Sharp bounds on the eigenvalues of trees

Let λ1(T) and λ2(T) be the largest and the second largest eigenvalues of a tree T, respectively. We obtain the following sharp lower bound for λ1(T): where di is the degree of the vertex vi and mi is the average degree of the adjacent vertices of vi. Equality holds if and only if T is a tree T(di,dj), where T(di,dj) is formed by joining the centers of di copies of K1,dj−1 to a new vertex vi, that is T(di,dj)−vi=diK1,dj−1. Let d1 and d2 be the highest and the second highest degree of T, respectively. Let r(T) be the maximum distance between the highest and the second highest degree vertices. We also show that if T is a tree of order (n> 2), then The equality holds if T is a tree T1 or a tree T2, or T is a tree T4 and d1=d2, where T1 is formed by joining the centers of K1,d1−1 and K1,d2−1 and T2 is formed by joining the centers of K1,d1−1 and K1,d2−1 to a new vertex, the T4 is formed by joining a 1-degree vertex of K1,d1 and K1,d2 to a new vertex.