Edge construction of molecular NSSDs

Abstract

A nonsingular graph with a singular deck, or NSSD, is a graph with weighted edges and no loops whose adjacency matrix is nonsingular and whose vertex-deleted subgraphs have singular adjacency matrices. When all the weights of the edges of a NSSD are equal to one, we obtain a molecular NSSD, which is an ipso omni-insulator molecular graph. We present necessary and sufficient conditions for a NSSD G to remain a NSSD after increasing the weight of one of its edges by w ∈ R \ {0}; this weight change may result in the addition or removal of that edge to or from G. Using this result, we construct molecular NSSDs of a fixed even order. This is accomplished by systematically introducing edges to, or removing edges from, a nonsingular tree.