On the spectral characterization of ?-shape trees

A ?-shape tree is a tree with exactly two of its vertices having the maximum degree 3. In this article, we classify the ?-shape trees into six types according to the number of their closed walks of length 6. Then we complete the spectral characterization for one type. We show that all graphs of one such type are determined by the spectrum. Another type i.e., W n is known to have the unique cospectral mate C 4???P n . Moreover, we find cospectral graphs of some graphs for the remaining four types.