On The Laplacian Coefficients And Signless Laplacian Spectral Radius Of Graphs

The theory of graph spectra as one of the main research directions of the algebraicgraph theory, mainly includes the adjacent spectral theory, Laplacian spectral theory,signless Laplacian spectral theory and normalized Laplacian spectral theory. The theoryof graph spectra can be seen as an attempt to discover the properties on structures ofgraphs and relations between the spectral and fixed parameters of graphs by means of thewell developed theory and technique of theories of graph and combinatorics. There areextensive applications in the fields of physics, quantum chemistry, computer science andcommunication network.The Laplacian coefcients have an important connection to many topological indicesof the molecule graph, which have the Wiener index and the hyper Wiener index. TheWiener index is generally regarded as one of the most valuable topological indices, andhas close relation with the physical characteristics and chemical characteristics of manycompounds, and plays a huge role in many fields of Chemistry. In addition, there is avery close relation between the Laplacian coefcients and Laplacian-like energy, whichhas similar characteristics to molecular graph energy.The spectral radius of signless Laplacian matrix is an important topic in the theoryof graph spectra, the definite solution of partial diferential equations and matrix theory,and has been applied widely in quantum chemistry, complex network and biology. It hasbeen applied widely to many field of graph theory. Meanwhile, the signless Laplacianmatrix is more closely relative to the graph structures than the adjacency matrix and theLaplacian matrix. It can show the some properties of graphs well. So, it has importanttheoretical and practical values to study the signless Laplacian spectral radius.The thesis focus on the Laplacian coefcient and the signless Laplacian spectral radiusin a given set of graphs. The main works can be summarized as follows:1. We give some transformations of connected graphs that decrease all Laplaciancoefcients. The tricyclic graph with smallest Laplacian coefcients among all the tricyclicgraphs is obtained.2. We give four transforms on graphs that decrease all Laplacian coefcients. Theunicyclic graphs with smallest Laplacian coefcients among all the unicyclic graphs withn vertices and m pendent vertices is obtained.3. We give some transformations of connected graphs that increase all Laplaciancoefcients. The bicyclic graph with maximum Laplacian coefcients among all the bi-cyclic graphs is obtained. Generally speaking, it is difcult to determine the maximum Laplacian coefcients of graphs. Therefore, there are only a few papers concerning themaximum Laplacian coefcients of trees and unicyclic graphs. Finally, we determine thegraph with the largest Laplacian-like energy among all the bicyclic graphs.4. According to the study of the change of the signless Laplacian spectral radius underthe various disturbance by the technique in the characteristic vector and the characteristicpolynomial, we determine graphs with the largest signless Laplacian spectral radius amongall the bicyclic graphs and tricyclic graphs with n vertices and fixed diameter d. As anapplication, we give first three graphs among all bicyclic graphs on n vertices, orderedaccording to their spectral radii in decreasing order.