Abstract :
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):imagewheredi and mi denote the degree of vertex vi and the average of the degrees of the vertices adjacent to vertex vi respectively. Equality holds if and only if T is a tree T(di,dj), where T(di,dj) is formed by joining the centres of di copies of K1,dj−1 to a new vertex vi, that is, T(di,dj)−vi=diK1,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, thenimagewhereE 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 K1,d1−1 and T2(d1) is formed by joining the centres of two copies of K1,d1−1 to a new vertex.
Moreover, we obtain the lower bounds for the sum of two largest Laplacian eigenvalues.
