Almost all graphs have no cospectral mates with height relative small to its order

The main result of this paper shows that almost all graphs of order $n$ have no cospectral mates with height $o((n\mathrm{/}\ln n{)}^{1\mathrm{/}10})$, improving an earlier result on cospectral mates with fixed level and covering the cospectral graphs obtained by GM-switching of relative small blocks.