Research on the Eigenvalues of Graphs and Mixed Graphs

This paper first elaborates the research situation and progress of Laplace characteristics and the eigenvalues value of graphs. The second is given an upper bound of characteristic value of a kind of special graph using the properties of similar matrices. At the same time, a new upper bound of Laplace characteristic values are given using properties of Laplace matrix and the similarity matrix, to improve the existing results. Then, we use the example of the upper bound of our results are more precise than some previous results. Finally the use Laplace non- zero eigenvalues of graph properties to give a bound expressions using the degree of square with a number of edges and the graph of the number, number of connected component expression map, it reflected the relationship between eigenvalues and the amount of Laplace.

On Edge Singularity and Eigenvectors of Mixed Graphs

Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.

LARGEST EIGENVALUE OF A UNICYCLIC MIXED GRAPH

The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U)∈(n,n+1] are characterized.

On the Inertia Index of a Mixed Graph in Terms of the Matching Number

A mixed graph G^(-) is obtained by orienting some edges of G, where G is the underlying graph of G^(-) . The positive inertia index, denoted by p~+( G), and the negative inertia index, denoted by n~-(G^(-) ), of a mixed graph G^(-) are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of G^(-) , respectively. In this paper, the positive and negative inertia indices of the mixed unicyclic graphs are studied. Moreover, the upper and lower bounds of the positive and negative inertia indices of the mixed graphs are investigated, and the mixed graphs which attain the upper and lower bounds are characterized respectively.

Causality Analysis of Futures Sugar Prices in Zhengzhou Based on Graphical Models for Multivariate Time Series

This paper presents a method of constructing a mixed graph which can be used to analyze the causality for multivariate time series.We construct a partial correlation graph at first which is an undirected graph.For every undirected edge in the partial correlation graph,the measures of linear feedback between two time series can help us decide its direction,then we obtain the mixed graph.Using this method,we construct a mixed graph for futures sugar prices in Zhengzhou（ZF）,spot sugar prices in Zhengzhou（ZS） and futures sugar prices in New York（NF）.The result shows that there is a bi-directional causality between ZF and ZS,an unidirectional causality from NF to ZF,but no causality between NF and ZS.