Yang, Xiuwen

English

Spectral graph theory is a very important research topic in algebraic graph theory and combinatorial matrix theory. Spectral graph theory primarily utilizes the spectral properties of graph matrices, including eigenvalues (or combinations of eigenvalues) and their associated eigenvectors, to study the structural properties of graphs. Digraphs are the natural extension of graphs, which can reflect the asymmetric relationship between research objects. In this thesis, with the use of methods such as topological ordering, Karamata’s inequality, equitable quotient matrix and Perron-Frobenius theorem, we mainly study the Laplacian matrix, the Aα-matrix and the eccentricity matrix of digraph with a fixed dichromatic number. The research in this thesis primarily holds theoretical value and significance, enriching the relevant theory of spectra of digraphs. The main research results are as follows:1. We characterize the digraphs which attain the minimal and maximal Laplacian energy within classes of digraphs with a fixed dichromatic number. We determine sharp bounds for the third Laplacian spectral moment among all join digraphs. 2. We extend the results about Laplacian spectral moments to Aα spectral moments. We find that the Laplacian matrix and the Aα-matrix have much in common with respect to the second and third spectral moments. 3. We characterize the digraph which has the maximal Aα spectral radius among all digraphs with a fixed dichromatic number, by using the equitable quotient matrix. We obtain the digraph which has the minimal Aα spectral radius of the join of in-trees with a fixed dichromatic number. 4. We mainly obtain lower bounds for the eccentricity spectral radius among all join digraphs with a fixed dichromatic number, and give upper bounds for the eccentricity spectral radius of some special join digraphs with a fixed dichromatic number.<br/

University of Twente Research Information

Spectral properties of digraphs with a fixed dichromatic number

https://ris.utwente.nl/ws/files/324413057/xyangphdthesis.pdf

Spectral properties of digraphs with a fixed dichromatic number

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University of Twente Research Information