Generalized Binomial Edge Ideals

• Generalized binomial edge ideals are determinantal ideals generated by 2×2 minors in an m×n matrix that extend classical binomial edge ideals to higher-rank settings.
• They encode connectivity information in generalized block graphs through clique decompositions and minimal cut-sets, leading to explicit formulas for depth and regularity.
• Inductive proof techniques based on leaf-peeling reveal how combinatorial structures directly influence homological invariants, opening new pathways for further research.

A generalized binomial edge ideal is a determinantal ideal constructed from a graph, extending the classical binomial edge ideals to higher-rank settings via 2-minors of an $m \times n$ matrix of indeterminates. This framework is particularly tractable and homologically rich for graph classes known as generalized block graphs, which generalize block graphs by relaxing intersection properties among maximal cliques while retaining strong chordality. Generalized binomial edge ideals encode granular connectivity information linked to the combinatorial structure of the underlying graph, yielding precise formulas for depth and regularity, and facilitating a powerful induction based on clique decompositions.

1. Definition and Construction

Let $K$ be a field, and fix integers $m, n \geq 2$. Let $S = K[x_{ij} : 1 \leq i \leq m,\, 1 \leq j \leq n]$ be the polynomial ring formed from an $m \times n$ matrix of indeterminates $X = (x_{ij})$. Given a simple graph $G$ on $[n]=\{1,\ldots,n\}$, its generalized binomial edge ideal $J_G \subset S$ is generated by all $2 \times 2$ minors of $X$ taken over pairs of columns indexed by edges of $G$: $$J_G = \left( P_{k\ell|ij} : 1 \leq k < \ell \leq m,\ \{i,j\} \in E(G),\ i<j \right)$$ where $$P_{k\ell|ij} = x_{k,i} x_{\ell,j} - x_{\ell,i} x_{k,j}$$ (Chaudhry et al., 2017). For $m=2$, this reduces to the classical binomial edge ideal introduced by Herzog–Hibi–Rauh.

2. Generalized Block Graphs: Combinatorics and Structure

A generalized block graph is a chordal graph $G$ with the property that, for every triple of distinct maximal cliques $F_i, F_j, F_k$ with nonempty intersection, all pairwise intersections coincide; that is, $F_i \cap F_j = F_i \cap F_k = F_j \cap F_k$ (Chaudhry et al., 2017). Block graphs are the special case where maximal cliques meet in at most one vertex.

Key associated data include:

• $\omega(G)$: clique number (size of largest clique).
• $\mathcal{D}_i(G)$: set of minimal cut-sets of size $i$.
• $a_i(G):=|\mathcal{D}_i(G)|$. For block graphs, $a_i(G)=0$ whenever $i>1$.

3. Depth and Regularity: Main Homological Formulas

For $m, n \geq 2$ and $G$ a connected generalized block graph on $n$ vertices, the homological invariants of $S/J_G$ admit closed combinatorial formulas (Chaudhry et al., 2017):

• Depth:

$\depth\, S/J_G = n + (m-1) - \sum_{i=2}^{\omega(G)-1} (i-1)\, a_i(G)$

This formula also holds for the initial ideal $\mathrm{in}_<(J_G)$ under the lex order $x_{1,1} > \ldots > x_{m,n}$.

• Castelnuovo–Mumford Regularity ($m \geq n$):

$\reg\, S/J_G = \reg\, S/\mathrm{in}_<(J_G) = n-1$

• Regularity ($m < n$):

$\reg(J_G) \leq \reg(\mathrm{in}_<(J_G)) \leq n-1$

These invariants are governed by the combinatorial cut-set data $a_i(G)$, quantifying the complexity of intersection structures among cliques.

4. Inductive Proof Techniques and Structural Decomposition

The key method is induction on the number of maximal cliques $r$ in $G$, performed via leaf-peeling (clique-splitting) strategies rooted in chordal graph theory:

• Dirac’s theorem ensures the clique complex admits a leaf order $F_1, \ldots, F_r$.
• Peeling the final leaf $F_r$ leads to a decomposition involving a minimal cut-set $A$, and the ideal breaks as:

$$J_G = J_{G'} \cap \left( (\{x_{1j},...,x_{mj} : j \in A\}) + J_{G''} \right)$$

where $G'$ merges $F_r$ and its branches, and $G''$ is the induced subgraph on $[n] \setminus A$.

• Both $G'$ and $G''$ remain generalized block graphs, allowing induction on depth and regularity.
• The Depth Lemma and regularity cut-and-paste techniques finalize the argument, tracking how $a_i(G)$ evolves under the split.

5. Explicit Example

Consider $G$ on five vertices with cliques $F_1 = \{1,2,3\}$, $F_2 = \{1,2,4\}$, $F_3 = \{1,2,5\}$, where intersections satisfy $$F_1 \cap F_2 \cap F_3 = F_1 \cap F_2 = F_1 \cap F_3 = F_2 \cap F_3 = \{1,2\}$$. Here, $a_2(G)=1$ (one minimal cut-set of size 2), $a_i(G)=0$ for $i \neq 2$, and $\omega(G)=3$. The depth formula gives

$\depth\, S/J_G = 5 + (m-1) - (2-1)\cdot 1 = m + 3$

For $m=2$, $\depth\, S/J_G = 5$; for $m \geq 5$, $\reg\, S/J_G=4$; for $m < 5$, the bound $\reg J_G \leq 4$ applies.

6. Unmixedness and Cohen–Macaulayness

In block graphs ($a_i(G)=0$ for $i>1$), the depth formula specializes to $n+(m-1)$. In this regime, $J_G$ is unmixed (all minimal primes of equal height) and Cohen–Macaulay precisely when $G$ is complete. The presence of non-singleton cut-sets ($a_i(G)>0$ for some $i>1$) signals the failure of unmixedness or Cohen–Macaulayness (Chaudhry et al., 2017, Kiani et al., 2015).

7. Context, Extensions, and Open Problems

For $m=2$, formulas recover and extend classical results for binomial edge ideals (Ene–Herzog–Hibi, Dokuyucu) with tight bounds via path lengths and explicit Betti number computations. For generalized block graphs with $m>2$, depth and regularity are now settled, but much remains open:

• Combinatorial classifications of graded Betti numbers for broader chordal families.
• Criteria for linear resolutions outside the complete graph case.
• The behavior of invariants under graph operations like clique gluing or vertex set amalgamation.

The foundation for extending these results arises from the methodology—splitting on minimal cut-sets, induction on clique number, and Gröbner basis control (Chaudhry et al., 2017). These offer a roadmap for future research in combinatorial commutative algebra concerning both binomial and generalized binomial edge ideals.

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References:

• On The Generalized Binomial Edge Ideals of Generalized Block Graphs (Chaudhry et al., 2017)