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 mcospectrality 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.
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