  A semiregular tree is a tree where all non-pendant vertices have the same
degree. Belardo et al. (MATCH Commun. Math. Chem. 61(2), pp. 503-515, 2009)
have shown that among all semiregular trees with a fixed order and degree, a
graph with index is a caterpillar. In this technical report we provide a
different proof for this theorem. Furthermore, we give counter examples that
show this result cannot be generalized to the class of trees with a given
(non-constant) degree sequence.

Combinatorics;Spectral Theory