Homological invariants of edge ideals of the multiple extended complete split-like graphs
2026-07-11 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study a specific family of graphs built by connecting independent sets to multiple copies of a particular small graph block. They use advanced algebraic tools like Hochster's formula and Betti number calculations to describe various properties of these graphs, including their independence complexes and algebraic invariants such as regularity and projective dimension. They also identify which graphs in this family have certain nice properties (well-covered and unmixed) and prove that none of them are Cohen–Macaulay. Additionally, they provide methods to compute important algebraic data related to these graphs.
graph joinindependence complexBetti numbersHochster's formulafree resolutionsregularityprojective dimensionCohen–Macaulaywell-covered graphsunmixed graphs
Authors
Bilal Ahmad Rather
Abstract
We study the graphs $MECS_{b,n}^a \cong \overline{K}_a \join \big(n(K_b+K_2)\big)$, obtained by attaching an independent set of size $a$ to $n$ disjoint copies of the block $K_b+K_2$. For $n=1$, we get $MECS_{b,1}^a$, and recover the results of one-block case studied in [Anand, Gupta, Rather and Singh, Homological invariants of some complete split-like graphs, Beitr. Algebra Geom. (2025)]. Using Hochster's formula, tensor products of minimal free resolutions over disjoint variable sets, and the Betti-number formula for graph joins, we derive explicit descriptions of the independence complex, independence polynomial and its analytic properties, Hilbert series, linear and quadratic Betti strands, regularity, projective dimension, and several structural invariants of $MECS_{b,n}^a$. We further classify the well-covered and unmixed members, compute induced matching numbers, show that the family is never Cohen--Macaulay, and record algorithmic procedures for evaluating the Betti data.