Some notes on graphs whose index is close to 2 Original Research Article

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 less-than-or-equals, slant k less-than-or-equals, slant d − 2), and the graphs from (ii) independently of k.