\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf R. Boulet, B. Jouve}
%
%
\medskip
\noindent
%
%
{\bf The Lollipop Graph is Determined by its Spectrum}
%
%
\vskip 5mm
\noindent
%
%
%
%
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

.
