Study On Graph Spectra Theory And Spectra Of Certain Classes Of Matrices With Their Combinatorial Characteristics

Special matrices, as the name suggests, are types of matrices with special structuresor properties. Special matrices have wide applications in many fields such as compu-tational mathematics, applied mathematics, economics, statistics, physics, biology andcomputer science. Therefore, it is more significant to have an investigation into spe-cial matrices than into general ones from both points of view of theoretical research andpractical application values. Spectra of graph, which has been one of the important is-sues in algebraic graph theory in recent decades, has extensive applications in disciplinesof theoretical chemistry, physics, communication network and information sciences etc.Additionally, Graph spectra theory also promotes and enhances study of graph theory andcombinatorial mathematics. It has already been an important aspect for study of combina-torial matrix theory. This thesis presents an intensive research on combinatorial propertiesof certain types of special matrices such as nonnegative matrices, generalized ultrametricmatrices, M-matrices, H-matrices, P(P₀)-matrices. At the same time, spectral estimatesof graphs, as one of the hot topics on theory of combinatorial matrices, are also studied.The thesis consists of four parts with six chapters.With the help of graph representations of generalized ultrametric matrices, i.e., bi-nary rooted tree structures of generalized ultrametric matrices, we study the closure prop-erties of generalized ultrametric matrices and derive some closure sufficient conditions ofHadamard product, generalized Perron complement and sum of generalized ultrametricmatrices. In addition, we also have a study on sub-direct sums of P₀-matrices and give apositive answer to a problem presented by S.M. Fallat and C.R. Johnson.Numerical characteristics of matrices are studied. We firstly give a new upper boundon the spectral radius of Hadamard product of nonnegative matrices. Applying this result,we present some upper and lower bounds of the minimum eigenvalues of M-matrices.Numerical examples exemplify that our results are better than some known results in somecircumstances. At the same time, a new lower bound for Fan product of M-matrices isalso obtained. Secondly, employing the associated digraphs of matrices, we get some newintervals for singular values of matrices, simplifying and improving some known results.We study closure properties of diagonal-Schur complements of H-matrices as well as its subclass, doubly diagonally dominant matrices. Moreover, we also give the corre-sponding eigenvalue distributions. For linear systems with strictly diagonally dominantmatrices, we present some new upper and lower bounds on the spectral radius of iterativematrices of the generalized AOR iterative method. Applying these results, we discussconvergence of the generalized AOR iterative method and give its convergence region.Numerical examples illustrate that our results are improvements on some known results.We give some new estimates on spectra of Laplacian matrices, adjacency matricesand distance matrices of graphs. Firstly, a new upper bound on the spectral radius foradjacency matrices of simple unoriented graphs is given and helps to obtain some newupper bounds for the Laplacian spectral radius of mixed graphs. Theoretical analysisshows that our results improve some known results. Secondly, a new lower bound for theLaplacian spectral radius of bipartite graphs is presented. Moveover, we also get someupper and lower bounds on sum of powers of the Laplacian eigenvalues of bipartite graphsand characterize the extremal graphs of these bounds. Next, applying both upper boundsof the sum of the squares of the degrees of graphs and edge density of graphs, we obtainsome bounds on the algebraic connectivity of graphs and present the extremal graphs ofthese bounds. Fourthly, a new upper bound on the spectral radius of adjacency matricesof weighted graphs is given, which generalizes some known results. Finally, a new lowerbound on the spectral radius of distance matrices of graphs is derived, making use ofwhich, a new upper bound for D-energy of graphs is given, improving a known result.