The (Signless) Laplacian Spread Of Quasi-tree Graphs And Quasi-unicyclic Graphs

The Laplacian spread of a graph G is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of G.Zhai et al.conjectured that for any graph G on n vertices,SL(G)≤n-1 with equal-ity if and only if G or (?) is isomorphic to the join of an isolated vertex and a discon-nected graph on n—1 vertices.There are considerable results with respect to SL(G)in the literature,and most of them are around this conjecture.Motivated by the defini-tion of adjacency and Laplacian spreads of a graph G,Liu et al.defined the signless Laplacian spread of a graph to be the difference between the largest eigenvalue and the smallest eigenvalue of the signless Laplacian matrix of G.Oliveira et al.conjectured that for any connected graph G with n ≥ 5 vertices,SQ(G)≤(?),the upper bound is attained if and only if G = PCn,1,1.A connected graph G =(V,E)is called a quasi-tree graph,if there exists a vertex v0 ∈V(G)such that G-v0 is a tree.A connected graph G =(V,E)is called a quasi-unicyclic graph,if there exists a vertex v0 ∈ V(G)such that G—v0 is a unicyclic graph.In this paper,we study the(signless)Laplacian spread of quasi-tree graphs and quasi-unicyclic graphs around the above two conjectures.The main contents are as follows:The first chapter mainly introduces the background and the main progress of the(signless)Laplacian spread,and summarizes the main results obtained in this paper.The second chapter introduces some notations,concepts and lemmas,and proves some new lemmas.The third chapter presents an upper bound on the Laplacian spread of quasi-tree graphs and quasi-unicyclic graphs respectively.Moreover,we characterize the corre-sponding extremal graphs.We extend the result of[Y.Xu,J.X.Meng,The Laplacian spread of quasi-tree graphs,Linear Algebra Appl.435(2011),60-66]and show the conjecture of Zhai et al.is correct for quasi-tree graphs and quasi-unicyclic graphs.The fourth chapter gives an upper bound on the signless Laplacian spread of quasi-tree graphs and quasi-unicyclic graphs respectively.Moreover,we characterize the corresponding extremal graphs.These results show that the conjecture of Oliveira et al.is correct for quasi-tree graphs and quasi-unicyclic graphs.The fifth chapter presents some upper bounds on the signless Laplacian spread of simple connected graphs and partially proves that the conjecture of Oliveira et al.is correct for certain graphs.