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

TL;DR

This paper proves that nearly all graphs of a given size lack cospectral mates within a certain small height, advancing previous results on cospectral graph properties.

Contribution

It improves existing bounds on the height of cospectral mates for almost all graphs, extending prior work on GM-switching and fixed-level cospectral graphs.

Findings

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

The result refines bounds on the height of cospectral mates for typical graphs.

It extends previous results on cospectral graphs obtained by GM-switching.

Abstract

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