The Minimal Skew Energy Of Digraphs

Spectral graph theory mainly studies the spectral of operators or the matrix of graphs(such as the adjacency matrix, the distance matrix and the Laplacian matrix etc), and establishes the relations between the topological structures and the eigenspaces of graphs, and studies the topological structure of graphs by using algebraic, geometric theory and probabilistic methods, or study the spectral properties in algebra and geometry by using the topological structure of graphs.The energy of a graph is a part of the spectral graph theory. The energy of a graph has closed links to chemistry. Since the quantum-chemical characteristic of a conjugated molecule is its total π-electron energy. A digraph G~σis a simple undirected graph G with an orientation, which assigns to each edge a direction so that G~σbecomes a directed graph. Let S(G~σ) be the skew-adjacency matrix of G~σand λ1, λ2, · · ·, λndenote all the eigenvalues of the S(G~σ). The skew energy of G~σis de?ned as the sum of the absolute values of all eigenvalues of S(G~σ). In recent years, Adiga, Balakrishnan and So investigate the skew energy of digraphs,computing the skew energies of directed cycles under di?erent orientations and showing that the skew energy of a directed tree is independent of its orientation,which is equal to the energy of its underlying tree. After that, more chemists and mathematicians began to focus on this issue, so there are many problems about the skew energy of digraphs have been resolved. Li and Lian have summarized most of the known results on the skew energy of oriented graphs. Of course, some open problems are also proposed for further study. In this paper, we discuss the minimal skew energy of digraphs G~σ(n, m). The organization of this thesis is as follows.Chapter One: We introduce a background of graph theory, graph energy,digraph skew energy, some conceptions and notations, and the research problems and results.Chapter Two: We discuss the bicyclic digraphs with the second minimal skew energy.Let G~σ(n, m) be the connected oriented graphs with n vertices and m(n ≤m < 2(n- 2)) arcs.Chapter Three: We determine digraphs G~σ(n, m) with the second and the third minimal skew energy.Chapter Four: We obtain a low bound for the skew energy of all connected digraphs G~σ(n, m).