Graded Betti numbers of generalized split--join graphs and applications

We determine the full graded Betti tables of graph families that subsume several classes studied recently in the literature, namely the generalized multiple complete split-like graphs and the generalized clique-star graphs with arbitrary clique block sizes.

The method combines Hochster's formula with a precise decomposition of the associated independence complexes into disjoint unions of simplices and iterated joins of discrete complexes.

This reduces every graded Betti number to an explicit coefficient extraction formula and yields closed expressions for the linear strand, higher strands, Hilbert series, regularity, projective dimension, and extremal Betti numbers.

In particular, we prove a sharp criterion for $2$-linear resolution and identify the regularity corner in terms of the number of nontrivial clique blocks.

As applications, we recover and extend earlier results on equal-block split-like graphs, obtain complete formulas for pineapple graphs, and derive consequences for power graphs of cyclic groups, elementary abelian groups, and prime-power dihedral groups.