Distance based indices in nanotubical graphs: part 2

Nanotubical graphs are obtained by wrapping a hexagonal grid, and then possibly closing the tube with caps. In this paper we show that the asymptotics for Balaban, Sum-Balaban, and Harary indices for all nanotubical graphs of type (k, l) on n vertices are \(\frac{9\pi (k+l)}{2n}\), \(\frac{9\sqrt{2}}{2}\sqrt{k+l}\cdot \log (1+\sqrt{2})\) and \((k+l)n\log (n)\), respectively. In all the cases, the leading term depends on the circumference of the nanotubical graph, but not on its specific type. Thus, we conclude that these distance based topological indices seem not to be the most suitable for distinguishing nanotubes with the same circumference as far as the leading term is concerned.     

Chemical graphs with degenerate topological indices based on information on distances

Recently, four new types of vertex invariants, namelyu, v, x, andy, were defined on the basis of information on graph distances. They were combined to give four highly selective topological indices:U, V, X, andY. The degeneracy, i.e. equal values for nonisomorphic graphs, of the four topological indices is investigated. A structural condition and a graphical method which gives pairs of molecular graphs with identicalU, V, X, andY topological indices are introduced. The smallest pair of 4-trees representing alkanes having degeneratedU, V, X, andY values consists of trees with eighteen vertices.     

PI indices of pericondensed benzenoid graphs

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., , where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of a class of pericondensed benzenoid graphs consisting of three rows of hexagonal of various lengths.     

New vertex invariants and topological indices of chemical graphs based on information on distances

By applying information theory to the set of topological distances from one vertex to all other graph vertices, one obtains four new types of vertex invariants (u _i,v _i,x _i,Y _i) which are real numbers (as opposed to integers). They may be combined in many ways to afford new topological indices. One such type leads to indicesU, V, X andY which show no degeneracy for alkanes with up to 15 vertices.     

Stereotopological indices for a family of chemical graphs

We describe a mathematical method that can be employed to define stereotopological indices of placements of certain graphs in space. These indices are applied to successfully distinguish between configurations in a chemically interesting family of knotted and/or linked four-valent oriented graphs in space. The methods are fundamentally algebraic and combinatorial in nature and are most readily understood in the context of calculations and the study of several key examples that are presented.     

Graphs whose vertices are graphs with bounded degree: Distance problems

By an f-graph we mean a graph having no vertex of degree greater than f. Let U(n,f) denote the graph whose vertex set is the set of unlabeled f-graphs of order n and such that the vertex corresponding to the graph G is adjacent to the vertex corresponding to the graph H if and only if H is obtainable from G by either the insertion or the deletion of a single edge. The distance between two graphs G and H of order n is defined as the least number of insertions and deletions of edges in G needed to obtain H. This is also the distance between two vertices in U(n,f). For simplicity, we also refer to the vertices in U(n,f) as the graphs in U(n,f). The graphs in U(n,f) are naturally grouped and ordered in levels by their number of edges. The distance nf/2 from the empty graph to an f-graph having a maximum number of edges is called the height of U(n,f). For f =2 and for f≥(n-1)/2, the diameter of U(n,f) is equal to the height. However, there are values of the parameters where the diameter exceeds the height. We present what is known about the following two problems: (1) What is the diameter of U(n,f) when 3≥f<(n-1)/2? (2) For fixed f, what is the least value of n such that the diameter of U(n,f) exceeds the height of U(n,f)?     

Computing weighted Szeged and PI indices from quotient graphs

The weighted (edge-)Szeged index and the weighted (vertex-)PI index are modifications of the (edge-)Szeged index and the (vertex-)PI index, respectively, because they take into account also the vertex degrees. As the main result of this article, we prove that if G is a connected graph, then all these indices can be computed in terms of the corresponding indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than the Θ^*-partition. If G is a benzenoid system or a phenylene, then it is possible to choose a partition of the edge set in such a way that the quotient graphs are trees. As a consequence, it is shown that for a benzenoid system, the mentioned indices can be computed in sublinear time with respect to the number of vertices. Moreover, closed formulas for linear phenylenes are also deduced.     

Wiener indices of trees and monocyclic graphs with given bipartition

The Wiener index of a connected graph is defined as the sum of distances between all unordered pairs of its vertices. It has found various applications in chemical research. We determine the minimum and the maximum Wiener indices of trees with given bipartition and the minimum Wiener index of monocyclic graphs with given bipartition, respectively. We also characterize the graphs whose Wiener indices attain these values. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Wiener, hyper-Wiener, detour and hyper-detour indices of bridge and chain graphs

Given a collection of connected graphs one may build bridge and chain graphs out of them. In this paper it is shown how the Wiener, hyper-Wiener, detour and hyper-detour indices for bridge and chain graphs are determined from the respective indices of the individual graphs. The results obtained are illustrated by some examples.     

Computer generation of acyclic graphs based on local vertex invariants and topological indices. Derived canonical labelling and coding of trees and alkanes

A new procedure (GENLOIS) is presented for generating trees or certain classes of trees such as 4-trees (graphs representing alkanes), identity trees, homeomorphical irreducible trees, rooted trees, trees labelled on a certain vertex (primary, secondary, tertiary, etc.). The present method differs from previous procedures by differentiating among the vertices of a given parent graph by means of local vertex invariants (LOVIs). New graphs are efficiently generated by adding points and/or edges only to non-equivalent vertices of the parent graph. Redundant generation of graphs is minimized and checked by means of highly discriminating, recently devised topological indices based either on LOVIs or on the information content of LOVIs. All trees onN + 1 (N + 1 < 17) points could thus be generated from the complete set of trees onN points. A unique cooperative labelling for trees results as a consequence of the generation scheme. This labelling can be translated into a code for which canonical rules were recently stated by A.T. Balaban. This coding appears to be one of the best procedures for encoding, retrieving or ordering the molecular structure of trees (or alkanes).Dedicated to Professor Alexandru T. Balaban on the occasion of his 60th anniversary.     

Photofragmentation of closo-carboranes part 1: Energetics of decomposition

The ionic fragmentation following B 1s and C 1s excitation of three isomeric carborane cage compounds [closo-dicarbadodecaboranes: orthocarborane (1,2-C2B10H12), metacarborane (1,7-C2B10H12), and paracarborane (1,12-C2B10H12)] is compared with the energetics of decomposition. The fragmentation yields for all three molecules are quite similar. Thermodynamic cycles are constructed for neutral and ionic species in an attempt to systemically characterize single-ion closo-carborane creation and fragmentation processes. Lower energy decomposition processes are favored. Among the ionic species, the photon-induced decomposition is dominated by BH+ and BH2(+) fragment loss. Changes in ion yield associated with core to bound excitations are observed.     

Generation of molecular graphs for QSAR studies: an approach based on supergraphs

A new algorithm for generation of substituted derivatives of a given structure and its software implementation are described. The program has deterministic and stochastic generation modes and efficiently supports various types of structural constraints. The problem of exhaustive and irredundant generation is discussed, and a new algorithm of the complete rejection of isomorphic molecular graphs is proposed. The main application of the generator is QSAR studies; however, applications in combinatorial chemistry are also possible.