Research On The Signless Laplacian Spectrum Of Graphs And Its Associated Parameters

The Signless Laplacian is an active and important field in graph theory. The Signless Laplacian are closely associated with the structure of graphs, and its eigenvalues are all invariants of graphs under isomorphism. This paper is written on the Signless Laplacian of graphs, and researches on the Signless Laplacian Spread of Graphs, the Signless Laplacian Estrada Index of Graphs and the Signless Laplacian Index of Graphs. It is organized as follows.In the first chapter, we first introduce some useful basic concepts and propositions, and then give an introduction to the background of the research and results in this paper.In the second chapter, we present some lower bounds for SQ(G) and SQ(G)+SQ(GC) in terms of the k-degree and the independent number, respectively.In the third chapter, we present some sharp lower bounds for SLLEE(G) in terms of the k-degree and the first Zagred index, respectively.In the fourth chapter, we show that for all the connected graphs of order n≥5with signless Laplacian inde q1(G) and radius rad(G), q1(G)·rad(G) is maximum for and only for the graph Bagn-2s+3,2s-1, where s=(?). This solves a conjecture in [24] on the signless Laplacian index involving the radius.Finally, conclusion and some problems to be studied further are proposed in the thesis.