Sharp lower bounds on the Laplacian eigenvalues of trees

Let lambda(1)(T) and lambda(2)(T) be the largest and the second largest Laplacian eigenvalues of a tree T. We obtain the following sharp lower bound for lambda(1)(T): lambda(1)(T) greater than or equal to max {(d(i) + m(i) + 1) + root(d(i) + m(i) + 1)(2) - 4(d(i)m(i) + 1)/2 : v(i) is an element of V}, where d(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,K-dj-1 to a new vertex v(i), that is, T(d(i),d(j)) - v(i) = d(i)K(1,dj-1). Let v(1) be the highest degree vertex of degree d(1) and v(2) be the second highest degree vertex of degree d(2). We also show that if T is a tree of order n > 2, then [GRAPHICS] where E is the set of edges. Equality holds if T = T-1(d(1)) or T = T-2(d(1)), where T-1(d(1)) is formed by joining the centres of two copies of K-1,K-d1-1 and T-2(d(1)) is formed by joining the centres of two copies of K-1,K-d1-1 to a new vertex. Moreover, we obtain the lower bounds for the sum of two largest Laplacian eigenvalues. (C) 2004 Published by Elsevier Inc.