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 ω,vertex covering number τ and diameter d.Also let μ1(G)≥μ2(G)≥…≥μn(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)=(?)μi(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 Δand 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(τ+1)+e(G)-ω(ω-1)/2,(1≤k≤n)in literature 27],and also we give an counterexample to disprove the theorem Sk(G)≤(τ-[d+2)k+e(G)-d+cos(kπ/d)+cos(π/d)sin(kπ/d)+sin(kπ)/d/sin(π/d)and Sk(C)≤k(τ+s2-s1)+e(G)-s1(s2-1)respectively,where Ks1,s2(s1≤s2)is the maximum complete bipartite subgraph of G.