Coding Of Graphs Research Articles

Zagreb Connection Indices of Molecular Graphs Based on Operations

Topological index (numeric number) is a mathematical coding of the molecular graphs that predicts the physicochemical, biological, toxicological, and structural properties of the chemical compounds that are directly associated with the molecular graphs. The Zagreb connection indices are one of the TIs of the molecular graphs depending upon the connection number (degree of vertices at distance two) appeared in 1972 to compute the total electron energy of the alternant hydrocarbons. But after that, for a long period, these are not studied by researchers. Recently, Ali and Trinajstic Mol. Inform. 372018,1−7 restudied the Zagreb connection indices and reported that the Zagreb connection indices comparatively to the classical Zagreb indices provide the better absolute value of the correlation coefficient for the thirteen physicochemical properties of the octane isomers (all these tested values have been taken from the website http://www.moleculardescriptors.eu). In this paper, we compute the general results in the form of exact formulae & upper bounds of the second Zagreb connection index and modified first Zagreb connection index for the resultant graphs which are obtained by applying operations of corona, Cartesian, and lexicographic product. At the end, some applications of the obtained results for particular chemical structures such as alkanes, cycloalkanes, linear polynomial chain, carbon nanotubes, fence, and closed fence are presented. In addition, a comparison between exact and computed values of the aforesaid Zagreb indices is also included.

Zagreb Connection Indices of Subdivision and Semi-Total Point Operations on Graphs

Representation or coding of the molecular graphs with the help of numerical numbers plays a vital role in the studies of physicochemical and structural properties of the chemical compounds that are involved in the molecular graphs. For the first time, the modified first Zagreb connection index appeared in the paper by Gutman and Trinajstic (1972) to compute total electron energy of the alternant hydrocarbons, but after that, for a long time, it has not been studied. Recently, Ali and Trinajstic (2018) restudied the first Zagreb connection index ZC1, the second Zagreb connection index ZC2, and the modified first Zagreb connection index ZC1∗ to find entropy and acentric factor of the octane isomers. They also reported that the values provided by the International Academy of Mathematical Chemistry show better chemical capability of the Zagreb connection indices than the ordinary Zagreb indices. Assume that S1 and S2 denote the operations of subdivision and semitotal point, respectively. Then, the S-sum graphs Q1+QS2 are obtained by the cartesian product of SQ1 and Q2, where S∈S1,S2, Q1andQ2 are any connected graphs, and SQ1 is a graph obtained after applying the operation S on Q1. In this paper, we compute the Zagreb connection indices (ZC1, ZC2, and ZC1∗) of the S-sum graphs in terms of various topological indices of their factor graphs. At the end, as an application of the computed results, the Zagreb connection indices of the S-sum graphs obtained by the particular classes of alkanes are also included.

Multiplicative Zagreb Indices of Molecular Graphs

Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as ∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.

Self-classification of assembly database using evolutionary method

PurposeThe purpose of the paper is to develop a method of automatic classification of the components of the assembly units. The method is crucial for developing an automatic ship assembly planning tools. The proposed method takes into account the assumptions specific for shipbuilding technology processes: high complexity of structures, difficult expert-based classification of components, fixed priority relations between connections resulting from geometrical constraints and demands of welding processes.Design/methodology/approachThe set of ex post determined liaisons and assembly sequences constitutes the database of structures which have been made-up earlier. The components classification problem is solved using matrix coding of graphs. Information in such form is stored in the database. The minimization of number of cycles in the graph of classes sequence and minimization of diversity of classes within all constructions has been proposed as criteria of optimization. The genetic algorithm has been proposed as a solution method.FindingsThe proposed method solves the problem of components’ classifications. It allows setting the pattern of priorities between classes of various connections. This gives a chance to determine the relationship constraints between the connections of new structures for which assembly sequences are not established.Research limitations/implicationsMathematical formulation of the database is quite laborious. The possibility of partial automation of this process should be considered. Owing to the complexity of the problem, a relatively simple objective function has been proposed. During a ship hull assembly, additional criteria should be taken into account, what will be the direction of further research.Practical implicationsAutomatic classification of components is dedicated for implementation in shipyards and similar assembly systems. Tests performed by the authors confirm efficiency of presented method in supporting management of the database and assembly of new structures planning. Suggested activity-oriented approach allows for easy conversion of any assembly unit structure to the form of a matrix.Originality/valueThe new approach for components classification according to its assembly features distinguishes the proposed method from others. The use of nilpotent matrix theory in an acyclicity of graphs analysis is also a unique achievement. Original crossover and mutation operators for assembly sequence were proposed in the article.

Algebraic operations on graphs with using numeric coding for simulating the systems of rocket and space technology

The paper adduces the relators for algebraic operations on graphs using the numeric codes of the graphs. The special algebra of codes has been devised with consideration of the principles of graph transformation. This paper demonstrates the relevance of numeric coding of graphs for solving the problems of enumeration, systematization, and compact representation of the information about the structural and functional characteristics of the systems of flow distribution and conversion of rocket and space technology and for conducting the modeling transformations of given systems in the course of the structural and functional studies as well.

The Rank-Width of Edge-Coloured Graphs

A C-coloured graph is a graph, that is possibly directed, where the edges are coloured with colours from the set C. Clique-width is a complexity measure for C-coloured graphs, for finite sets C. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss some possible extensions of the notion of rank-width to C-coloured graphs. There is not a unique natural notion of rank-width for C-coloured graphs. We define two notions of rank-width for them, both based on a coding of C-coloured graphs by \({\mathbb{F}}^{*}\)-graphs—\(\mathbb {F}\)-coloured graphs where each edge has exactly one colour from \(\mathbb{F}\setminus \{0\},\ \mathbb{F}\) a field—and named respectively \(\mathbb{F}\) -rank-width and \(\mathbb {F}\) -bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for \(\mathbb{F}^{*}\)-graphs and prove that \(\mathbb{F}^{*}\)-graphs of bounded \(\mathbb{F}\)-rank-width are characterised by a list of \(\mathbb{F}^{*}\)-graphs to exclude as vertex-minors (this list is finite if \(\mathbb{F}\) is finite). An algorithm that decides in time O(n 3) whether an \(\mathbb{F}^{*}\)-graph with n vertices has \(\mathbb{F}\)-rank-width (resp. \(\mathbb{F}\)-bi-rank-width) at most k, for fixed k and fixed finite field \(\mathbb{F}\), is also given. Graph operations to check MSOL-definable properties on \(\mathbb{F}^{*}\)-graphs of bounded \(\mathbb{F}\)-rank-width (resp. \(\mathbb{F}\)-bi-rank-width) are presented. A specialisation of all these notions to graphs without edge colours is presented, which shows that our results generalise the ones in undirected graphs.

Open string diagrams I: Topological type

An arbitrary Feynman graph for string field theory interactions is analyzed and the homeomorphism type of the corresponding world sheet surface is completely determined even in the nonorientable cases. Algorithms are found to mechanically compute the topological characteristics of the resulting surface from the structure of the signed oriented graph. Whitney’s permutation-theoretic coding of graphs is utilized.

Canonical numbering and coding of reaction center graphs and reduced reaction center graphs abstracted from imaginary transition structures. A novel approach to the linear coding of reaction types

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Generating the Nine-Point Graphs

A program has been written which recently generated all the (unlabelled) nine-point graphs. Written in MACRO-10 assembly language and run on a 165K PDP-10, it generates the complete set of 274,668 graphs in less than six hours. The algorithm on which this program is based is discussed with an emphasis on coding of graphs and various programming techniques designed to save space and time during execution. The methods developed may have applications in other combinatorial generating problems.

Generating the nine-point graphs

A program has been written which recently generated all the (unlabelled) nine-point graphs. Written in MACRO-10 assembly language and run on a 165K PDP-10, it generates the complete set of 274,668 graphs in less than six hours. The algorithm on which this program is based is discussed with an emphasis on coding of graphs and various programming techniques designed to save space and time during execution. The methods developed may have applications in other combinatorial generating problems.