Graphs cospectral with starlike trees

A tree which has exactly one vertex of degree greater than two is said to be starlike. In spite of seemingly simple structure of these trees, not much is known about their spectral properties. In this paper, we introduce a generalization of the notion of cospectrality called m-cospectrality which turns out to be useful in constructing cospectral graphs. Based on this, we construct cospectral mates for some starlike trees. We also present a set of necessary and sufficient conditions for divisibility of the characteristic polynomial of a starlike tree by the characteristic polynomial of a path.