Incidence Energy Of Graphs

Let G be a simple and connected graph with n vertices. The energy of a graph G is defined as the sum of the absolute values of all the eigenvalues of G. Nikiforov (2007) generalized the graph energy to the energy of an arbitrary matrix. It is of great significance to study the incidence energy of graphs and the skew energy of digraph in the chemical graph theory. In recent years, the research on these two kinds of energies is attracting more and more concern of mathematicians.Jooyandeh et al. (2009) introduced the incidence energy IE(G) of a graph G, which is defined as the sum of the singular values of the incidence matrix I(G), i.e., IE(G)=∑i=1nσi, where σ1,σ2, …,σn are the singular values of I(G), namely the square roots of the eigenvalues of I(G)I(G)t with I(G)t denoting the transpose of I(G).Adiga et al. (2010) defined the skew energy of digraph as the sum of the absolute values of all the eigenvalues of the skew adjacency matrix S(Gσ), i.e εs(Gσ) =∑ i=1n|λi(Gσ)|, where λ1(Gσ),λ2(Gσ),…m λn(Gσ) are the eigenvalues of S(Gσ).In this thesis, we mainly study the incidence energy of graphs and the skew energy of digraphs. For the incidence energy, we consider the graphs with the minimal incidence energy in three classes of graphs: (ⅰ) the (n, m)-graphs without even cycles, (ⅱ) the unicyclic bipartite graphs, and (ⅲ) the bicyclic bipartite graphs. For the skew energy of digraphs, we study the properties of skew energies of an oriented graph. The main results of the thesis are outlined as follows.1) We first propose a new transformation to compare the signless Laplacian coefficients of two graphs. By establishing a mapping and making use of the partial orderings and the relationship between the signless Laplacian co-efficients and the incidence energy, we derive the graph with the minimal incidence energy among (n, m)-graphs without even cycles.2) By using the transformations and by comparing the signless Laplacian co-efficients of two graphs, we deduce the graphs with the minimal incidence energy among the unicyclic bipartite graphs and the bicyclic bipartite graph-s, respectively.3) By using a singular value inequality, we obtain some properties of the skew energies of an oriented graph when edges are deleted.