Nonbonding orbitals in fullerenes : nuts and cores in singular polyhedral graphs

A zero eigenvalue in the spectrum of the adjacency matrix of the graph representing an unsaturated carbon framework indicates the presence of a nonbonding π orbital (NBO). A graph with at least one zero in the spectrum is singular; nonzero entries in the corresponding zero-eigenvalue eigenvector(s) (kernel eigenvectors) identify the core vertices. A nut graph has a single zero in its adjacency spectrum with a corresponding eigenvector for which all vertices lie in the core. Balanced and uniform trivalent (cubic) nut graphs are defined in terms of (−2, +1, +1) patterns of eigenvector entries around all vertices. In balanced nut graphs all vertices have such a pattern up to a scale factor; uniform nut graphs are balanced with scale factor one for every vertex. Nut graphs are rare among small fullerenes (41 of the 10 190 782 fullerene isomers on up to 120 vertices) but common among the small trivalent polyhedra (62 043 of the 398 383 nonbipartite polyhedra on up to 24 vertices). Two constructions are described, one that is conjectured to yield an infinite series of uniform nut fullerenes, and another that is conjectured to yield an infinite series of cubic polyhedral nut graphs. All hypothetical nut fullerenes found so far have some pentagon adjacencies:  it is proved that all uniform nut fullerenes must have such adjacencies and that the NBO is totally symmetric in all balanced nut fullerenes. A single electron placed in the NBO of a uniform nut fullerene gives a spin density distribution with the smallest possible (4:1) ratio between most and least populated sites for an NBO. It is observed that, in all nut-fullerene graphs found so far, occupation of the NBO would require the fullerene to carry at least 3 negative charges, whereas in most carbon cages based on small nut cubic polyhedra, the NBO would be the highest occupied molecular orbital (HOMO) for the uncharged system.peer-reviewe

Coalesced and embedded nut graphs in singular graphs

A nut graph has a non-invertible (singular) 0-1 adjacency matrix with non-zero entries in every kernel eigenvector. We investigate how the concept of nut graphs emerges as an underlying theme in the theory of singular graphs. It is known that minimal configurations (MCs) are necessarily found as subgraphs of singular graphs. We construct MCs having nut graphs as subgraphs. Nut graphs can be coalesced with singular graphs at particular vertices or grown into a family of core graphs of larger nullity by adding a vertex at a time. Moreover, we propose a construction of nut line graph of trees by coalescence and a local enlargement of nut fullerenes and tetravalent nut graphs.peer-reviewe

Omni-conducting and omni-insulating molecules

The source and sink potential model is used to predict the existence of omni-conductors (and omni-insulators): molecular conjugated π systems that respectively support ballistic conduction or show insulation at the Fermi level, irrespective of the centres chosen as connections. Distinct, ipso, and strong omni-conductors/omni-insulators show Fermi-level conduction/insulation for all distinct pairs of connections, for all connections via a single centre, and for both, respectively. The class of conduction behaviour depends critically on the number of non-bonding orbitals (NBO) of the molecular system (corresponding to the nullity of the graph). Distinct omni-conductors have at most one NBO; distinct omni-insulators have at least two NBO; strong omni-insulators do not exist for any number of NBO. Distinct omni-conductors with a single NBO are all also strong and correspond exactly to the class of graphs known as nut graphs. Families of conjugated hydrocarbons corresponding to chemical graphs with predicted omni-conducting/insulating behaviour are identified. For example, most fullerenes are predicted to be strong omni-conductors

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 K2 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.peer-reviewe

A spectral view of fullerenes

Fullerenes, a third family of allotropes of carbon ( C), exist as large stable clusters of C atoms. The eigenvalues of the adjacency matrix A of a graph, with the same structure as a fullerene, estimate the energies of the 1r-electrons in these unsaturated systems and the eigenvectors of A model the 1r-molecular orbitals. The eigenvalue zero of A indicates the presence of NBOs with no net stabilization or destabilization. Zero energy levels are rare in fullerenes. We study the substructures in fullerenes and other trivalent polyhedra that determine the presence of the eigenvalue zero. Together with the symmetry group of the graph, they shed new light on singular graphs and on singular polyhedra in particular.peer-reviewe

A characterization of singular graphs

Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgraphs.peer-reviewe

Zooming in on fullerenes

Carbon does not appear only as diamond and graphite. Fullerenes, forming a third family of allotropes of carbon (C), exist as large stable clusters of C atoms. A trivalent polyhedron P is a cubic graph which may be embedded on a convex 3-D surface of genus zero and a fullerene, Cn, is P with twelve pentagons, the remaining faces being hexagons. We introduce the concept of nut fullerenes, so called because their skeleton is a nut graph that implies one NBO with the charge contributed by the NBO electron being shared among all the C-centres. The charge distribution over all the framework of the molecule has strong chemical consequences. We study the substructures in fullerenes and other trivalent polyhedra, that determine the presence of a NBO. Together with the symmetry group of the graph, they shed new light on singular graphs and polyhedra in particular.peer-reviewe

A note on Cayley nut graphs whose degree is divisible by four

A nut graph is a non-trivial simple graph such that its adjacency matrix has a one-dimensional null space spanned by a full vector. It was recently shown by the authors that there exists a $d$-regular circulant nut graph of order $n$ if and only if $4 \mid d, \, 2 \mid n, \, d > 0$, together with $n \ge d + 4$ if $d \equiv_8 4$ and $n \ge d + 6$ if $8 \mid d$, as well as $(n, d) \neq (16, 8)$ [arXiv:2212.03026, 2022]. In this paper, we demonstrate the existence of a $d$-regular Cayley nut graph of order $n$ for each $4 \mid d, \, d > 0$ and $2 \mid n, \, n \ge d + 4$, thereby resolving the existence problem for Cayley nut graphs and vertex-transitive nut graphs whose degree is divisible by four