PARITY OF AN ODD DOMINATING SET

For a simple graph G with vertex set V (G) = {v1, ..., vn}, we de-fine the closed neighborhood set of a vertex u as N[u] = {v ??? V (G) | v is adja-cent to u or v = u} and the closed neighborhood matrix N(G) as the matrix obtained by setting to 1 all the diagonal entries of the adjacency matrix of G. 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 an odd dominating set of G is equal to the parity of the rank of G, where the 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.