A curious family of convex benzenoids and their altans

The altan graph of G, a(G, H), is constructed from graph G by choosing an attachment set H from the vertices of G and attaching vertices of H to alternate vertices of a new perimeter cycle of length 2|H|. When G is a polycyclic plane graph with maximum degree 3, the natural choice for the attachment set is to take all perimeter degree-2 vertices in the order encountered in a walk around the perimeter. The construction has implications for the electronic structure and chemistry of carbon nanostructures with molecular graph a(G, H), as kernel eigenvectors of the altan correspond to non-bonding π molecular orbitals of the corresponding unsaturated hydrocarbon. Benzenoids form an important subclass of carbon nanostructures. A convex benzenoid has a boundary on which all vertices of degree 3 have exactly two neighbours of degree 2. The nullity of a graph is the dimension of the kernel of its adjacency matrix. The possible values for the excess nullity of a(G, H) over that of G are 2, 1, or 0. Moreover, altans of benzenoids have nullity at least 1. Examples of benzenoids where the excess nullity is 2 were found recently. It has been conjectured that the excess nullity when G is a convex benzenoid is at most 1. Here, we exhibit an infinite family of convex benzenoids with 3-fold dihedral symmetry (point group D3h) where nullity increases from 2 to 3 under altanisation. This family accounts for all known examples with the excess nullity of 1 where the parent graph is a singular convex benzenoid

Enumerating molecules.

This report is a comprehensive review of the field of molecular enumeration from early isomer counting theories to evolutionary algorithms that design molecules in silico. The core of the review is a detail account on how molecules are counted, enumerated, and sampled. The practical applications of molecular enumeration are also reviewed for chemical information, structure elucidation, molecular design, and combinatorial library design purposes. This review is to appear as a chapter in Reviews in Computational Chemistry volume 21 edited by Kenny B. Lipkowitz

Computing Fifth Geometric-Arithmetic Index for Circumcoronene series of benzenoid Hk

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. The geometric-arithmetic index is a topological index was introduced by Vukicevic and Furtula in 2009 and defined as  in which degree of vertex u denoted by dG(u) (or du for short). In 2011, A. Graovac et al defined a new version of GA index as  where  The goal of this paper is to compute the fifth geometric-arithmetic index for "Circumcoronene series of benzenoid Hk (k≥1)"

Computing Atom-Bond Connectivity (ABC4) index for Circumcoronene Series of Benzenoid

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. In such a simple molecular graph, vertices represent atoms and edges represent bonds. The Atom-Bond Connectivity (ABC) index is a topological index was defined as  where dv denotes degree of vertex v. In 2010, a new version of Atom-Bond Connectivity (ABC4) index was defined by M. Ghorbani et. al as  where and NG(u)={vV(G)|uvE(G)}. The goal of this paper is to compute the ABC4 index for Circumcoronene Series of Benzenoi

GRAPH-THEORETICAL STUDIES ON FLUORANTHENOIDS AND FLUORENOIDS - ENUMERATION OF SOME CATACONDENSED SYSTEMS

Precise definitions are given for some classes of molecular graphs with one pentagon and otherwise hexagons: the monopentapolyhexes. The fluoranthenoid and fluorenoid systems belong to monopentapolyhexes. Complete mathematical solutions, using combinatorial summations on the one hand and generating functions on the other hand, are given for the numbers of catacondensed simply connected monopentapolyhexes (catafluorenoids and the corresponding helicenic systems). Generating functions and numerical values are included

Atlas Kekuléovih valentnih struktura Buckminsterfulerena

The 12,500 Kekulé valence structures of Buckminsterfullerene C60, of which only 158 are symmetry distinct, show large variation in their forms and characteristics. We display all 158 symmetry distinct Kekulé valence structures which have been classified according to their innate degree of freedom (df), which varies from the maximal value of df = 10 to df = 5. The most important and unique Kekulé valence structure in which CC double bonds are exocyclic to pentagonal faces belongs to df = 10, while there are 36 symmetry distinct Kekulé structures associated with df = 5.Od 12.500 Kekuléovih valentnih struktura Buckminsterfullerena samo je 158 različitih ako se razmatraju samo strukture koje su različite u odnosu na operacije simetrije. Kekuléove valentne strukture klasificirane su preme stupnju slobode (df), koji se mjenja od maksimalne vrijednosti df = 10 pa do df = 5. Najvažnija i jedinstvena Kekuléova valentna struktura u kojoj su sve CC dvostrukle veze izvan peteročlanih prstena ima stupanj slobode 10, dok broj različitih Kekuléovoh struktura obzirom na simetriju ima 36 kojim pripada stupanj slobode 5

Wiener Index Calculation on the Benzenoid System: A Review Article

The Weiner index is considered one of the basic descriptors of fixed interconnection networks because it provides the average distance between any two nodes of the network. Many methods have been used by researchers to calculate the value of the Wiener index. starting from the brute force method to the invention of an algorithm to calculate the Wiener index without calculating the distance matrix. The application of the Wiener index is found in the molecular structure of organic compounds, especially the benzenoid system. The value of the Wiener index of a molecule is closely related to its physical and chemical properties. This paper will show a comprehensive bibliometric survey of peer-reviewed articles referring to the Wiener index of benzenoid. The Wiener index values of several benzenoid compounds using cubic polynomial are also reported. The Wiener index of benzenoid supports much of the research and provides productive citations for citing the study.   Keywords: Wiener index, benzenoid, distance matrix, chemical properties, cubic polynomial, topological

Kekulé Structures of Fullerene C70

Despite that, besides Buckminsterfullerene C60, fullerene C70 is the next most stable structure, there are considerable differences in structural properties of these two most common fullerenes. The paper reports on numerous mathematical properties of the set of Kekulé structures of C70. Of over 50,000 Kekulé structures of fullerene C70, only 2780 Kekulé valence structures are distinct, while all the others are symmetry related. The subset of distinct Kekulé valence structures was examined and classified into six classes according to the degree of freedom (df), varying from df = 5 to df = 11. Enumeration of conjugated circuits R1, R2 and R3 points to two symmetry related dominant Kekulé structures having the maximal number of 20 R1. There are 16 distinct symmetry unrelated Kekulé structures of C70 that have no conjugated circuits R1 at all