Spectral characterization of the complete graph removing a path of small length

A graph $G$ is said to be \emph{determined by its spectrum} if any graph having the same spectrum as $G$ is isomorphic to $G$. Let $K_n \setminus P_{\ell}$bethegraphobtainedfrom$K_n$byremovingedgesof$P_\ell$,where$P_\ell$isapathoflength$\ell-1$whichisasubgraphofacompletegraph$K_n$\mathrm{.}C\acute{a}maraandHaemers\text{cite}MCconjecturedthat$K_n \backslash P_{\ell}$isdeterminedbyitsadjacencyspectrumforevery$2\leq \ell \leq n$\mathrm{.}Inthispaperweshowthattheconjectureistruefor$7\leq \ell \leq9$.