Emms–Hancock–Severini–Wilson conjecture on positive supports of Grover walks

Let $G$ and $H$ be strongly regular graphs. Let $U_G$ and $U_H$ denote the time evolution operators of the Grover walks induced by $G$ and $H$, respectively, and for a real matrix $M$ let $S(M)$ be its positive support, defined by

$$S(M)_{b,a}=\begin{cases}1&\text{if }M_{b,a}>0,\\0&\text{if }M_{b,a}\leq 0.\end{cases}$$

Let $\sigma(M)$ denote the spectrum of a matrix $M$.

Emms–Hancock–Severini–Wilson conjecture.

$$G\cong H\Longleftrightarrow\sigma\bigl(S(U_G^3)\bigr)=\sigma\bigl(S(U_H^3)\bigr).$$

The conjecture proposes that the spectrum of the positive support of the cube of the Grover-walk operator distinguishes non-isomorphic strongly regular graphs, and is motivated by the connection between Grover-walk positive supports, graph spectra, and graph isomorphism. Finding the class of strongly regular graphs for which the conjecture holds remains an interesting open problem.