The lollipop graph is determined by its spectrum

TL;DR

This paper proves that both even and odd lollipop graphs are uniquely identified by their spectra, extending previous results and answering a specific open question in spectral graph theory.

Contribution

The authors generalize the spectral determination from odd to even lollipop graphs using a counting closed walks method.

Findings

Both even and odd lollipop graphs are determined by their spectra.

The proof is based on counting closed walks in the graph.

The result answers an open question in spectral graph theory.

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. We revisit the proof for odd lollipop, generalize it for even lollipop and therefore answer the question. Our proof is essentially based on a method of counting closed walks.