Sharp lower bounds on the Laplacian eigenvalues of trees

Let λ₁(T) and λ₂(T) be the largest and the second largest Laplacian eigenvalues of a tree T. We obtain the following sharp lower bound for λ₁(T):λ₁(T) ≥max{d_i+m_i+1)+√(d_i+m _i+1)²-4(d_im_i+1)/2:v_i ∈ V}, whered_i and m_i denote the degree of vertex v_i and the average of the degrees of the vertices adjacent to vertex v_i respectively. Equality holds if and only if T is a tree T(d _i,d_j), where T(d_i,d_j) is formed by joining the centres of d_i copies of K_1,dj-1 to a new vertex v_i, that is, T(d_i,d_j)-v _i=d_iK_1,dj-1. Let v₁ be the highest degree vertex of degree d₁ and v₂ be the second highest degree vertex of degree d₂. We also show that if T is a tree of order n>2, thenλ₂(T)≥{d₂ifv₁v ₂ ∈ E,(d₂+1)+√(d₂+1)²-4/2 ifv₁v₂ ∉ E, where E is the set of edges. Equality holds if T = T₁(d₁) or T = T₂(d₁), where T₁(d₁) is formed by joining the centres of two copies of K_1,dj-1 and T₂(d₁) is formed by joining the centres of two copies of K_1,dj-1 to a new vertex. Moreover, we obtain the lower bounds for the sum of two largest Laplacian eigenvalues.