The Lollipop Graph is Determined by its Spectrum

Abstract

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.