Some Results On The Sum Of Laplacian And Signless Laplacian Eigenvalues Of Graphs

Let G be a simple graph with order n,matching number m(G),clique num-ber ?,vertex covering number ? and diameter d.Also let ?1(G)??2(G)?…??n(G)=0 and q1(G)? q2(G)?…?qn(G)?0 be the Laplacian eigenvalues and signless Laplacian eigenvalues of G respectively.For any 1?k?n,let Sk(G)=(?)?i(G),Sk-(G)=(?)qi(G).In this paper,we primarily focus on the Brouwer's conjecture Sk(G)?<e(G)+(2 k+1)(k=1,2,…,n)and Brouwer's conjecture for the signless Laplacian eigenvalues Sk-(G)?e(G)+(2 k+1)(k=1,2,…,n).We prove that for the graphs with m(G)=1,2,where 3?k?n and m(G)=3,where 5 ?k?n,both of the two conjectures are true.And also we prove that for the bipartite graph Gs1,s2 with s1<s2,where 2s1+1 ?k?n;for the graph with girth g,where 1?k?[g/5],specially for the graphs satisfying 3+(?)/2?[g/5]where 1?k?n;for the graph with maximum degree ?and p pendent vertices are adjacent to the maximum degree vertex,where 5+(?)/2?k?n,specially for the c-cycles graph satisfying ?>c+p+4,where 1?k?n;for the complete split graph,where 1?k?n,the latter conjecture is also true.We give another proof for the theorem Sk(G)?k(?+1)+e(G)-?(?-1)/2,(1?k?n)in literature 27],and also we give an counterexample to disprove the theorem Sk(G)?(?-[d+2)k+e(G)-d+cos(k?/d)+cos(?/d)sin(k?/d)+sin(k?)/d/sin(?/d)and Sk(C)?k(?+s2-s1)+e(G)-s1(s2-1)respectively,where Ks1,s2(s1?s2)is the maximum complete bipartite subgraph of G.