Batal, Mehmet2022-12-292022-12-292022https://doi.org/10.31801/cfsuasmas.1051208http://hdl.handle.net/20.500.12575/86611For a simple graph G with vertex set V ( G ) = { v 1 , . . . , v n } , we define the closed neighborhood set of a vertex u as \\ N [ u ] = { v ∈ V ( G ) | v is adjacent to u or v = u } and the closed neighborhood matrix N ( G ) as the matrix whose i th column is the characteristic vector of N [ v i ] . We say a set S is odd dominating if N [ u ] ∩ S is odd for all u ∈ V ( G ) . We prove that the parity of the cardinality of an odd dominating set of G is equal to the parity of the rank of G , where rank of G is defined as the dimension of the column space of N ( G ) . Using this result we prove several corollaries in one of which we obtain a general formula for the nullity of the join of graphs.enLights out, all-ones problem, odd dominating set, parity domination, domination numberArticle714102310281303-5991