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{\bf R. Boulet, B. Jouve}
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{\bf The Lollipop Graph is Determined by its Spectrum}
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An even (resp.\ odd) lollipop is the coalescence of a cycle of even
(resp.\ odd) length and a path with pendant vertex as distinguished
vertex. It is known that the odd lollipop is determined by its
spectrum and the question is asked by W.~Haemers, X.~Liu and Y.~Zhang
for the even lollipop. A private communication of Behruz Tayfeh-Rezaie
pointed out that an even lollipop with a cycle of length at least $6$
is determined by its spectrum but the result for lollipops with a
cycle of length $4$ is still unknown. We give an unified proof for
lollipops with a cycle of length not equal to $4$, generalize it for
lollipops with a cycle of length $4$ and therefore answer the
question. Our proof is essentially based on a method of counting
closed walks.

\bye

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