On The Distance Laplacian And Distance Signless Laplacian Eigenvalues Of Graphs

Spectral Graph Theory is an important research domain of Algebra Graph Theory and Combinatorial Matrix Theory, which has been developed fast and attracted much attention in recent decades. In addition, what the Spectral Graph Theory involves is the various of matrices and corresponding eigenvalues of graphs. The distance Laplacian matrix and distance signless Laplacian matrix, as the generalization of Laplacian matrix and signless Laplacian matrix, are proposed by M. Aouchche and P. Hansen in2013. The definitions of them are as follows: L(G) = diag(T r)- D(G) and Q(G) =diag(T r) + D(G), where diag(T r) is a diagonal matrix of the vertex transmissions of G and D(G) denotes the distance matrix of G.In this paper, we mainly prove the following 5 conjectures proposed by M.Aouchche and P. Hansen:Conjecture 1 If T is a tree with order n(≥ 5), then ?2(T) ≥ 2n- 1 with equality if and only if T = Sn.Conjecture 2 Let T be a tree on n(≥ 4) vertices and q2 be the second largest eigenvalue of Q(T). Then q2≥ 2n- 5, with equality if and only if T = Sn.Conjecture 3 For any graph G on n(≥ 4) vertices, ?2(G) ≥ n with equality if and only if G = Kn, or G = Kn- e for some e ∈ E(Kn).Conjecture 4 Let G be a unicyclic graph with order n(≥ 6), then ?1(G) ≥ ?1(S+n)with equality if and only if G = S+n.Conjecture 5 Let G be a unicyclic graph with order n(≥ 6), then ?2(G) ≥ ?2(S+n)with equality if and only if G = S+n, where ?2(G) denotes the second largest distance Laplacian eigenvalue of G.Moreover, we claim that, in Conjecture 5, the order of the unicyclic graph must be n ≥ 7, not n ≥ 6.