Planar polycyclic graphs and their Tutte polynomials

We consider several classes of planar polycyclic graphs and derive recurrences satisfied by their Tutte polynomials. The recurrences are then solved by computing the corresponding generating functions. As a consequence, we obtain values of several chemically and combinatorially interesting enumerative invariants of considered graphs. Some of them can be expressed in terms of values of Chebyshev polynomials of the second kind.     

CMMSE-on the first general Zagreb index

The aim of this paper is to obtain new inequalities involving the first general Zagreb index, and characterize graphs which are extremal with respect to them. We also obtain inequalities involving the forgotten and second general Zagreb indices.     

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 reciprocal molecular topological index

We report some properties of the reciprocal molecular topological index RMTI of a connected graph, and, in particular, its relationship with the first Zagreb index M₁. We also derive the upper bounds for RMTI in terms of the number of vertices and the number of edges for various classes of graphs, including K _r+1 -free graphs with r ≥ 2, quadrangle-free graphs, and cacti. Additionally, we consider a Nordhaus-Gaddum-type result for RMTI.     

The connective eccentricity index of graphs and its applications to octane isomers and benzenoid hydrocarbons

The connective eccentricity index (CEI) of a graph G is defined as , where ε_G(.) denotes the eccentricity of the corresponding vertex. The CEI obligates an influential ability, which is due to its estimating pharmaceutical properties. In this paper, we first characterize the extremal graphs with respect to the CEI among k-connected graphs (k-connected bipartite graphs) with a given diameter. Then, the sharp upper bound on the CEI of graphs with given connectivity and minimum degree (independence number) is determined. Finally, we calculate the CEI of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons. We compare their CEI with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the CEI is better than or as good as the models corresponding to the other distance-based indices.     

The characteristic polynomial of a chemical graph

We list uses of, and the computational methods for the characteristic polynomial of a (chemical) graph. Pour computational methods are singled out for more detailed presentation. These are the graphical methods of Sachs, the recurrence formulae for several classes of simple graphs, the method based on Ulam subgraphs, and the Le Verrier — Faddeev — Franic recursive method. The latter method appears, at present, to be the most efficient procedure for the computation of the characteristic polynomials of graphs of sizes with up to even a few hundred sites.Dedicated to Dennis H. Rouvray, the friend and one of the foremost popularizers of chemical graph theory in our time.Research supported by the Robert A. Welch Foundation of Houston, Texas, USA.     

Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index

Recently, Araujo and De la Pe?a gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which are trees and the cyclomatic number of the graphs. The ramification index of a tree is first shown to be equal to the number of pending vertices minus 2. Then, in view of extremal graphs obtained with the system AutoGraphiX, all bounds of Araujo and De la Pe?a are improved, yielding tight bounds, and in one case corrected. Moreover, chemical trees of a given order and a number of pending vertices with minimum and with maximum connectivity index are characterized.     

On the ABC index of connected graphs with given degree sequences

The atom-bond connectivity (ABC) index of a graph G is defined to be \(ABC(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}\) where d(u) is the degree of a vertex u. The ABC index plays a key role in correlating the physical–chemical properties and the molecular structures of some families of compounds. In this paper, we describe the structural properties of graphs which have the minimum ABC index among all connected graphs with a given degree sequence. Moreover, these results are used to characterize the extremal graphs which have the minimum ABC index among all unicyclic and bicyclic graphs with a given degree sequence.     

Topological analysis of some special classes of graphs. II. steps,ladders, cylinders

A general strategy is proposed for generating the eigenvectors and the eigenvalues of some special classes of graphs from the well-known chemical graphs such as lines and cycles which are isomorphic to the hydrogen-suppressed linear and cyclic polyenes. This method is applied to step graphs, ladders, cylinders, etc. Net sign analyses are then performed for all these special classes of graphs.     

Reference energy extremal optimization: a stochastic search algorithm applied to computational protein design

We adapt a combinatorial optimization algorithm, extremal optimization (EO), for the search problem in computational protein design. This algorithm takes advantage of the knowledge of local energy information and systematically improves on the residues that have high local energies. Power-law probability distributions are used to select the backbone sites to be improved on and the rotamer choices to be changed to. We compare this method with simulated annealing (SA) and motivate and present an improved method, which we call reference energy extremal optimization (REEO). REEO uses reference energies to convert a problem with a structured local-energy profile to one with more random profile, and extremal optimization proves to be extremely efficient for the latter problem. We show in detail the large improvement we have achieved using REEO as compared to simulated annealing and discuss a number of other heuristics we have attempted to date.     

Energies of some Non-regular Graphs

The energy of a graph G is the sum of the absolute values of its eigenvalues. In this paper, we study the energies of some classes of non-regular graphs. Also the spectrum of some non-regular graphs and their complements are discussed.     

Subgraphs of pair vertices

Subgraphs obtained by applying several fragmentation criteria are investigated. Two well known criteria (Szeged and Cluj), and two new others are defined and characterized. An example is given for the discussed procedures. The matrix and polynomial representations of vertices composing each type of subgraphs were also given. Analytical formulas for the polynomials of several classes of graphs are derived. The newly introduced subgraphs/fragments, called MaxF and CMaxF, appear to have interesting properties, which are demonstrated.     

CMMSE: A new approximation to the geometric–arithmetic index

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric–arithmetic index \(GA_1\) and characterize graphs extremal with respect to them.     

New isomerization operations for fullerene graphs

The problem of topological isomerization operations for fullerenes is considered. Two basic classes of isomerization transformations admissible for fullerene graphs are proposed: rotation and mirror reflection of a chosen part of fullerene graph. Pyracylene and generalized Stone-Wales rearrangements are only the specific cases of extended isomerization operations for the fullerene graphs.