A Square-freeness Condition for Graphs Determined by Their Generalized Spectra

A graph $G$ is said to be determined by the generalized spectrum (DGS), if for any graph $H$, whenever $H$ and $G$ are cospectral and their complements are also cospectral, then $H$ is isomorphic to $G$. Previously, we show a graph $G$ is DGS provided $2^{-\lfloor n/2\rfloor}\det W(G)$ is odd and square-free, where $W(G)$ denotes the walk-matrix of $G$. In this talk, we show that the oddness assumption can actually be removed, thereby confirming a two-decades conjecture of the second author. Joint work with Quanyu Tang and Hao Zhang.