Classify group-theoretic information preserved by Proper POE Complement Graphs

Determine which group-theoretic information is preserved by the Proper POE Complement Graph construction $G\mapsto D^*(G)$, and classify all pairwise non-isomorphic finite groups that give rise to a fixed graph $X_{h,r}$ under this construction.

Background

The paper establishes that the graph D∗(G)D^*(G) can fail to determine the isomorphism type of its realizing finite group. In particular, distinct Abelian and non-Abelian finite $2$-groups can produce isomorphic graphs X2m,2d−1X_{2^m,2^d-1} and consequently identical adjacency spectra.

Although the paper gives explicit families of such collisions and characterizes realizability for the considered graph family, it does not determine the complete group-theoretic information encoded by D∗(G)D^*(G) or classify every non-isomorphic group realizing a given Xh,rX_{h,r}. The unresolved problem therefore concerns the full inverse theory of the Proper POE Complement Graph construction.

References

Hence the family provides infinitely many explicit examples in which distinct algebraic structures are indistinguishable at the level of the associated graph and its adjacency spectrum. This raises the broader inverse problem of determining which group-theoretic information is preserved by $D*(G)$, and of classifying all non-isomorphic groups that give rise to a fixed graph $X_{h,r}$.

— Structural Characterizations and Algebraic Realizations of a Family of Regular Integral Graphs  (2609.25957 - Manna et al., 22 Sep 2026) in Remark \ref{rem:inverse-problem}, Section 5, “Group Realizations of the Graph $X_{h,r}$”