  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. However, an explicit description of the spectrum is
unknown. In this paper we prove that for any integer $k\geqslant 0$ there
exists $n_0$ such that for any $n\geqslant n_0$ and any $m \in \{0, \dots,
k\}$, $m$ is an eigenvalue of $T_n$. In particular, it is proved that zero is
an eigenvalue of $T_n$ for any $n\neq2$, and one is an eigenvalue of $T_n$ for
any odd $n\geqslant 7$ and for any even $n \geqslant 14$. We also present exact
values of the third and the fourth largest eigenvalues of $T_n$ with their
multiplicities.
