Analytic Properties Of Combinatorial Matrices

Combinatorial matrix is the basic research object in combinatorics.we investigate the an-alytic properties of combinatorial matrices whose eigenvalues are all real,such as the total positivity,the asymptotic normality and the(p,q)-interlacing property.This thesis is organized as follows.In the first part,we consider analytic properties of Delannoy triangle.There are many similarities between the Delannoy triangle and the Pascal triangle,their entries have rich combinatorial interpretations.Firstly,we investigate the zeros of Delannoy polynomials.We show that all the zeros are real and dense in a closed interval,and therefore the sequence in each row of Delannoy triangle forms a Polya frequency sequence.Secondly,we prove that the Delannoy triangle,as a double-index sequence,is asymptotically normal.Finally,we obtain the total positivity of Delannoy-like triangles.As consequences,the Pascal triangle,the Fibonacci triangle and the Delannoy triangle are all totally positive.In the second part,we consider the(p,q)-interlacing property of graph matrices,which contains the adjacency matrix,the Laplacian matrix and the normalized Laplacian matrix.We present a characterization of eigenvalues inequalities between two Hermitian matrices by means of inertia indices.As applications,we deal with some classical eigenvalue inequalities for Hermitian matrices,including the Cauchy interlacing theorem and the Weyl inequality,in a simple and unified approach.We also obtain a necessaiy and sufficient conditions forjudging the(p,q)-interlacing property of matrices,and then study the(p,g)-interlacing property of graph matrices under some graph operations.Finally,in the third part,we consider the(p,q)-interlacing property of Hermitian matrices.People study properties of digraph in terms of some Hermitian matrices,such as the Hermitian Laplacian matrix and the Hermitian normalized Laplacian matrix.We study the interlacing of Hermitian Laplacian matrix and the compatibility of Hermitian normalized Laplacian matrix for the edge perturbation of digraphs.On the other hand,Weyl’s eigenvalue inequality describe the(p,q)-interlacing property of a Hermitian matrix for additive Hermitian perturbations.We show that the converse of Weyl’s eigenvalue inequality holds.