Investigation on Singularity,Signature Matrix and Spectrum of Mixed Graphs

The spectral theory of graph is an important branch of graph theory, and the main part of this theory is the connection between the spectral properties and the structural properties, characterization of the structural properties of graphs. We discuss the problems about singularity, signature matrix and spectrum of mixed graphs. Without loss of generality, parallel edges and loops are permitted in mixed graphs. Let G_1 and G_2 be connected mixed graphs which are obtained from an underlying graph G. When G_1 and G_2 have the same singularity, the number of induced cycles in G_i(i=1, 2) is l(l=1,l1), the length of the smallest induced cycles is 1, 2, at least 3. According to conclusions and mathematics induction, we find that the singularity of corresponding induced cycles in G_1 and G_2 are the same if and only if there exists a signature matrix D such that L(G_2)=D~TL(G_1)D. D may be the product of some signature matrices. If L(G_2)=D~TL(G_1)D, G_1 and G_2 have the same spectrum.