2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/146282The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained. Using this connection, a few results in the minimum rank problem are derived by applying some known results from projective geometry.23 pages, 5 figures, 1 Sage programCombinatorics05C50, 05C75, 15A03, 05B25, 51E20text