Sharp bounds on the eigenvalues of trees

Let λ1(T)λ1(T) and λ2(T)λ2(T) be the largest and the second largest eigenvalues of a tree TT, respectively. We obtain the following sharp lower bound for λ1(T)λ1(T): λ1(T)≥max{di+mi−1}, where didi is the degree of the vertex vivi and mimi is the average degree of the adjacent vertices of vivi. Equality holds if and only if TT is a tree T(di,dj)T(di,dj), where T(di,dj)T(di,dj) is formed by joining the centers of didi copies of K1,dj−1K1,dj−1 to a new vertex vivi, that is T(di,dj)−vi=diK1,dj−1T(di,dj)−vi=diK1,dj−1.Let d1d1 and d2d2 be the highest and the second highest degree of TT, respectively. Let r(T)r(T) be the maximum distance between the highest and the second highest degree vertices. We also show that if TT is a tree of order (n>n> 2), then λ2(T)≥{d1+d2−1−(d1+d2−1)2−4(d1−1)(d2−1)2ifr(T)=1,d1+d2−(d1−d2)2+42ifr(T)=2,d1−1ifr(T)=3andd1=d2,d2otherwise . The equality holds if TT is a tree T1T1 or a tree T2T2, or TT is a tree T4T4 and d1=d2d1=d2, where T1T1 is formed by joining the centers of K1,d1−1K1,d1−1 and K1,d2−1K1,d2−1 and T2T2 is formed by joining the centers of K1,d1−1K1,d1−1 and K1,d2−1K1,d2−1 to a new vertex, the T4T4 is formed by joining a 1-degree vertex of K1,d1K1,d1 and K1,d2K1,d2 to a new vertex.