Indices of Trees with a Prescribed Diameter

Let G = (V (G), E(G)) be a simple graph, and let A be its adjacency matrix. The characteristic polynomial det(xI −A) of A is called the characteristic polynomial of G, and is denoted by φ(G, x). The eigenvalues of A (i.e. the zeros φ(G, x)) are called the eigenvalues of G. The index of a graph G is the largest eigenvalue of G, denoted by ρ(G). It has been studied extensively in the literature [1, 3]; see also [2]. In the class of all trees with n ≥ 6 vertices, M. Hofmeister [7] determined the trees whose indices are the k–th largest, for k = 1, 2, . . . , 5. Let Tn,d be the class of trees on n vertices and diameter d, with 2 ≤ d ≤ n−1. Obviously, if T ∈ Tn,2 then T is a star (= K1,n−1), and if T ∈ Tn,n−1 then T is a path (= Pn). In addition, if d = 3 or d = n− 2 then (as noted in Section 3) we can easily order all trees in non-increasing order with respect to the index. So we can assume further (as mentioned in the abstract) that 4 ≤ d ≤ n− 3. The main purpose of this paper is to identify the trees in Tn,d having the k-th largest index, for all k up to ⌊ d 2 ⌋ + 1, or ⌊ d 2 ⌋ if d = n− 3.