On Characterization Of Some Extremal Graphs With Respect To The Spectral Radius

In recent years, the(distance) spectral radius of a connected graph has been studied extensively. Based on the previous study, we studied some problems about some spectral radius of bicyclic graphs and bipartite graphs. Firstly, this paper introduced the achievement and background of spectral theory, the spectrum of distance matrix, distance signless Laplacian matrix and distance Laplacian matrix of graphs.Suppose that the vertex set of a graph G is V(G) = {v1, · · ·, vn}. Then we denote by T rG(vi) the sum of distances between viand other vertices of G. Let T r(G) be the n × n diagonal matrix with its(i, i)-entry equal to T rG(vi) and D(G) be the distance matrix of G. Then LD(G) = T r(G)- D(G) is the distance Laplacian matrix of G. The distance Laplacian spectral radius of G is the spectral radius of LD(G).Let A(G) be the adjacent matrix of G and D(G) be the n × n diagonal matrix with its(i, i)-entry equal to the degree d(vi) of vi. Then QA(G) = D(G) + A(G) and LA(G) =D(G)- A(G) are the signless Laplacian matrix and Laplacian matrix of G, respectively.The largest eigenvalues of QA(G) and LA(G) are signless Laplacian and Laplacian spectral radius of G, respectively.And then we introduced our main results in the next four chapters.Firstly, in Chapter II, we determined the graph with minimum distance Laplacian spectral radius among bicyclic graphs with fixed numbers of vertices.Secondly, in Chapter III, we denoted by Bmnthe class of all bipartite graphs of order n with matching number m, and Bs nthe class of all bipartite graphs of order n with vertex connectivity s. Then we determined the graphs with minimum distance Laplacian spectral radius in Bmnand Bsn, respectively.Thirdly, in Chapter IV, we determined the graphs with maximum(signless) Laplacian spectral radius in Bmnand Bsn, respectively.Fourthly, in Chapter V, we determined the graphs with maximum spectral radius among all trees, and all bipartite unicyclic, bicyclic, tricyclic, tetracyclic, pentacyclic and quasi-tree graphs, respectively.