Characterization of graphs produced by weighted edge-partition tensor constructions

Characterize all graphs that arise by partitioning an edge set into subgraphs with two distinct eigenvalues, weighting the edges so that the subgraphs share the same two eigenvalues, and applying the corresponding tensor-product construction.

Background

The paper constructs graphs with q(G) = 2 by partitioning edge sets into pieces whose associated 0–1 matrices have the same two eigenvalues, then combining these matrices through tensor products. The authors observe that Lemma 2.3 of [3] allows any graph with q(G) = 2 to realize any prescribed pair of eigenvalues, suggesting a broader construction in which the edges of subgraphs with q = 2 are weighted so that their associated matrices share a common pair of eigenvalues.

For unweighted constructions, Lemma 6 determines the resulting matrix pattern from the sum of the component matrices. With weighted edges, however, that lemma cannot be applied directly, and the authors state that the pattern of the resulting graph is not clear. The unresolved problem is therefore to characterize precisely the graphs generated by this weighted edge-partition and tensor-product method.