Edge-adjacency Matrix Research Articles

New molecular structure based models for estimation of the CO2 solubility in different choline chloride-based deep eutectic solvents (DESs)

In this study, CO2 solubility in different choline chloride-based deep eutectic solvents (DESs) has been investigated using the Quantitative Structure–Property Relationship (QSPR). In this regard, the effect of different structures of the hydrogen bond donor (HBD) in choline chloride (ChCl) based deep eutectic solvents (DESs) has been studied in different temperatures and different molar ratios of ChCl as hydrogen bond acceptor (HBA) to HBD. 12 different datasets with 390 data on the CO2 solubility were chosen from the literature for the model development. Eight predictive models, which contain the pressure and one structural descriptor, have been developed at the fixed temperature (i.e. 293, 303, 313, or 323 K), and the constant molar ratio of ChCl to HBD equal to 1:3 or 1:4. Moreover, two models were also introduced, which considered the effects of pressure, temperature, and HBD structures, simultaneously in the molar ratios equal to 1:3 or 1:4. Two additional datasets were used only for the further external validation of these two models at new temperatures, pressures, and HBD structures. It was identified that CO2 solubility depends on the “EEig02d” descriptor of HBD. “EEig02d” is a molecular descriptor derived from the edge adjacency matrix of a molecule that is weighted by dipole moments. This descriptor is also related to the molar volume of the structure. The statistical evaluation of the proposed models for the unfixed and fixed temperature datasets confirmed the validity of the developed models.

A new Technique in Protein Structure Quantitative Identification

Graph distance is also named approximate graph isomorphism, error-tolerant graph matching, it is a measure of similarity (or dissimilarity) between two graphs. The graph distance function is an elusive question remaining a pressing problem in a wide range of fields. In general, graphs are usually enriched with node and edge attributes, namely, heterogeneous graphs, our focus is encapsulating node and edge identities in a graph. In addition, an adequate representation of the graph-adjacency matrix can capture the connectivity of the graph structure, while an adjacency matrix bears high-order combinatorial nature. Solving these challenges in the effective analysis of these graphs is of utmost importance. In this paper, structural and semantic features are both preserved by graphs, so the analysis of row sum and eigenspectra for vertex and edge adjacency matrix has captured the affinity interactions within graphs locally and globally. Therefore, we estimate the geometrical and semantic dissimilarities/distances between graphs extensively and systemically. Several illustrative chemical compound cases implement in practical scenarios, thus, the theory could be easily understood. Our method defines a quantitative distance (or dissimilarity) measure for comparing graphs, it has a good measure of expressiveness and a suited polynomial computation complexity, which paves the way for graph analysis.

New bound for edge spectral radius and edge energy of graphs

Let $ X(V,E) $ be a simple graph with $ n $ vertices and $ m $ edges without isolated vertices. Denote by $ B = (b_{ij})_{mtimes m} $ the edge adjacency matrix of $ X $. Eigenvalues of the matrix $ B $, $mu_1, mu_2, cdots, mu_m $, are the edge spectrum of the graph $ X $. An important edge spectrum-based invariant is the graph energy, defined as $ E_e(X) =sum_{i=1}^{m} vert mu_i vert $. Suppose $ B^{'} $ be an edge subset of $ E(X) $ (set of edges of $ X $). For any $ e in B^{'} $ the degree of the edge $ e_i $ with respect to the subset $ B^{'} $ is defined as the number of edges in $ B^{'} $ that are adjacent to $ e_i $. We call it as $ varepsilon $-degree and is denoted by $ varepsilon_i $. Denote $ mu_1(X) $ as the largest eigenvalue of the graph $ X $ and $ s_i $ as the sum of $ varepsilon $-degree of edges that are adjacent to $ e_i $. In this paper, we give lower bounds of $ mu_1(X) $ and $ mu_1^{D^{'}}(X) $ in terms of $ varepsilon $-degree. Consequently, some existing bounds on the graph invariants $ E_e(X) $ are improved.

A polynomial‐time algorithm for simple undirected graph isomorphism

A polynomial‐time algorithm for simple undirected graph isomorphism

On the Energy of Benzenoid Hydrocarbons

In chemistry, the energy of a graph is of interest since it can be used to approximate the total π-electron energy of molecules. The spectral moments of the edge adjacency matrix and adjacency matrix recently have been successfully employed in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies of alkanes, alkyl halides, benzyl alcohols, cycloalkanes, and benzenoid hydrocarbons. Let G be a molecular graph and L(G) be a line graph, then the eigenvalues of G and L(G) are denoted by, and respectively. The energies A and L(G) are defined as and In this paper, we examined both energies with spectral moments of the edge adjacency matrix and adjacency matrix of benzenoid hydrocarbons. These theoretical conclusions provide practical guiding significance for pharmaceutical engineering, complex network and quantify the degree of folding of long organic molecules.

On Tosha-degree of an Edge in a Graph

In an earlier paper, we have introduced the Tosha-degree of an edge in a graph without multiple edges and studied some properties. In this paper, we extend the definition of Tosha-degree of an edge in a graph in which multiple edges are allowed. Also, we introduce the concepts - zero edges in a graph, $T$-line graph of a multigraph, Tosha-adjacency matrix, Tosha-energy, edge-adjacency matrix and edge energy of a graph $G$ and obtain some results.

Results on the Estrada Indices of Benzenoid Hydrocarbons

The spectral moments of the edge adjacency matrix and adjacency matrix recently have been successfully employed in quantitative structure–property relationship and quantitative structure–activity relationship studies of alkanes, alkyl halides, benzyl alcohols,cycloalkanes and benzenoid hydrocarbons. Let G be a molecular graph with n vertices, m edges and G(L) be a line graph. Both graphs G and G(L) can be represented by their adjacency matrixes A and E, respectively. The eigenvalues of G and G(L) are denoted by, and respectively. The Estrada index A and G(L) are defined as and In this paper, we examined both Estrada indices with spectral moments of the edge adjacency matrix and adjacency matrix of benzenoid hydrocarbons. These theoretical conclusions provide practical guiding significance for pharmaceutical engineering, complex network and quantify the degree of folding of long organic molecules.

A polynomial‐time algorithm for simple undirected graph isomorphism

SummaryThe graph isomorphism problem is to determine two finite graphs that are isomorphic which is not known with a polynomial‐time solution. This paper solves the simple undirected graph isomorphism problem with an algorithmic approach as NP=P and proposes a polynomial‐time solution to check if two simple undirected graphs are isomorphic or not. Three new representation methods of a graph as vertex/edge adjacency matrix and triple tuple are proposed. A duality of edge and vertex and a reflexivity between vertex adjacency matrix and edge adjacency matrix were first introduced to present the core idea. Beyond this, the mathematical approval is based on an equivalence between permutation and bijection. Because only addition and multiplication operations satisfy the commutative law, we propose a permutation theorem to check fast whether one of two sets of arrays is a permutation of another or not. The permutation theorem was mathematically approved by Integer Factorization Theory, Pythagorean Triples Theorem, and Fundamental Theorem of Arithmetic. For each of two n‐ary arrays, the linear and squared sums of elements were respectively calculated to produce the results.

Bond-extended stochastic and nonstochastic bilinear indices. I. QSPR/QSAR applications to the description of properties/activities of small-medium size organic compounds

Bond-extended stochastic and nonstochastic bilinear indices are introduced in this article as novel bond-level molecular descriptors (MDs). These novel totals (whole-molecule) MDs are based on bilinear maps (forms) similar to use defined in linear algebra. The proposed nonstochastic indices try to match molecular structure provided by the molecular topology by using the kth Edge(Bond)-Adjacency Matrix (Ek, designed here as a nonstochastic E matrix). The stochastic parameters are computed by using the kth stochastic edge-adjacency matrix, ESk, as matrix operators of bilinear transformations. This new edge (bond)-adjacency relationship can be obtained directly from Ek and can be considered like a new matrix-transformation strategic to obtain new relations for a molecular graph. In both set of MDs, chemical information is codified by using different pair combinations of atomic weightings (in this case four atomic-labels: atomic mass, polarizability, van der Waals volume, and electronegativity). In addition, a local-fragment (bond-type) formalism was also developed. The kth bond-type bilinear indices are calculated by summing the kth bond bilinear indices of all bonds of the same bond type in the molecules. The new set of MDs can be easily and quickly calculated in our in house software TOMOCOMD-CARDD (topological molecular computational design computer-aided-rational-drug design). The reported application and utilization of these MDs for predictive capability correlations of structure with physicochemical and pharmacological properties are reviewed. Three benchmark datasets have been used to evaluate the QSPR/QSAR behavior of the new bond-level TOMOCOMD-CARDD MDs. We developed the QSPR models to describe several physicochemical properties of octane isomers (First Case) and, to analyze of the boiling point of 28 alkyl-alcohols (Second Case) and to examine of the specific rate constant (log k), the partition coefficient (log P), as well as the antibacterial activity of 34 derivatives of 2-furylethylenes (Third Case). For these three rounds, the quantitative models found are significant from a statistical point of view and permit a clear interpretation of the studied properties in terms of the structural features of molecules. A leave-one-out cross-validation procedure revealed that the regression models had a good predictability. The comparison with other approaches reveals good performance of the method proposed. Therefore, it is clearly demonstrated that this suitability is higher than that shown by other 2D/3D well-known sets of MDs. The approach described here appears to be a very promising structural invariant, useful for QSPR/QSAR studies and shown to provide an excellent alternative or guides for discovery and optimization of new lead compounds, reducing the time and cost of the traditional procedure. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Bond-based linear indices of the non-stochastic and stochastic edge-adjacency matrix. 1. Theory and modeling of ChemPhys properties of organic molecules

Novel bond-level molecular descriptors are proposed, based on linear maps similar to the ones defined in algebra theory. The kth edge-adjacency matrix (E(k)) denotes the matrix of bond linear indices (non-stochastic) with regard to canonical basis set. The kth stochastic edge-adjacency matrix, ES(k), is here proposed as a new molecular representation easily calculated from E(k). Then, the kth stochastic bond linear indices are calculated using ES(k) as operators of linear transformations. In both cases, the bond-type formalism is developed. The kth non-stochastic and stochastic total linear indices are calculated by adding the kth non-stochastic and stochastic bond linear indices, respectively, of all bonds in molecule. First, the new bond-based molecular descriptors (MDs) are tested for suitability, for the QSPRs, by analyzing regressions of novel indices for selected physicochemical properties of octane isomers (first round). General performance of the new descriptors in this QSPR studies is evaluated with regard to the well-known sets of 2D/3D MDs. From the analysis, we can conclude that the non-stochastic and stochastic bond-based linear indices have an overall good modeling capability proving their usefulness in QSPR studies. Later, the novel bond-level MDs are also used for the description and prediction of the boiling point of 28 alkyl-alcohols (second round), and to the modeling of the specific rate constant (log k), partition coefficient (log P), as well as the antibacterial activity of 34 derivatives of 2-furylethylenes (third round). The comparison with other approaches (edge- and vertices-based connectivity indices, total and local spectral moments, and quantum chemical descriptors as well as E-state/biomolecular encounter parameters) exposes a good behavior of our method in this QSPR studies. Finally, the approach described in this study appears to be a very promising structural invariant, useful not only for QSPR studies but also for similarity/diversity analysis and drug discovery protocols.

Bond-based global and local (bond, group and bond-type) quadratic indices and their applications to computer-aided molecular design. 1. QSPR studies of diverse sets of organic chemicals

The concept of atom-based quadratic indices is extended to a series of molecular descriptors (MDs) (both total and local) based on adjacency between edges. The kth edge-adjacency matrix (E ( k )) denotes the matrix of bond-based quadratic indices (non-stochastic) with respect to the canonical basis set. The kth "stochastic" edge-adjacency matrix, ES ( k ), is here proposed as a new molecular representation easily calculated from E ( k ). Then, the kth stochastic bond-based quadratic indices are calculated using ES ( k ) as operators of quadratic transformations. The study of six representative physicochemical properties of octane isomers was used to compare the ability of both series of MDs to produce significant quantitative structure-property relationship (QSPR) models. Moreover, the general performance of the new MDs in this QSPR study has been evaluated with respect to other 2D/3D well-known sets of indices and the obtained results shown a quite satisfactory behavior of the present method. The novel bond-level MDs were also used for the description and prediction of the boiling point of 28 alkyl-alcohols and to the modeling of the specific rate constant (log k) of 34 derivatives of 2-furylethylenes. These models were statistically significant and showed very good stability to data variation in leave-one-out (LOO) cross-validation experiment. The comparison with other approaches (edge- and vertices-based connectivity indices, total and local spectral moments, and quantum chemical descriptors as well as E-state/biomolecular encounter parameters) expose a good behavior of our method in this QSPR studies. The approach described in this report appears to be a very promising structural invariant, useful for QSPR/QSAR studies, similarity/diversity analysis, and computer-aided "rational" molecular (drug) design.

Tenth Spectral Moment for Molecular Graphs of Phenylenes

Our investigations are motivated by recent papers concerning the spectral moments of the edge-adjacency matrix, which were successfully employed in QSAR and QSPR studies of different classes of compounds [Estrada, E. J. Chem. Inf. Comput. Sci. 1996, 36, 844−849; 1997, 37, 320−328; 1998, 38, 23−27]. In this work, the evaluation of the 10th spectral moment of the vertex-adjacency matrix for phenylenes is considered. This is achieved by using the method of Hall [Hall, G. G. Theor. Chim. Acta 1986, 70, 323−332], which was originally applied to benzenoid systems and then adopted for molecular graphs of phenylenes [Marković, S.; Stajković, A. Theor. Chem. Acc. 1997, 96, 256−260]. It is found that the 10th spectral moment can be expressed in terms of 6 mutually independent invariants that can be easily deduced from the molecular graph.

Spectral Moments of the Edge Adjacency Matrix in Molecular Graphs. Benzenoid Hydrocarbons

The spectral moments of the edge adjacency matrix recently have been successfully employed in quantitative structure−property relationship (QSPR) and quantitative structure−activity relationship (QSAR) studies of alkanes, alkyl halides, benzyl alcohols, and cycloalkanes.1-3 In this work we examined spectral moments of the edge adjacency matrix of benzenoid hydrocarbons. We designed combinatorial formulas for spectral moments up to the seventh order. Our expressions depend on counts of certain structural features that can easily be deduced from the molecular graph. The first few spectral moments of the edge adjacency matrix and of the vertex adjacency matrix depend on the very same structural parameters, implying that the two sets of moments are equivalent from the vewpoint of QSPR and QSAR studies. We demonstrate this for the total π-electron energy.

Spectral Moments of the Edge Adjacency Matrix in Molecular Graphs. 3. Molecules Containing Cycles

A substructural approach to quantitative structure−property relationships based on the spectral moments of the edge adjacency matrix is extended to molecules containing cycles. Spectral moments are expressed as linear combinations of structural fragments of any kind of nonweighted graphs. The boiling points of a series of 80 cycloalkanes was well-described by the present approach. The predictive power of the model was proved by using a test set of another 26 compounds. An equation that expresses the contribution of the different fragments of the molecules to the boiling point was obtained.

Spectral Moments of the Edge-Adjacency Matrix of Molecular Graphs. 2. Molecules Containing Heteroatoms and QSAR Applications

A novel substructural approach to structure−property and structure−activity relationships studies is proposed. The method is based on the computation of spectral moments of the edge-weighted adjacency matrix. In the present work, bond distances are used as bond weights in the diagonal entries of edge matrix. Spectral moments of the edge-weighted adjacency matrix are expressed in terms of structural fragments present in molecules via algebraic expressions. Boiling points of 58 alkyl halides were well described by the present approach, and this property was expressed as a linear combination of structural fragments of the studied molecules. QSAR applications of the novel approach are explored by studying the antifungal activity of benzyl alcohols. An equation expressing contributions of each substituent in benzyl alcohols to the antifungal activity is obtained. The methodology proposed here permits the rational design of novel compounds with improved activity, which is illustrated for benzyl alcohol derivatives.

Spectral Moments of the Edge Adjacency Matrix in Molecular Graphs. 1. Definition and Applications to the Prediction of Physical Properties of Alkanes

A novel graph theoretical invariant based on the spectral moments of the edge adjacency matrix (E) is proposed. Spectral moments of the E matrix are used to describe seven physical properties of alkanes. All the regression models found are very significant from the statistical point of view. The spectral moments are expressed as linear combinations of the different structural fragments of the molecular graph. The use of the substructural approach for the description of seven physical properties of alkanes is also proved. The results obtained are interpreted in term of structural features of molecules.