Homological invariants of edge ideals of the multiple extended complete split-like graphs

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.