MATCH Communications in Mathematical and in Computer Chemistry

Online ISSN: 0340-6253
Publications
Fragmentation process (1) 
Article
This paper describes a procedure for the discovery of recurrent substrings in amino acid sequences of proteins, and its application to fungal cell walls. The evolutionary origins of fungal cell walls are an open biological question. This question can be approached by studies of similarity among the sequences and sub-sequences of fungal wall proteins and by comparison to proteins in animals. We describe here how we have discovered building blocks, represented as recurrent sequence motifs (sub-sequences), within fungal cell wall proteins. These motifs have not been systematically identified before, because the low Shannon entropy of the cell wall sequences has hindered searches for local sequence similarities by sequence alignments. Nonetheless, our new, composition-based scoring matrices for local alignment searches now support statistically valid alignments for such low entropy sequences (Coronado et al. 2006. Euk. Cell 5: 628-637). We have now searched for similarities in a set of 171 known and putative cell wall proteins from baker's yeast, Saccharomyces cerevisiae. The aligned segments were repeatedly subdivided and catalogued to identify 217 recurrent sequence motifs of length 8 amino acids or greater. 95% of these motifs occur in more than one cell wall protein. The median length of the motifs is 22 amino acid residues, considerably shorter than protein domains. For many cell wall proteins, these motifs collectively account for more than half of their amino acids. The prevalence of these motifs supports the idea of fungal cell wall proteins as assemblies of recurrent building blocks.
 
Article
We find simple saturated alcohols with the given number of carbon atoms and the minimal normal boiling point. The boiling point is predicted with a weighted sum of the generalized first Zagreb index, the second Zagreb index, the Wiener index for vertex-weighted graphs, and a simple index caring for the degree of a carbon atom being incident to the hydroxyl group. To find extremal alcohol molecules we characterize chemical trees of order $n$, which minimize the sum of the second Zagreb index and the generalized first Zagreb index, and also build chemical trees, which minimize the Wiener index over all chemical trees with given vertex weights.
 
Article
A conjecture of AutoGraphiX on the relation between the Randi\'c index $R$ and the algebraic connectivity $a$ of a connected graph $G$ is: $$\frac R a\leq (\frac{n-3+2\sqrt{2}}{2})/(2(1- \cos {\frac{\pi}{n}})) $$ with equality if and only if $G$ is $P_n$, which was proposed by Aouchiche and Hansen [M. Aouchiche and P. Hansen, A survey of automated conjectures in spectral graph theory, {\it Linear Algebra Appl.} {\bf 432}(2010), 2293--2322]. We prove that the conjecture holds for all trees and all connected graphs with edge connectivity $\kappa'(G)\geq 2$, and if $\kappa'(G)=1$, the conjecture holds for sufficiently large $n$. The conjecture also holds for all connected graphs with diameter $D\leq \frac {2(n-3+2\sqrt{2})}{\pi^2}$ or minimum degree $\delta\geq \frac n 2$. We also prove $R\cdot a\geq \frac {8\sqrt{n-1}}{nD^2}$ and $R\cdot a\geq \frac {n\delta(2\delta-n+2)} {2(n-1)}$, and then $R\cdot a$ is minimum for the path if $D\leq (n-1)^{1/4}$ or $\delta\geq \frac n 2-1$.
 
Article
This is the first time that trigonometrically fitted Adams-Bashforth- Moulton predictor-corrector (P-C) methods are used to efficiently solve the resonance problem of the Schrödinger equation. Our new trigonometrically fitted P-C schemes are based on the well known Adams-Bashforth-Moulton methods. In particular, they are based on the fourth order Adams-Bashforth scheme (as predictor) and on the fifth order Adams-Moulton scheme (as corrector). We tested the efficiency of our newly developed schemes against well known methods, with excellent results. The numerical experimentations indicate that at least one of our schemes is noticeably more efficient compared to other methods, which are specially designed for the numerical solution of the Schrödinger equation.
 
Not all patches with internal 6-cycles are benzenes. The patch in this figure is a helicene. 
(k, 1)-nanotube 
Article
Recently a class of molecular graphs, called altans, became a focus of attention of several theoretical chemists and mathematicians. In this paper we study primary iterated altans and show, among other things, their connections with nanotubes and nanocaps. The question of classification of bipartite altans is also addressed. Using the results of Gutman we are able to enumerate Kekul\'e structures of several nanocaps of arbitrary length.
 
Article
The anti-forcing number of a connected graph $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching. In this paper, we show that the anti-forcing number of every fullerene has at least four. We give a procedure to construct all fullerenes whose anti-forcing numbers achieve the lower bound four. Furthermore, we show that, for every even $n\geq20$ ($n\neq22,26$), there exists a fullerene with $n$ vertices that has the anti-forcing number four, and the fullerene with 26 vertices has the anti-forcing number five.
 
Article
The standard two-step model of homogeneous-catalyzed reactions had been theoretically analyzed at various levels of approximations from time to time. The primary aim was to check the validity of the quasi-steady-state approximation, and hence emergence of the Michaelis-Menten kinetics, with various substrate-enzyme ratios. But, conclusions vary. We solve here the desired set of coupled nonlinear differential equations by invoking a new set of dimensionless variables. Approximate solutions are obtained via the power-series method aided by Pade approximants. The scheme works very successfully in furnishing the initial dynamics at least up to the region where existence of any steady state can be checked. A few conditions for its validity are put forward and tested against the findings. Temporal profiles of the substrate and the product are analyzed in addition to that of the complex to gain further insights into legitimacy of the above approximation. Some recent observations like the reactant stationary approximation and the notions of different timescales are revisited. Signatures of the quasi-steady-state approximation are also nicely detected by following the various reduced concentration profiles in triangular plots. Conditions for the emergence of Michaelis-Menten kinetics are scrutinized and it is stressed how one can get the reaction constants even in the absence of any steady state.
 
Article
In the statistical description of dynamical systems, an indication of the irreversibility of a given state change is given geometrically by means of a (pre-)ordering of state pairs. Reversible state changes of classical and quantum systems are shown to be represented by isometric state transformations. An operational distinction between reversible and irreversible dynamics is given and related to the geometric characterisation of the associated state transformations.
 
A 3-repetition in a vertex v of degree 6 
Article
A novel self-assembly strategy for polypeptide nanostructure design was presented in [Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments, Nature Chemical Biology 9 (2013) 362--366]. The first mathematical model (polypeptide nanostructure can naturally be presented as a skeleton graph of a polyhedron) from [Stable traces as a model for self-assembly of polypeptide nanoscale polyhedrons, MATCH Commun. Math. Comput. Chem. 70 (2013) 317-330] introduced stable traces as the appropriate mathematical description, yet we find them deficient in modeling graphs with either very small (less or equal to 2) or large (greater or equal to 6) degree vertices. We introduce strong traces which remedy both of the above mentioned drawbacks. We show that every connected graph admits a strong trace by studying a connection between strong traces and graph embeddings. Further we also characterize graphs which admit parallel (resp. antiparallel) strong traces.
 
Trees: T B 12 , 5 with degree balance, T B 14 , 5 with almost perfect degree balance and T B 16 , 5 
Tree transformations from Cases 1, 2 and 3 of Theorem 2. 
Tree transformations from Lemmas 6, 7 and 8 respectively. 
Article
In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit formulas for the value of augmented eccentric connectivity index are derived.
 
Article
In theory, the molecular formula of an unknown compound can be calculated from its exact molecular mass. However, even with highly accurate modern mass spectrometers with an accuracy of 1 ppm or lower, it is generally not possible to determine the molecular formula uniquely from measurements. Intensities of isotopic peaks are typically used as additional information to narrow down possible formulas associated with a mass spectrum (MS) peak, but this is not sufficient for larger compounds. Here, we introduce a method that takes information from fragment peak masses of the MS/MS into account to improve the reliability of formula determination. Matchvalues that reflect the consistency with MS isotope peaks and MS/MS fragment patterns are computed for candidate molecular formulas. We demonstrate that these matchvalues outperform methods based on isotope peak intensities alone. In test cases with medium sized organic molecules (< 1000 u) the true molecular formula achieves the highest matchvalues using the MS/MS data.
 
Article
For a graph $G$ of order $n$ and with eigenvalues $\lambda_1\geqslant\cdots\geqslant\lambda_n$, the HL-index $R(G)$ is defined as $R(G) ={\max}\left\{|\lambda_{\lfloor(n+1)/2\rfloor}|, |\lambda_{\lceil(n+1)/2\rceil}|\right\}.$ We show that for every connected bipartite graph $G$ with maximum degree $\Delta\geqslant3$, $R(G)\leqslant\sqrt{\Delta-2}$ unless $G$ is the the incidence graph of a projective plane of order $\Delta-1$. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index.
 
Article
The aim of this paper is to present a generalization of Lunn-Senior's mathematical model of isomerism in organic chemistry. The main idea of Lunn and Senior is that if the type of isomerism is fixed, a molecule with a fixed skeleton and d univalent substituents has a symmetry group $W\leq S_d$ which is generally not the molecule's 3-dimensional symmetry group. The unit character of W induces a representation of the symmetric group $S_d$ which governs the combinatorics of the isomers of the given molecule. Lunn-Senior's thesis is that certain non-negative integers established by this representation are upper boundaries of the corresponding numbers, yielded by the experiment (and often coincide with them). Moreover, the authors define (in a particular case) a partial order among the objects of the model, such that some simple substitution reactions correspond to inequalities. These two groups of data determine the group W, and produce so called "type properties" of the molecule (properties which do not depend on the nature of the univalent substituents). Our hypothesis is that if we replace the unit character of $W$ by any one-dimensional character of $W$ (thus we count only a part of the isomers - those having a maximum property), we also get a type property of the molecule. An instance of that is the inventory of the stereoisomers called chiral pairs. The formalism can be generalized naturally and produces some preliminary chemical results. Especially the partial order is defined and studied in the general case and indicates the possible genetic relations among the corresponding molecules. An important result of E. Ruch which connects the dominance order among partitions and the existence of chiral pairs is obtained as a consequence of a more general statement.
 
Article
The second part of this paper is devoted to the following important question in organic chemistry: given two isomers of a molecule, how to identify them with their structural formulae using only type properties of that molecule? A classical answer of this question is given for benzene by the identification of its di-substituted (para, ortho, and meta), and tri-substituted (asymmetric, vicinal, and symmetric) derivatives via the Korner substitution reactions among them. Here we develop a machinery within the framework of the Lunn-Senior's mathematical model of isomerism in organic chemistry, which, in principle, answers this question. In particular, it is shown that the members of a chiral pair cannot be distinguished via substitution reactions. The examples of ethene, benzene, and cyclopropane are discussed.
 
(a) A Clar cover C of pyrene, (b) The four perfect matchings in the image f (C)
Article
The Clar covering polynomial (also called Zhang-Zhang polynomial in some chemical literature) of a hexagonal system is a counting polynomial for some types of resonant structures called Clar covers, which can be used to determine Kekul\'e count, the first Herndon number and Clar number, and so on. In this paper we find that the Clar covering polynomial of a hexagonal system H coincides with the cube polynomial of its resonance graph R(H) by establishing a one-to-one correspondence between the Clar covers of H and the hypercubes in R(H). Accordingly, some applications are presented.
 
Graph of the regression analysis of the inner pseudo-product 〈 g, g〉 a versus the molecular weight MW. The 95% Confidence Interval for regression coefficient is: Lower bound, 18.258, and upper bound 18.344.
The average of inner pseudo-product between codon subsets XAZ, XUZ, XCZ and XGZ. Behind each codon subset, for example, XUZ there are 16 realizations.
Article
A new n-dimensional vector space of the DNA sequences on the Galois field of the 64 codons (GF(64)) is proposed. In this vector space gene mutations can be considered linear transformations or translations of the wild type gene. In particular, the set of translations that preserve the chemical type of the third base position in the codon is a subgroup which describes the most frequent mutations observed in mutational variants of four genes: human phenylalanine hydroxylase (PAH), human beta globin (HBG), HIV-1 Protease (HIVP) and HIV-1 Reverse transcriptase (HIVRT). Furthermore, an inner pseudo-product defined between codons tends to have a positive value when the codons code to similar amino acids and a negative value when the codons code to amino acids with extreme hydrophobic properties. Consequently, it is found that the inner pseudo-product between the wild type and the mutant codons tends to have a positive value in the mutational variants of the genes: PAH, HBG, HIVP, HIVRT.
 
Caterpillar C n,d .  
Correspondence between caterpillars C n,2r−1 and C n,2r .  
Correspodence between caterpillars C n,d and C(a 0 , a 1 , . . . , a d ).  
Graphs with n = 18 and d = 4 that maximize c 16 and c 15 .  
Graphs with n = 17 and r = 5 that maximize c 15 and c 14 .  
Article
Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k = 0}^n (-1)^k c_k \lambda^{n - k}$. It is well known that for trees the Laplacian coefficient $c_{n-2}$ is equal to the Wiener index of $G$. Using a result of Zhou and Gutman on the relation between the Laplacian coefficients and the matching numbers in subdivided bipartite graphs, we characterize first the trees with given diameter and then the connected graphs with given radius which simultaneously minimize all Laplacian coefficients. This approach generalizes recent results of Liu and Pan [MATCH Commun. Math. Comput. Chem. 60 (2008), 85--94] and Wang and Guo [MATCH Commun. Math. Comput. Chem. 60 (2008), 609--622] who characterized $n$-vertex trees with fixed diameter $d$ which minimize the Wiener index. In conclusion, we illustrate on examples with Wiener and modified hyper-Wiener index that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution.
 
Article
Let $G=(V,E)$ be a simple graph with $n = |V|$ vertices and $m = |E|$ edges. The first and second Zagreb indices are among the oldest and the most famous topological indices, defined as $M_1 = \sum_{i \in V} d_i^2$ and $M_2 = \sum_{(i, j) \in E} d_i d_j$, where $d_i$ denote the degree of vertex $i$. Recently proposed conjecture $M_1 / n \leqslant M_2 / m$ has been proven to hold for trees, unicyclic graphs and chemical graphs, while counterexamples were found for both connected and disconnected graphs. Our goal is twofold, both in favor of a conjecture and against it. Firstly, we show that the expressions $M_1/n$ and $M_2/m$ have the same lower and upper bounds, which attain equality for and only for regular graphs. We also establish sharp lower bound for variable first and second Zagreb indices. Secondly, we show that for any fixed number $k\geqslant 2$, there exists a connected graph with $k$ cycles for which $M_1/n>M_2/m$ holds, effectively showing that the conjecture cannot hold unless there exists some kind of limitation on the number of cycles or the maximum vertex degree in a graph. In particular, we show that the conjecture holds for subdivision graphs.
 
Irreversible reaction network with a deficiency of 4
Article
This paper presents new results about the optimization based generation of chemical reaction networks (CRNs) of higher deficiency. Firstly, it is shown that the graph structure of the realization containing the maximal number of reactions is unique if the set of possible complexes is fixed. Secondly, a mixed integer programming based numerical procedure is given for computing a realization containing the minimal/maximal number of complexes. Moreover, the linear inequalities corresponding to full reversibility of the CRN realization are also described. The theoretical results are illustrated on meaningful examples.
 
Article
The Hosoya polynomial of a graph encompasses many of its metric properties, for instance the Wiener index (alias average distance) and the hyper-Wiener index. An expression is obtained that reduces the computation of the Hosoya polynomials of a graph with cut vertices to the Hosoya polynomial of the so-called primary subgraphs. The main theorem is applied to specific constructions including bouquets of graphs, circuits of graphs and link of graphs. This is in turn applied to obtain the Hosoya polynomial of several chemically relevant families of graphs. In this way numerous known results are generalized and an approach to obtain them is simplified. Along the way several misprints from the literature are corrected.
 
Two cospectral graphs but not cospectral with regard to A
Article
The aim of this paper is to study some parameters of simple graphs related with the degree of the vertices. So, our main tool is the $n\times n$ matrix ${\cal A}$ whose ($i,j$)-entry is $$ a_{ij}= \left\lbrace \begin{array}{ll} \frac{1}{\sqrt{\delta_i\delta_j}} & {\rm if }\quad v_i\sim v_j ; \\ 0 & {\rm otherwise,} \end{array} \right. $$ where $\delta_i$ denotes the degree of the vertex $v_i$. We study the Randi\'{c} index and some interesting particular cases of conditional excess, conditional Wiener index, and conditional diameter. In particular, using the matrix ${\cal A}$ or its eigenvalues, we obtain tight bounds on the studied parameters.
 
3: Tricyclic graph P 6,6,6 n . 
Article
For a given simple graph $G$, the energy of $G$, denoted by $\mathcal {E}(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix, which was defined by I. Gutman. The problem on determining the maximal energy tends to be complicated for a given class of graphs. There are many approaches on the maximal energy of trees, unicyclic graphs and bicyclic graphs, respectively. In this paper, we study the maximal energy of tricyclic graphs. Let $P^{6,6,6}_n$ denote the graph with $n\geq 20$ vertices obtained from three copies of $C_6$ and a path $P_{n-18}$ by adding a single edge between each of two copies of $C_6$ to one endpoint of the path and a single edge from the third $C_6$ to the other endpoint of the $P_{n-18}$. Very recently, Aouchiche et al. [M. Aouchiche, G. Caporossi, P. Hansen, Open problems on graph eigenvalues studied with AutoGraphiX, {\it EURO J. Comput. Optim.} {\bf 1}(2013), 181--199] put forward the following conjecture: let $G$ be a tricyclic graphs on $n$ vertices with $n=20$ or $n\geq22$, then $\mathcal{E}(G)\leq \mathcal{E}(P_{n}^{6,6,6})$ with equality if and only if $G\cong P_{n}^{6,6,6}$. We partially solve this conjecture.
 
Article
In the first part of this paper, we propose new optimization-based methods for the computation of preferred (dense, sparse, reversible, detailed and complex balanced) linearly conjugate reaction network structures with mass action dynamics. The developed methods are extensions of previously published results on dynamically equivalent reaction networks and are based on mixed-integer linear programming. As related theoretical contributions we show that (i) dense linearly conjugate networks define a unique super-structure for any positive diagonal state transformation if the set of chemical complexes is given, and (ii) the existence of linearly conjugate detailed balanced and complex balanced networks do not depend on the selection of equilibrium points. In the second part of the paper it is shown that determining dynamically equivalent realizations to a network that is structurally fixed but parametrically not can also be written and solved as a mixed-integer linear programming problem. Several examples illustrate the presented computation methods.
 
Article
The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. Let $S_2$ be the star of order 2 (or $K_2$) and $Q$ be the graph obtained from $S_2$ by attaching two pendent edges to each of the end vertices of $S_2$. Majstorovi\'c et al. conjectured that $S_2$, $Q$ and the complete bipartite graphs $K_{2,2}$ and $K_{3,3}$ are the only 4 connected graphs with maximum degree $\Delta \leq 3$ whose energies are equal to the number of vertices. This paper is devoted to giving a confirmative proof to the conjecture. Comment: 17 pages
 
Article
The eccentric connectivity index $\xi^c$ is a distance--based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. We prove that the broom has maximum $\xi^c$ among trees with a fixed maximum vertex degree, and characterize such trees with minimum $\xi^c$\,. In addition, we propose a simple linear algorithm for calculating $\xi^c$ of trees.
 
Article
The eccentric connectivity index, proposed by Sharma, Goswami and Madan, has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature. We now report mathematical properties of the eccentric connectivity index. We establish various lower and upper bounds for the eccentric connectivity index in terms of other graph invariants including the number of vertices, the number of edges, the degree distance and the first Zagreb index. We determine the n-vertex trees of diameter with the minimum eccentric connectivity index, and the n-vertex trees of pendent vertices, with the maximum eccentric connectivity index. We also determine the n-vertex trees with respectively the minimum, second-minimum and third-minimum, and the maximum, second-maximum and third-maximum eccentric connectivity indices for Comment: 18 pages, 2 figures
 
Article
We consider topological indices I that are sums of f(deg(u)) f(deg(v)), where {u,v} are adjacent vertices and f is a function. The Randi{\'c} connectivity index or the Zagreb group index are examples for indices of this kind. In earlier work on topological indices that are sums of independent random variables, we identified the correlation between I and the edge set of the molecular graph as the main cause for correlated indices. We prove a necessary and sufficient condition for I having zero covariance with the edge set.
 
Article
The Wiener index of a connected graph is the sum of topological distances between all pairs of vertices. Since Wang gave a mistake result on the maximum Wiener index for given tree degree sequence, in this paper, we investigate the maximum Wiener index of trees with given degree sequences and extremal trees which attain the maximum value. Comment: 19 pages, 2 figures
 
An example of Huffman tree construction  
The counterexample for non-monotone weights  
Article
In 1997 Klav\v{z}ar and Gutman suggested a generalization of the Wiener index to vertex-weighted graphs. We minimize the Wiener index over the set of trees with the given vertex weights' and degrees' sequences and show an optimal tree to be the, so-called, Huffman tree built in a bottom-up manner by sequentially connecting vertices of the least weights.
 
Article
The Randi\'c index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randi\'c index $R(G)$ with relations to the diameter $D(G)$ and the average distance $\mu(G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $\delta(G)$, if $\delta(G)\geq 5$, then $R(G)-D(G)\geq \sqrt 2-\frac{n+1} 2$; if $\delta(G)\geq n/5$ and $n\geq 15$, $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq \mu(G)$. Furthermore, for any arbitrary real number $\varepsilon \ (0<\varepsilon<1)$, if $\delta(G)\geq \varepsilon n$, then $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq \mu(G)$ hold for sufficiently large $n$. Comment: 7 pages
 
Article
For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix, and the distance energy is defined as the sum of the absolute values of the eigenvalues of its distance matrix. We establish lower and upper bounds for the distance spectral radius of graphs and bipartite graphs, lower bounds for the distance energy of graphs, and characterize the extremal graphs. We also discuss upper bounds for the distance energy.
 
Article
Let $G$ be a simple graph and $\alpha$ a real number. The quantity $s_{\alpha}(G)$ defined as the sum of the $\alpha$-th power of the non-zero Laplacian eigenvalues of $G$ generalizes several concepts in the literature. The Laplacian Estrada index is a newly introduced graph invariant based on Laplacian eigenvalues. We establish bounds for $s_{\alpha}$ and Laplacian Estrada index related to the degree sequences.
 
Article
Let $G$ be a simple graph with $n$ vertices and let $\mu_1 \geqslant \mu_2 \geqslant...\geqslant \mu_{n - 1} \geqslant \mu_n = 0$ be the eigenvalues of its Laplacian matrix. The Laplacian Estrada index of a graph $G$ is defined as $LEE (G) = \sum\limits_{i = 1}^n e^{\mu_i}$. Using the recent connection between Estrada index of a line graph and Laplacian Estrada index, we prove that the path $P_n$ has minimal, while the star $S_n$ has maximal $LEE$ among trees on $n$ vertices. In addition, we find the unique tree with the second maximal Laplacian Estrada index.
 
CRN realization of eq. (27) containing the vertices of the only nontrivial strong component of the dense realization
Article
An algorithm is given in this paper for the computation of dynamically equivalent weakly reversible realizations with the maximal number of reactions, for chemical reaction networks (CRNs) with mass action kinetics. The original problem statement can be traced back at least 30 years ago. The algorithm uses standard linear and mixed integer linear programming, and it is based on elementary graph theory and important former results on the dense realizations of CRNs. The proposed method is also capable of determining if no dynamically equivalent weakly reversible structure exists for a given reaction network with a previously fixed complex set.
 
Construction of the cylindrical part of a (4, 3)-nanotube. The hexagons with the same name overlap.  
All possible types of characterizing rings for a (4, 3)-infinite (hexagonal) tube. The shaded hexagons on the dashed lines are overlapping.  
For a white vertex v which receives charge −1 from a bad or transition hexagon h, there is always another hexagon, which sends positive charge to v.
Article
The saturation number of a graph $G$ is the cardinality of any smallest maximal matching of $G$, and it is denoted by $s(G)$. Fullerene graphs are cubic planar graphs with exactly twelve 5-faces; all the other faces are hexagons. They are used to capture the structure of carbon molecules. Here we show that the saturation number of fullerenes on $n$ vertices is essentially $n/3$.
 
Article
Gutman and Wagner proposed the concept of the matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of its matching polynomial. The matching energy of $G$ is defined to be the sum of the absolute values of $\mu_{i}\ (i=1,2,\ldots,n)$. Gutman and Cvetkoi\'c determined the tricyclic graphs on $n$ vertices with maximal number of matchings by a computer search for small values of $n$ and by an induction argument for the rest. Based on this result, in this paper, we characterize the graphs with the maximal value of matching energy among all tricyclic graphs, and completely determine the tricyclic graphs with the maximal matching energy. We prove our result by using Coulson-type integral formula of matching energy, which is similar as the method to comparing the energies of two quasi-order incomparable graphs.
 
Article
Heydari \cite{heydari2013} presented very nice formulae for the Wiener and terminal Wiener indices of generalized Bethe trees. It is pity that there are some errors for the formulae. In this paper, we correct these errors and characterize all trees with the minimum terminal Wiener index among all the trees of order $n$ and with maximum degree $\Delta$.
 
Article
Let $G$ be a simple undirected $n$-vertex graph with the characteristic polynomial of its Laplacian matrix $L(G)$, $\det (\lambda I - L (G))=\sum_{k = 0}^n (-1)^k c_k \lambda^{n - k}$. Laplacian--like energy of a graph is newly proposed graph invariant, defined as the sum of square roots of Laplacian eigenvalues. For bipartite graphs, the Laplacian--like energy coincides with the recently defined incidence energy $IE (G)$ of a graph. In [D. Stevanovi\' c, \textit{Laplacian--like energy of trees}, MATCH Commun. Math. Comput. Chem. 61 (2009), 407--417.] the author introduced a partial ordering of graphs based on Laplacian coefficients. We point out that original proof was incorrect and illustrate the error on the example using Laplacian Estrada index. Furthermore, we found the inverse of Jacobian matrix with elements representing derivatives of symmetric polynomials of order $n$, and provide a corrected elementary proof of the fact: Let $G$ and $H$ be two $n$-vertex graphs; if for Laplacian coefficients holds $c_k (G) \leqslant c_k (H)$ for $k = 1, 2, ..., n - 1$, then $LEL (G) \leqslant LEL (H)$. In addition, we generalize this theorem and provide a necessary condition for functions that satisfy partial ordering based on Laplacian coefficients.
 
Article
We predict the number of hexagonal systems consisting of 24 and 25 hexagons to be H 24 = 122237774262384 and H 25 = 606259305418149, with 6 and 5 significant digits, respectively. Further estimates for H n up to n = 31 are also given. Hexagonal Systems Informally speaking, a hexagonal system can be viewed as a connected arrangement of hexagonal cells packed in the same way as the typical honeycomb arrangement in a beehive. More formally, it is a finite connected plane graph with no cut-vertices, in which all interior regions are mutually congruent regular hexagons [1]. Hexagonal systems have from time to time attracted the attention of mathematicians (and were named "hexagonal animals", "honeycomb systems", "polyhexes", etc.), in connection with statistical physics and applications to lattice gas models [2, 3, 4]. But the main interest in them comes from chemistry: hexagonal systems are the natural graph representations of benzenoid hydrocarbons, whence the names "benzenoid graphs", ...
 
Article
. Let (G; OmegaGamma be a permutation group of degree n. Let V (G; OmegaGamma be the set of all square matrices of order n which commute with all permutation matrices corresponding to permutations from (G; OmegaGamma : V (G; OmegaGamma is a matrix algebra which is called the centralizer algebra of (G; OmegaGamma : In this paper we introduce the combinatorial analogue of centralizer algebras, namely coherent (cellular) algebras and consider the properties of these algebras. It turns out that coherent algebras provide a very helpful tool for the investigation of the symmetries of graphs of different kinds, in particular, of molecular graphs. 1 Partially supported by DAAD allowance for study visit to Germany 2 Supported by grant #I-0333-263.06/93 from the G. I.F. , the German-Israeli Foundation for Scientific Research and Development Contents 1 Introduction 1 2 The subject of algebraic combinatorics 6 3 Problems related to the perception of the symmetry of chemical graphs...
 
Article
A novel technique for chemical synthesis in drug research is combinatorial chemistry, where usually a set of building-block molecules is attached to a core structure in all the combinatorially possible ways. The resulting set of compounds (called a library) can then be systematically screened for a desired biological activity. In this paper we discuss ways and limits of a mathematical simulation of this procedure. At first, two methods for selecting the building-blocks from a given structure pool are presented with the objective to obtain only dissilimar library entries. Next an algorithm is described for the exhaustive and redundancy-free generation of a combinatorial library, illustrated by a single-step and a multi-component reaction. Finally equations for the enumeration of the library sizes are derived and the limits of the virtual combinatorial chemistry, i.e. purely in computer and without experiment, are discussed. 1
 
Article
In this paper a software package is described that allows using the MM2 force field to compute the energy of three-dimensional conformations of molecules; this energy is minimized, the resulting structures are automatically classified. Thus we get an impression about the set of all low-energy three-dimensional conformations of a given molecule defined in terms of two dimensional connectivity information. The accuracy of the resulting information can be tuned by changing input parameters for the method presented. This is a hybrid which is built from a method for classification of three-dimensional molecules, from a conjugate gradient method for local minimization and from operators stemming from evolutionary algorithms. The latter have proven successful in the solution of di#cult optimization tasks (not only) in mathematical chemistry. They are of special importance for the approximate solution of problems where global optima of multimodal functions in high dimensional spaces are sought. and in Computer Chemistry - match, no. 38, October 1998, pp. 137--159. # This work was supported by the DFG under grant Ke 201/16-1. ## e-mail: clemens.frey@uni-bayreuth.de The paper is organized in three sections. The first one gives a formalization of the problem and shows in which way this problem was solved up to now. In the second section the evolutionary algorithm is introduced after a short presentation of a general formalization for this kind of algorithms. Each operator of our method is shown in a separate subsection. A definition of the function which determines the transition from one generation to the next generation concludes this section. The last section shows trials and corresponding results obtained with the presented method. The summary of trials is followed by list...
 
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The total Z-transformation graph and digraph of perfect matchings of a plane bipartite graph are defined. For a plane elementary bipartite graph it is shown that its total Z-transformation graph and digraph are 2-connected and strongly connected respectively. As an immediate consequence, we have that a plane bipartite graph is elementary if and only if its total Z-transformation digraph is strongly connected.
 
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The probability distribution (density) function for the experimental signal-to-noise ratio (SNR) defined as x̄/s, where x̄ is the sample mean and s is the customary sample standard deviation, has been derived and found to be in excellent agreement with accurate Monte Carlo simulation results. The SNR probability distribution function is a hypergeometric function which has no closed-form expression in elementary functions. The same applies to the probability distribution function for the relative standard deviation. In contrast, the probability distribution function for the approximate SNR defined by μ/s‘, where μ is the population mean parameter and s‘ ≡ s[(N − 1)/N]1/2, has a closed-form expression but is inaccurate for small numbers of measurements. The experimental SNR is a biased estimator of the true SNR, but the bias is easily correctable. Monte Carlo simulation methods were used to derive critical value tables for comparison of experimental SNRs and relative standard deviations. The critical value tables presented herein are accurate to about 1% for confidence levels of 75%, 90%, and 95%, and to about 5% for 99% confidence level.
 
Search tree for Algorithm 1 
Search tree for Algorithm 2 
Search tree for Algorithm 3 
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The best barbeque problem asks for the largest intersection of n sets that are taken one from each of n given collections of subsets of some universal set. This combinatorial optimization problem arises in the context of discovering so-called cis-regulatory modules in regulatory DNA sequences. There are some similarities between this problem and the problem of finding cliques in graphs. These similarities are here utilized to develop novel branch-and-bound algorithms for the best barbeque problem. The algorithms are evaluated using random data. Compared with previously used algorithms, the new algorithms are capable of solving substantially larger instances, which are crucial for some application scenarios.
 
A connected graph G with D(G) = {a, b, c}. The edges that should be removed are drown with dashed lines. 
G(l, k, s) 
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{Recently Hansen and Vukicevic proved that the inequality $M_1/n \leq M_2/m$, where $M_1$ and $M_2$ are the first and second Zagreb indices, holds for chemical graphs, and Vukicevic and Graovac proved that this also holds for trees. In both works is given a distinct counterexample for which this inequality is false in general. Here, we present some classes of graphs with prescribed degrees, that satisfy $M_1/n \leq M_2/m$: Namely every graph $G$ whose degrees of vertices are in the interval $[c; c + \sqrt c]$ for some integer $c$ satisies this inequality. In addition, we prove that for any $\Delta \geq 5$, there is an infinite family of graphs of maximum degree $\Delta$ such that the inequality is false. Moreover, an alternative and slightly shorter proof for trees is presented, as well\ as for unicyclic graphs.
 
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The Wiener index of a graph, which is the sum of the distances between all pairs of vertices, has been well studied. Recently, Sills and Wang in 2012 proposed two conjectures on the maximal Wiener index of trees with a given degree sequence. This note proves one of the two conjectures and disproves the other.
 
W 1,5 = C 5 K 1
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The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.
 
The wREFERASS user interface. 
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The derivation of steady-state equations is necessary for the interpretation of enzyme kinetic studies. We have implemented a C++ library and its graphical interface for MS-Windows, named wREFERASS, to facilitate the study of complex reaction mechanisms at the strict steady-state. The C++ library is freely available and could be used directly in any program language as a library function call. We have tested the program with very complex enzyme-catalyzed reaction mechanisms and found its performance to be satisfactory. The program wREFERASS along with instructions and many examples, can be downloaded from http://oretano.iele-ab.uclm.es/~fgarcia/wREFERASS/. Copyright 2011 Elsevier B.V.
 
S(C 5 )
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Eigenvalue of a graph is the eigenvalue of its adjacency matrix. The energy of a graph is the sum of the absolute values of its eigenvalues. In this note we obtain analytic expressions for the energy of two classes of regular graphs.
 
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Ivan Gutman
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Kinkar Das
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Boris Furtula
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