Some notes on graphs whose spectral radius is close to

In Belardo et al. [F. Belardo, E.M. Li Marzi, S.K. Simić, Some notes on graphs whose index is close to 2, Linear Algebra Appl. 423 (2007) 81–89] the authors considered 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; by assuming that each graph within these classes has two vertices of degree 3 at distance k, they order by the spectral radius the graphs from (i) for any fixed k(1⩽k⩽d-2) and the graphs from (ii) for . In this paper we consider two classes of graphs denoted by (or simply ) and , containing respectively the classes (i) and (ii). The graphs in the first class () are paths of length n (called main paths) having two hanging paths of length m at vertices i and j. The graphs in the second class () are cycles of girth g having two hanging paths of length s and t at vertices at distance k (). For graphs in these latter two classes we give an ordering, with respect to the spectral radius, extending the one shown in Belardo et al. (2007). Furthermore we give an upper bound for the spectral radius of the graphs in and a lower and an upper bound for graphs in , following the limit point technique used in Belardo et al. [F. Belardo, E.M. Li Marzi, S.K. Simić, Path-like graphs ordered by the index, Int. J. Algebra 1 (3) (2007) 113–128].