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}. Also let ^μ1(G)^μ2(G)μn-₁(G)^μn(G)=0 and ^q1(G)^q2(G)^qn(G)0 be the Laplacian eigenvalues and signless Laplacian eigenvalues of G, respectively. In this paper we obtain ^μi(G)bmax{|NH(^vk)^NH(^vj)|:^vkvjE(H)}, where ^NH(^vk) is the set of neighbors of vertex ^vk in V(H)=V(G)\^Ui, ^Ui is any (i-1)-subset of V(G) (here, we agree that i{1,⋯,n-1} and ^μi(G)i-1 if E(H)=). For any graph G, this bound does not exceed the order of G. Moreover, we prove thatmax{(G)};maxin{^dG(^vk)+^vjNG(^vk)N^dG(^vj)^dG(^vk)}, where is the i-th largest degree of G and N.