to be the sum of the absolute values of all eigenvalues of its adjacent matrix. In 1977, the famous mathematic-chemist Gutman first proposed the theory of graph energy which has been evolved into one of the very active directions on graph theory ever since. Moreover, it has been rather widely concerned by theoretical chemist and mathematicians. Particularly since 2000, graph energy has been deeply developed in which many important results have been discovered. This aspect has had many conclusions.Besides graph energy of the adjacency matrix, a few other related matrix of the graph energy have been proposed and widely studied, such as Laplacian energy, Signless Laplacian energy, incidence energy, distance energy, and the skew energy of oriented graph. Analogous to the definition of the energy of a simple undirected graph, the skew energy of an oriented graph G is defined as the sum of the absolute values of all eigenvalues of the corresponding skew-adjacency matrix.Using the method of coefficient comparion and the theorem of zero points of function, we investigate the ordering of the oriented unicyclic graphs in term of their minimal skew energies and the oriented unicyclic graphs with minimal skew energies among all oriented unicyclic graphs with given girth. The main results are as follows.(1)We first determine the third-minimal to the ninth-minimal oriented uni-cyclic graphs according to their minimal skew energies;(2)We then characterize the oriented unicyclic graphs with the second-minimal skew energy by given girth.