On The Upper Bound Of Spectral Radius Of Irregular Weighted Graphs

The theory of graph spectra is an important research field in Algebraic Graph Theory,in which various relationships between the spectra and structural properties of graphs are established through the matrices associated with graphs and by means of the results and methods in Matrix Theory and Graph Theory.Here,the matrices associated with graphs mainly include adjacency matrix,Laplacian matrix,signless Laplacian matrix and distance matrix,while the spectrum of a graph generally refers to the whole eigenvalues of a certain matrix of the graph.In the study of the theory of graph spectra,estimating the bounds on the spectral radius of various matrices of graphs is an important research topic.In the thesis,we mainly study the upper bounds on the spectral radius of the adjacency matrix(also called adjacent spectral radius)and the spectral radii of the signless Laplacian matrix(also called signless Laplacian spectral radius)of irregular weighted graphs,and obtain the following results:In Chapter 3,upper bounds on the adjacent spectral radius of an irregular weighted graph are given in terms of the number of vertices,diameter and maximum vertex weight.At the same time,an upper bound on the adjacent spectral radius of a k-connected irregular weighted graph is also established in terms of the number of vertices,connectivity,maximum vertex weight,minimum edge weight and sum of vertex weights.In Chapter 4,upper bounds on the signless Laplacian spectral radius of an irregular weighted graph are given in terms of the number of vertices,diameter and maximum vertex weight.At the same time,an upper bound on the signless Laplacian spectral radius of a k-connected irregular weighted graph is also established in terms of the number of vertices,connectivity,maximum vertex weight,minimum edge weight and sum of vertex weights.An unweighted graph can be naturally regarded as a special weighted graph with each edge weight being one and hence,the results obtained in this thesis generalize some known results about the upper bounds on the adjacent spectral radius and signless Laplacian spectral radius of irregular unweighted graphs.