2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115475We study the combinatorial analogues of the classical invariants of measurable equivalence relations. We introduce the notion of cost and $β$-invariants (the analogue of the first $L^2$-Betti number introduced by Gaboriau) for sequences of finite graphs with uniformly bounded vertex degrees and examine the relation of these invariants and the rank gradient resp. mod $p$ homology gradient invariants introduced by Lackenby for residually finite groups.Group TheoryCombinatorics20F65text