On the geometric–arithmetic index by decompositions-CMMSE

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. There are many papers studying different kinds of indices (as Wiener, hyper–Wiener, detour, hyper–detour, Szeged, edge–Szeged, PI, vertex–PI and eccentric connectivity indices) under particular cases of decompositions. The main aim of this paper is to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the so-called primary subgraphs obtained by a general decomposition of G. Furthermore, using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied to the computation of the geometric-arithmetic index of Spiro chain of hexagons, polyphenylenes and polyethene.     

On acyclic molecular graphs with maximal Hosoya index, energy, and short diameter

The total number of matchings of a graph is defined as its Hosoya index. Conjugated and non-conjugated acyclic graphs that have maximal Hosoya index and short diameter are characterized in this paper, explicit expressions of the Hosoya indices of these extremal graphs are also presented.     

On sombor indices of tetraphenylethylene,terpyridine rosettes and QSPR analysis on fluorescence properties of several aromatic hetero-cyclic species

Aromatic heterocyclic compounds have received a lot of interest due to their various important medicinal and biological applications. The broad synthetic investigation and functional usefulness of heterocyclic molecules is driving a surge in research interest. They are found in more than 90% of innovative medications and bridge the gap between biology and chemistry, where so much scientific discovery and application happens. Heterocycles are also useful in a variety of domains, including pharmaceutical chemistry, biochemistry, and others. In this article, quantitative structure-property relationship (QSPR) models is developed using sombor indices to predict fluorescence properties of aromatic hetero-cyclic species based on their structural features. This allows researchers to estimate the fluorescence behavior of new molecules without performing experimental measurements. As an application, we have computed the sombor indices for self-assembled supramolecular graphs made of terpyridine (TPE) and tetraphenylethylene (TPY) molecules that are produced as rosette cycles. This form of rosettes graph is used in electrical sensors, light emitting diodes, bioimaging and photoelectric devices, and so on. Tetraphenylethylene can be used to make fluorescent probes for next-generation sensing applications with typical induced aggregative emission behavior.     

On valency based topological properties of the starphene St[n,m,l] and fenestrene F[n,m]

In theoretical chemistry the quantitative parameters which are used to describe the atomic topology of graphs are termed as topological indices. Through these topological indices many physical and chemical characteristics such as melting point, entropy, energy generation and vaporisation enthalpy of chemical compounds can be predicted. The theory of graphs has a significant use in measuring the relationship of certain associated graphs with various topological indices. In this paper, we compute novel topological indices based on eV- and ve-degrees for starphene $\mathit{St}[n,m,l]$ and fenestrene $F[n,m]$. A Maple-based algorithm is proposed for the calculation of ve and eV-degree based topological indices from the graph adjacency matrix.     

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.     

Degeneracy of some matrix graph invariants

New matrices associated with graphs and induced global and local topological indices of molecular graphs were proposed recently by Diudea, Minailiuc and Balaban. These matrices in canonical form are matrix graph invariants. A combined degeneracy of such invariants is considered. For every case of degeneracy corresponding graphs are presented.     

Generalized Wiener indices of zigzagging pentachains

The “pentachains” studied in this paper are graphs formed of concatenated 5-cycles. Explicit formulas are obtained for the Schultz and modified Schultz indices of these graphs, as well as for generalizations of these indices. In the process we give a more refined version of the procedure that earlier was reported for the ordinary Wiener index.     

Novel shape descriptors for molecular graphs.

We report on novel graph theoretical indices which are sensitive to the shapes of molecular graphs. In contrast to the Kier's kappa shape indices which were based on a comparison of a molecular graph with graphs representing the extreme shapes, the linear graph and the "star" graph, the new shape indices are obtained by considering for all atoms the number of paths and the number of walks within a graph and then making the quotients of the number of paths and the number of walks the same length. The new shape indices show much higher discrimination among isomers when compared to the kappa shape indices. We report the new shape indices for smaller alkanes and several cyclic structures and illustrate their use in structure-property correlations. The new indices offer regressions of high quality for diverse physicochemical properties of octanes. They also have lead to a novel classification of physicochemical properties of alkanes.     

Doubly-Colored Graphs as Graphical Models for Subductions of Coset Representations, Double Cosets, and Unit Subduced Cycle Indices

The concept of doubly-colored graphs is proposed to model subductions of coset representations, double cosets, and unit subduced cycle indices, which have been mathematically formulated in coset algebraic theory developed by Fujita (Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin, 1991).     

QSPR modeling: graph connectivity indices versus line graph connectivity indices

Five QSPR models of alkanes were reinvestigated. Properties considered were molecular surface-dependent properties (boiling points and gas chromatographic retention indices) and molecular volume-dependent properties (molar volumes and molar refractions). The vertex- and edge-connectivity indices were used as structural parameters. In each studied case we computed connectivity indices of alkane trees and alkane line graphs and searched for the optimum exponent. Models based on indices with an optimum exponent and on the standard value of the exponent were compared. Thus, for each property we generated six QSPR models (four for alkane trees and two for the corresponding line graphs). In all studied cases QSPR models based on connectivity indices with optimum exponents have better statistical characteristics than the models based on connectivity indices with the standard value of the exponent. The comparison between models based on vertex- and edge-connectivity indices gave in two cases (molar volumes and molar refractions) better models based on edge-connectivity indices and in three cases (boiling points for octanes and nonanes and gas chromatographic retention indices) better models based on vertex-connectivity indices. Thus, it appears that the edge-connectivity index is more appropriate to be used in the structure-molecular volume properties modeling and the vertex-connectivity index in the structure-molecular surface properties modeling. The use of line graphs did not improve the predictive power of the connectivity indices. Only in one case (boiling points of nonanes) a better model was obtained with the use of line graphs.     

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     

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.     

Generalized Wiener indices in hexagonal chains

The Wiener index, or the Wiener number, also known as the “sum of distances” of a connected graph, is one of the quantities associated with a molecular graph that correlates nicely to physical and chemical properties, and has been studied in depth. An index proposed by Schultz is shown to be related to the Wiener index for trees, and Ivan Gutman proposed a modification of the Schultz index with similar properties. We deduce a similar relationship between these three indices for catacondensed benzenoid hydrocarbons (graphs formed of concatenated hexagons, or hexagonal chains, or sometimes acenes). Indeed, we may define three families of generalized Wiener indices, which include the Schultz and Modified Schultz indices as special cases, such that similar explicit formulae for all generalized Wiener indices hold on hexagonal chains. We accomplish this by first giving a more refined proof of the formula for the standard Wiener index of a hexagonal chain, then extending it to the generalized Wiener indices via the notion of partial Wiener indices. Finally, we discuss possible extensions of the result. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

On the great success of bond-additive topological indices such as Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a quantitative refinement of the distance nonbalancedness and also a peripherality measure in molecular graphs and networks. In this direction, we introduce other variants of bond-additive indices, such as edge-Mostar and total-Mostar indices. The present article explores a computational technique for Mostar, edge-Mostar, and total-Mostar indices with respect to the strength-weighted parameters. As an application, these techniques are applied to compute the three indices for the family of coronoid and carbon nanocone structures.     

Structure-resonance theory for homoaromatic bridged annulenes

The π systems of bridged annulenes can ben represented by π molecular graphs with homoconjugative interactions at the bridge positions. Nonbonding MO's of odd molecular graphs derived from the parent bridged annulene graphs can be used to carry out structure-resonance theory calculations. A general outline of possible applications is given. Specific comparisons are made between calculated bond orders and bond lengths, and between calculated and experimental ionization potentials. The concept of neutral homoaromaticity is supported by good agreement between calculated and experimental properties.