The Adjacency Matrices of Complete and Nutful Graphs

A real symmetric matrix G with zero entries on its diagonal is an adjacency matrix associated with a graph G (with weighted edges and no loops) if and only if the non-zero entries correspond to edges of G. An adjacency matrix G belongs to a generalized-nut graph G if every entry in a vector in the nullspace of G is non-zero. A graph G is termed NSSD if it corresponds to a non-singular adjacency matrix G with a singular deck {G - v}, where G - v is the submatrix obtained from G by deleting the vth row and column. An NSSD G whose deck consists of generalized-nut graphs with respect to G is referred to as a G-nutful graph. We prove that a G-nutful NSSD is equivalent to having a NSSD with G(-1) as the adjacency matrix of the complete graph. If the entries of G for a G-nutful graph are restricted to 0 or 1, then the graph is known as nuciferous, a concept that has arisen in the context of the quantum mechanical theory of the conductivity of non-singular Carbon molecules according to the SSP model. We characterize nuciferous graphs by their inverse and the nullities of their one- and two-vertex deleted subgraphs. We show that a G-nutful graph is a NSSD which is either K-2 or has no pendant edges. Moreover, we reconstruct a labelled NSSD either from the nullspace generators of the ordered one-vertex deleted subgraphs or from the determinants of the ordered two-vertex deleted subgraphs.