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 K 2 , 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. (Irene Sciriha), alex.farrugia@um.edu.mt (Alexander Farrugia) c b This work is licensed under http://creativecommons.org/licenses/by/3.0/

Molecular graphs and molecular conduction: the d-omni-conductors

“…K2 is the trivial nucifer with no intra devices, and unique TLA CXI. All non-trivial nucifers are non-chemical; 38 vertex-transitive examples are known with n ≥ 24, 39 and in the present work we were able to find smaller examples, with n ≥ 18, by considering graphs with two orbits of vertices. Hence, the class CCI with η = 0 is in fact populated, though not by chemical graphs.…”