On the spectral characterization of the p-sun and the (p,q)-double sun

In 1973 Schwenk [7] proved that almost every tree has a cospectral mate. Inspired by Schwenk's result, in this paper we study the spectrum of two families of trees. The p-sun of order 2p+1 is a star K1,p with an edge attached to each pendant vertex, which we show to be determined by its spectrum among connected graphs. The (p,q)-double sun of order 2(p+q+1) is the union of a p-sun and a q-sun by adding an edge between their central vertices. We determine when the (p,q)-double sun has a cospectral mate and when it is determined by its spectrum among connected graphs. Our method is based on the fact that these trees have few distinct eigenvalues and we are able to take advantage of their nullity to shorten the list of candidates.