Three Extremal Graphs Determined By Their Spectral Radii

Spectral theory is the correspondence between the research on spectral distribution ofthe graph structure and the corresponding graph labeling problem. In recent years, it hasa rapid development and becomes an active research topic. It is also a part of the combi-natorial matrix. Theoretically speaking, this problem deepens the characterization of theinherent relationship on the discrete structure. At the same time, it has far-reaching practicalapplication background in network optimization and design, integrated circuit design andoperations research aspects.Graph spectrum usually consists of three kinds: the spectrum of adjacent matrix, thespectrum of Laplacian matrix and the spectrum of signless Laplacian matrix. In our paper,we mainly study the Laplacian spectral radius of bicyclic graph with given independentnumber, the spectral radius of tree with given domination number, and the spectral radius oftricyclic graph. At last, we determine their extremal graphs, respectively.The paper consists of four sections. In the first section, we introduced the researchbackground on Laplacian matrix and study the Laplacian spectral radius of bicyclic graphswith given independence number and characterize the extremal graphs completely. In thesecond section, we study the minimum eigenvalue for trees with given domination numberand characterize the extremal graph completely. In the third section, we determine the ex-tremal graph with minimum least eigenvalue among all connected tricyclic graphs of ordern.