Some notes on graphs whose index is close to 2

Abstract

We consider two classes of graphs: (i) trees of order n and diameter d =n - 3 and (ii) unicyclic graphs of order n and girth g = n - 2. Assuming that each graph within these classes has two vertices of degree 3 at distance k, we order by the index (i.e. spectral radius) the graphs from (i) for any fixed k (1 ≤ k ≤ d - 2), and the graphs from (ii) independently of k.