  Transposition graph $T_n$ is defined as a Cayley graph over the symmetric
group generated by all transpositions. It is known that all eigenvalues of
$T_n$ are integers. Moreover, zero is its eigenvalue for any $n\geqslant 4$.
But the exact distribution of the spectrum of the graph $T_n$ is unknown. In
this paper we prove that integers from the interval $[-\frac{n-4}{2},
\frac{n-4}{2}]$ lie in the spectrum of $T_n$ if $n \geqslant 19$.
