Indices for special classes of trees

In this article we study the index (the largest eigenvalue of the adjacency matrix) in two special classes of trees, namely: starlike trees and double brooms  . For each class, we determine conditions for the index to be integer. We examine conditions in order to compare the index of a starlike tree with Δ+1, where Δ is the maximum degree of the graph. We also prove that indices of a broom-like tree and a double broom, except for some cases, are between Δ and Δ+1, characterizing when occurs one equality. Furthermore, we build in these classes, infinite families of non-integral trees with integer index.