Spectral ordering of trees with small index

Studying graphs by means of the largest eigenvalue of the adjacency matrix (the index) has been a problem largely investigated in Spectral Graph Theory. Here, we consider trees with index in the real interval (2,√(2+√5)) and their ordering with respect to the index. For such graphs, we prove three indices comparisons whose order has been conjectured in [F. Belardo, E.M. Li Marzi, S.K. Simić, Ordering graphs with index in the interval (2,√(2+√5)), Discrete Appl. Math., 156 (2008), pp. 1670-1682]. By doing so, we prove a conjectured ordering of trees with smallest indices, open since 2002, and we obtain the first 8 trees of even order with largest index.