Coalesced and embedded nut graphs in singular graphs

  title={Coalesced and embedded nut graphs in singular graphs},
  author={Irene Sciriha},
  journal={Ars Math. Contemp.},
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… 

Figures from this paper

Generation and properties of nut graphs
A new algorithm is presented for the exhaustive generation of non-isomorphic nut graphs using the position of the zero eigenvalue in the graph spectrum, and the dispersion in magnitudes of kernel eigenvector entries ($r: the ratio of maximum to minimum magnitude of entries).
On 12-regular nut graphs
A nut graph is a simple graph whose adjacency matrix is singular with 1-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each d
Construction of larger singular and nonsingular graphs using a path
A singular graph G has an adjacency matrix A(G) with nullity η(G)> 0. Vertices of singular graphs are classified as core and noncore vertices. There are two types of noncore vertices: noncore
Existence of Regular Nut Graphs for Degree at Most 11
The problem of determining the orders n for which d-regular nut graphs exist is solved and the complete lists of all d-Regular nut graphs of order n for small values of d and n are determined.
On circulant nut graphs
Coalescing Fiedler and core vertices
The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex
Charting the space of chemical nut graphs
Molecular graphs of unsaturated carbon frameworks or hydrocarbons pruned of hydrogen atoms, are chemical graphs. A chemical graph is a connected simple graph of maximum degree $3$ or less. A nut
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
Existence of regular nut graphs and the fowler construction
In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler?s Construction which is a local enlargement applied to a vertex in a graph is introduced to
Nullspace vertex partition in graphs
To maximize the number of edges for optimal network graphs with a specified nullity, this work determines which perturbations make up sufficient conditions for the core vertex set of the adjacency matrix of a graph to be preserved on adding edges.


Nut graphs : maximally extending cores
A graphG is singular if there is a non-zero eigenvector v0 in the nullspace of its adjacency matrix A. Then Av0 = 0. The subgraph induced by the vertices corresponding to the nonzero components of v0
On the construction of graphs of nullity one
Minimal configuration trees
A graph is singular of nullity η if zero is an eigenvalue of its adjacency matrix with multiplicity η. If η(G)=1, then the core of G is the subgraph induced by the vertices associated with the
On the Rank of Graphs 1
The properties of singular graphs obtained in a previous paper "On the construction of graphs of nullity one", lead to the characterization of graphs of small rank. The minimal conflgurations that
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
Nonbonding Orbitals in Fullerenes: Nuts and Cores in Singular Polyhedral Graphs
It is proved that all uniform nut fullerenes must have such adjacencies and that the NBO is totally symmetric in all balanced nut fulLErenes.
How can one actually compute the eigenvalues of a graph? In principal, there are three methods. Namely, (1) we can search for p orthogonal eigenvectors, (2) we can determine the first p moments by
An Atlas of Fullerenes
Introduction. 1: Fullerene cages. 2: Electronic structure. 3: Steric strain. 4: Symmetry and spectroscopy. 5: Fullerene isomerisation. 6: Carbon gain and loss. Appendix: The spiral computer program.
Zur modernen Theorie ungesättigter und aromatischer Verbindungen
Inhaltsubersicht 1 Heuristische Gesichtspunkte zur theoretischen Behandlung. 2 Allgemeines zur quantentheoretischen Behandlung. 3 Die erste Methode (HLS-method; HLSP oder VB-method).
Minimal configurations and interlacing