Constructing segments of quadratic length in $Spec(T_n)$ through segments of linear length

A Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group $Sym_n$ generated by all transpositions. It is known that the spectrum of $T_n$ consists of integers, but it is not known exactly how these numbers are distributed. In this paper we prove that integers from the segment $[-n, n]$ lie in the spectrum of $T_n$ for any $n\geqslant 31$. Using this fact we also prove the main result of this paper that a segment of quadratic length with respect to $n$ lies in the spectrum of $T_n$.