No chemical graph on more than two vertices is nuciferous

Abstract

A simple graph is nuciferous if its 0-1 adjacency matrix is nonsingular and if its inverse has zero entries on its main diagonal and a non-zero entry at each off-diagonal position. A nuciferous graph is a molecular graph that represents an ipso omni-insulating but distinct omni-conducting molecule. It has been conjectured in 2012 that only K2, the complete graph on two vertices, is nuciferous. We show that this conjecture is true for chemical graphs, that is, graphs whose vertex degree is at most three.