arXiv subjects

Nino Bašić

Publications and source records attributed to Nino Bašić.

27 records · Page 2Linked to original sources

arXiv2020

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 graph is a connected simple graph with a singular adjacency matrix that has one zero eigenvalue and a non-trivial kernel eigenvector without zero entries. Nut graphs have no vertices of degree $1$: they are leafless. The intersection of these two sets, the chemical nut graphs, is of interest in applications in chemistry and molecular physics, corresponding to structures with fully distributed radical reactivity and omniconducting behaviour at the Fermi level. A chemical nut graph consists of $v_2 \ge 0$ vertices of degree $2$ and an even number, $v_3 > 0$, of vertices of degree $3$. With the aid of systematic local constructions that produce larger nut graphs from smaller, the combinations $(v_3, v_2)$ corresponding to realisable chemical nut graphs are characterised. Apart from a finite set of small cases, and two simply defined infinite series, all combinations $(v_3, v_2 )$ with even values of $v_3 > 0$ are realisable as chemical nut graphs. Of these combinations, only $(20,0)$ cannot be realised by a planar chemical nut graph. The main result characterises the ranges of edge counts for chemical nut graphs of all orders $n$.

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arXiv2020

Convexity deficit of benzenoids

In 2012, a family of benzenoids was introduced by Cruz, Gutman, and Rada, which they called convex benzenoids. In this paper we introduce the convexity deficit, a new topological index intended for benzenoids and, more generally, fusenes. This index measures by how much a given fusene departs from convexity. It is defined in terms of the boundary-edges code. In particular, convex benzenoids are exactly the benzenoids having convexity deficit equal to 0. Quasi-convex benzenoids form the family of non-convex benzenoids that are closest to convex, i.e., they have convexity deficit equal to 1. Finally, we investigate convexity deficit of several important families of benzenoids.

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arXiv2019

Point-ellipse and some other exotic configurations

In this paper we introduce point-ellipse configurations and point-conic configurations. We study some of their basic properties and describe two interesting families of balanced point-ellipse, respectively point-conic $6$-configurations. The construction of the first family is based on Carnot's theorem, whilst the construction of the second family is based on the Cartesian product of two regular polygons. Finally, we investigate a point-ellipse configuration based on the regular $24$-cell.

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arXiv2018

Splittable and unsplittable graphs and configurations

We prove that there exist infinitely many splittable and also infinitely many unsplittable cyclic $(n_3)$ configurations. We also present a complete study of trivalent cyclic Haar graphs on at most 60 vertices with respect to splittability. Finally, we show that all cyclic flag-transitive configurations with the exception of the Fano plane and the Möbius-Kantor configuration are splittable.

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arXiv2017

On the Clar Number of Benzenoid Graphs

A Clar set of a benzenoid graph $B$ is a maximum set of independent alternating hexagons over all perfect matchings of $B$. The Clar number of $B$, denoted by ${\rm Cl}(B)$, is the number of hexagons in a Clar set for $B$. In this paper, we first prove some results on the independence number of subcubic trees to study the Clar number of catacondensed benzenoid graphs. As the main result of the paper we prove an upper bound for the Clar number of catacondensed benzenoid graphs and characterize the graphs that attain this bound. More precisely, it is shown that for a catacondensed benzenoid graph $B$ with $n$ hexagons ${\rm Cl}(B) \leq [(2n+1)/3]$.

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arXiv2017

Sizes of Pentagonal Clusters in Fullerenes

Stability and chemistry, both exohedral and endohedral, of fullerenes are critically dependent on the distribution of their obligatory 12 pentagonal faces. It is well known that there are infinitely many IPR-fullerenes and that the pentagons in these fullerenes can be at an arbitrarily large distance from each other. IPR-fullerenes can be described as fullerenes in which each connected cluster of pentagons has size 1. In this paper we study the combinations of cluster sizes that can occur in fullerenes and whether the clusters can be at an arbitrarily large distance from each other. For each possible partition of the number 12, we are able to decide whether the partition describes the sizes of pentagon clusters in a possible fullerene, and state whether the different clusters can be at an arbitrarily large distance from each other. We will prove that all partitions with largest cluster of size 5 or less can occur in an infinite number of fullerenes with the clusters at an arbitrarily large distance of each other, that 9 partitions occur in only a finite number of fullerene isomers and that 15 partitions do not occur at all in fullerenes.

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arXiv2016

An algebraic approach to enumerating non-equivalent double traces in graphs

Recently designed biomolecular approaches to build single chain polypeptide polyhedra as molecular origami nanostructures have risen high interest in various double traces of the underlying graphs of these polyhedra. Double traces are walks that traverse every edge of the graph twice, usually with some additional conditions on traversal direction and vertex neighborhood coverage. Given that double trace properties are intimately related to theefficiency of polypeptide polyhedron construction, enumerating all different possible double traces and analyzing their properties is an important step in the construction. In the paper, we study the automorphism group of double traces and present an algebraic approach to this problem, yielding a branch-and-bound algorithm.

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arXiv2015

Coronoids, Patches and Generalised Altans

In this paper we revisit coronoids, in particular multiple coronoids. We consider a mathematical formalisation of the theory of coronoid hydrocarbons that is solely based on incidence between hexagons of the infinite hexagonal grid in the plane. In parallel, we consider perforated patches, which generalise coronoids: in addition to hexagons, other polygons may also be present. Just as coronoids may be considered as benzenoids with holes, perforated patches are patches with holes. Both cases, coronoids and perforated patches, admit a generalisation of the altan operation that can be performed at several holes simultaneously. A formula for the number of Kekulé structures of a generalised altan can be derived easily if the number of Kekulé structures is known for the original graph. Pauling Bond Orders for generalised altans are also easy to derive from those of the original graph.

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arXiv2015

Iterated altans and their properties

Recently a class of molecular graphs, called altans, became a focus of attention of several theoretical chemists and mathematicians. In this paper we study primary iterated altans and show, among other things, their connections with nanotubes and nanocaps. The question of classification of bipartite altans is also addressed. Using the results of Gutman we are able to enumerate Kekulé structures of several nanocaps of arbitrary length.

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