Generalized binomial edge ideals of whisker graphs via an extension of generalized corona products

In this paper, we initiate a systematic study of generalized binomial edge ideals of whisker graphs by working within a substantially broader class of graphs. We extend the notion of generalized corona products, and through this enlarged framework, investigate fundamental algebraic invariants such as depth, (Castelnuovo-Mumford) regularity, and the Cohen-Macaulay property. In particular, we establish a sharp lower bound on the depth of generalized binomial edge ideals for our extended class, and further obtain explicit depth formula for a broad subclass of this family, which in turn recovers the depth formula for whisker graphs. We also establish sharp upper bounds for the regularity, and in the case of binomial edge ideals of whisker graphs over gap-free graphs, determine the exact value of the regularity. Finally, for our extended class, we provide a combinatorial classification of all Cohen-Macaulay binomial edge ideals, which in turn yields a new construction of Cohen-Macaulay binomial edge ideals.

💡 Research Summary

The paper initiates a systematic investigation of generalized binomial edge ideals (GBEIs) associated with whisker graphs, but does so within a much broader family of graphs obtained by extending the notion of generalized corona products. The authors define two new classes of graphs, G₁ and G₂. G₁ consists of graphs obtained by attaching a whisker to each vertex in a chosen subset S of a base graph G, while G₂ allows arbitrary graphs H₁,…,Hₖ to be attached to the vertices of S. This construction generalizes the ordinary corona product (when all Hᵢ are equal) and the classical whisker graph (when each Hᵢ is a single vertex K₁).

The main contributions are threefold: depth, regularity, and Cohen‑Macaulay classification.

Depth.
Theorem 3.3 provides a sharp lower bound for the depth of the GBEI of any graph D ∈ G₂: \