Indices of trees with a prescribed diameter

Abstract

The index of a graph is is the largest eigenvalue of its adjacency matrix. Let Tn,d be the class of trees with n vertices and diameter d. For all integers n and d with 4 ≤ d ≤ n - 3 we identify in Tn,d the tree with the k-th largest index for all k up to + 1 if d ≤ n - 4, or for all k up to if d = n - 3.