The DGA$_\alpha$-spectral determination conjecture for graphs in $\mathfrak{F}_n$

Let $\mathfrak{F}_n$ denote the class of graphs used in the paper, and let $n$ be the order of the graph. For $\alpha\in[0,1)$, let $c_\alpha$ be the associated parameter and let DGA$_\alpha$S mean that the graph is determined by its generalized $A_\alpha$-spectrum.

DGA$_\alpha$-spectral determination conjecture. If $G\in\mathfrak{F}_n$, $n$ is even, and $c_\alpha\geqslant 3$ is odd, then $G$ is DGA$_\alpha$S.

The conjecture addresses the case excluded from the paper's main theorem, which treats the arithmetic criterion for determining graphs by their generalized $A_\alpha$-spectrum. Its resolution is not provided here.