Upper bounds on the (signless) Laplacian eigenvalues of graphs

Let G be a simple graph of order n   with vertex set V={v1,v2,…,vn}V={v1,v2,…,vn}. Also let μ1(G)≥μ2(G)≥⋯≥μn−1(G)≥μn(G)=0μ1(G)≥μ2(G)≥⋯≥μn−1(G)≥μn(G)=0 and q1(G)≥q2(G)≥⋯≥qn(G)≥0q1(G)≥q2(G)≥⋯≥qn(G)≥0 be the Laplacian eigenvalues and signless Laplacian eigenvalues of G  , respectively. In this paper we obtain μi(G)≤i−1+minUi⁡max⁡{|NH(vk)∪NH(vj)|:vkvj∈E(H)}μi(G)≤i−1+minUi⁡max⁡{|NH(vk)∪NH(vj)|:vkvj∈E(H)}, where NH(vk)NH(vk) is the set of neighbors of vertex vkvk in V(H)=V(G)\UiV(H)=V(G)\Ui, UiUi is any (i−1)(i−1)-subset of V(G)V(G) (here, we agree that i∈{1,…,n−1}i∈{1,…,n−1} and μi(G)≤i−1μi(G)≤i−1 if E(H)=∅E(H)=∅). For any graph G, this bound does not exceed the order of G. Moreover, we prove thatmax⁡{μi(G),qi(G)}≤maxi≤k≤n⁡{dG(vk)+∑vj∈NG(vk)∩NdG(vj)dG(vk)}≤2dG(vi), where dG(vi)dG(vi) is the i-th largest degree of G   and N={vi,vi+1,…,vn}N={vi,vi+1,…,vn}.