The Bound For The Signless Laplacian Energy Of A Graph

The theory of graph spectra originated in the 1950s,mainly studies the structural and chemical properties of graphs by studying the relations between spectra and graphs.In fact,the theory of graph spectra have great significance and value in the fields of chemistry,computer science,image processing and information theory,etc.In 1978,Gutman I.formally proposed the energy of graph in the field of mathematics,and the energy of graphs is defined by graph spectra.The research on the energy of graphs is mainly divided into two aspects:one is the study on the bounds of the energy of graphs;the other is the extremal graph depicting the time when the bounds are reached.In 2006,Gutman I.and Zhou B.proposed the Laplacian energy of graphs,and the signless Laplacian energy of graphs is similar to the Laplacian energy of graphs.Therefore,the study on the signless Laplacian energy bound of graphs is also very valuable among many researches.For a simple graph G with n vertices,m edges and signless Laplacian eigenvalues q1≥q2≥…qn≥0,its signless Laplacian energy QE(G)is defined as QE(G)=∑i=1n|qi-d|,where d=2m/n is the average vertex degree of G.In this paper,we use different algebraic inequalities to study the bound for the signless Laplacian energy of a graph,and obtain some new bounds of signless Laplacian energy of a graph,which improving some known results.This paper is divided into three parts.The first part contains the research background,basic concepts and the existing results of the bound for the signless Laplacian energy of a graph.The second part states some of the following the results related to the signless Laplacian eigenvalues of the graph that needs to be used.The third part proves new lower bounds(Theorem3.2 and Theorem 3.6)and the new upper bound(Theorem3.9)for the signless Laplacian energy of a graph and the corresponding extremal graphs.In addition,gives some bounds of QE(G)for regular graph G and the corresponding extremal graphs.Finally,some concrete examples are calculated to verify these new bounds and improve the existing results.