In this note, we consider connected graphs with exactly two main eigenvalues. We will give several constructions for them, and as a consequence we show a family of those graphs with an unbounded number of distinct va-lencies.
The †-shape tree is the coalescence of the star K 1,4 and the path P n−4 with respect to two pendent vertices. In this paper, it is showed that the †-shape tree is determined by its adjacency spectrum if and only if n = 2k + 9 (k = 0, 1,. . .). Furthermore, all the cospectral mates of the †-shape tree are found when n = 2k + 9.