Characteristic polynomials of large graphs. On alternate form of characteristic polynomial [1]

A recursion exists between the absolute magnitudes of the coefficients of the characteristic polynomials of certain families of cyclic and acyclic graphs which makes their computation quite easy for very large graphs using a pencil-and-a-paper approach. Structural requirements are given for such families of graphs which are of interest to the problem of recognition defined in [1].     

Analysis of the topological dependency of the characteristic polynomial in its chebyshev expansion

The structural dependency (effect of branching and cyclisation) of an alternative form, the Chebyshev expansion, for the characteristic polynomial were investigated systematically. Closed forms of the Chebyshev expansion for an arbitrary star graph and a bicentric tree graph were obtained in terms of the “structure factor” expressed as the linear combination of the “step-down operator”. Several theorems were also derived for non-tree graphs. Usefulness and effectiveness of the Chebyshev expansion are illustrated with a number of examples. Relation with the topological index (Z _G ) was discussed. Operated for the U.S. Department of Energy by ISU under contract no. W-ENG-7405-82. Supported in part by the Office of Director     

Minimal energy of unicyclic graphs of a given diameter

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. For a given positive integer d with , we characterize the graphs with minimal energy in the class of unicyclic graphs with n vertices and a given diameter d.      

On the matching polynomials of graphs with small number of cycles of even length

Suppose that G is a simple graph. We prove that if G contains a small number of cycles of even length then the matching polynomial of G can be expressed in terms of the characteristic polynomials of the skew adjacency matrix A(G^e) of an arbitrary orientation G^e of G and the minors of A(G^e). In addition to a formula previously discovered by Godsil and Gutman, we obtain a different formula for the matching polynomial of a general graph. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Graph theoretical procedure for obtaining analytical expressions of eigenspectra of linear chains and cycles with alternant vertex weights and same edge weight: Application to some complicated graphs

A graph theoretical procedure for obtaining eigenvalues of linear chains and cycles having alternant vertex weights (h₁, h₂, h₁, h₂, h₁, h₂, …) and the same edge weight (k) have been developed. The eigenvalues of some complicated graphs, such as graphs of linear polyacenes, methylene‐substituted linear polyacenes and cylindrical polyacene strips, stack graphs, and reciprocal graphs have been shown to be generated in closed analytical forms by this procedure. Many such graphs represent chemically important molecules or radicals. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

On the centrality of vertices of molecular graphs

For acyclic systems the center of a graph has been known to be either a single vertex of two adjacent vertices, that is, an edge. It has not been quite clear how to extend the concept of graph center to polycyclic systems. Several approaches to the graph center of molecular graphs of polycyclic graphs have been proposed in the literature. In most cases alternative approaches, however, while being apparently equally plausible, gave the same results for many molecules, but occasionally they differ in their characterization of molecular center. In order to reduce the number of vertices that would qualify as forming the center of the graph, a hierarchy of rules have been considered in the search for graph centers. We reconsidered the problem of “the center of a graph” by using a novel concept of graph theory, the vertex “weights,” defined by counting the number of pairs of vertices at the same distance from the vertex considered. This approach gives often the same results for graph centers of acyclic graphs as the standard definition of graph center based on vertex eccentricities. However, in some cases when two nonequivalent vertices have been found as graph center, the novel approach can discriminate between the two. The same approach applies to cyclic graphs without additional rules to locate the vertex or vertices forming the center of polycyclic graphs, vertices referred to as central vertices of a graph. In addition, the novel vertex “weights,” in the case of acyclic, cyclic, and polycyclic graphs can be interpreted as vertex centralities, a measure for how close or distant vertices are from the center or central vertices of the graph. Besides illustrating the centralities of a number of smaller polycyclic graphs, we also report on several acyclic graphs showing the same centrality values of their vertices. © 2013 Wiley Periodicals, Inc.     

Properties of “structure factor” of characteristic polynomial and a proof of Hosoya-Randić conjectures

Some properties of structure factor of characteristic polynomial are discussed and Hosoya-Randi conjectures are proved rigorously.     

Calculation of the coefficients of the characteristic polynomial from its subgraphs

The characteristic polynomial associated with π-electrons of conjugated molecules are discussed by using subgraphs derived from molecular graphs as a basis for their construction. A practical method has been developed for evaluating the coefficient a_K of conjugated molecules. Applying this method, the general formulas of evaluating the coefficient a_K for homologous conjugated molecules have been obtained. The approach is illustrated on a few simple conjugated systems, including also a few polymeric systems. © 1996 John Wiley & Sons, Inc.     

Computational algorithms for matching polynomials of graphs from the characteristic polynomials of edge-weighted graphs

Computational algorithms are described which provide for constructing the set of associated edge-weighted directed graphs such that the average of the characteristic polynomials of the edge-weighted graphs gives the matching polynomial of the parent graph. The weights were chosen to be unities or purely imaginary numbers so that the adjacency matrix is hermitian. The computer code developed earlier by one of the authors (K.B.) is generalized for complex hermitian matrices. Applications to bridged and spirographs, some lattices and all polycyclic graphs containing up to four cycles are considered.     

On the calculation of the acyclic polynomial

Graphical methods are developed for recursive evaluation of the acyclic polynomial. Analytical formulas of the acyclic polynomials for several specific series of graphs are given. Mathematical properties of the derivatives of the acyclic polynomial are given.     

Voronota: A fast and reliable tool for computing the vertices of the Voronoi diagram of atomic balls

The Voronoi diagram of balls, corresponding to atoms of van der Waals radii, is particularly well‐suited for the analysis of three‐dimensional structures of biological macromolecules. However, due to the shortage of practical algorithms and the corresponding software, simpler approaches are often used instead. Here, we present a simple and robust algorithm for computing the vertices of the Voronoi diagram of balls. The vertices of Voronoi cells correspond to the centers of the empty tangent spheres defined by quadruples of balls. The algorithm is implemented as an open‐source software tool, Voronota. Large‐scale tests show that Voronota is a fast and reliable tool for processing both experimentally determined and computationally modeled macromolecular structures. Voronota can be easily deployed and may be used for the development of various other structure analysis tools that utilize the Voronoi diagram of balls. Voronota is available at: http://www.ibt.lt/bioinformatics/voronota . © 2014 Wiley Periodicals, Inc.     

Properties and derivation of the fourth and sixth coefficients of the characteristic polynomial of molecular graphs—New graphical invariants

Some previously unknown relationships for determining the a ₄ and a ₆ coefficients of the characteristic polynomial for polycyclic aromatic hydrocarbons are presented for the first time. The structural information contained in these coefficients is more fully revealed. The equations derived for a ₄ and a ₆ allow one to determine the characteristic polynomial by inspection for many small molecular graphs. Some relationships for a ₈ and a ₁₀ are presented. The set of known graphical invariants (GI) or properties that remain unchanged in isomeric PAH6s is now shown to be GI={a ₄, a ₆+n ₀+2r ₆, a ₈ ^c , d _s+N_Ic, N_c, N_h, N_Ic+N_Pc, q, r}.Part VIII: A periodic table for polycyclic aromatic hydrocarbons     

Functional groups expressed as graphs extracted from the Laplacian of the electron density

The valence charge concentration shell, as determined by the Laplacian of the electron density, is used as a source of quantum topological graphs, called L‐graphs. A considerable number of such graphs are extracted from the ab initio wave functions of 31 molecules calculated at the B3LYP/6‐311+G(2d,p)//B3LYP/6‐311+G(2d,p) level, covering common functional groups in organic chemistry. We show how L‐graphs can be constructed from a largely transferable subgraph called atomic L‐graph. We investigate the topological stability of the L‐graphs as a function of the basis set. Reliable and consistent atomic L‐graphs are only obtained with basis sets of triple‐zeta quality or higher. The recurrence of invariant motifs or subgraphs in the L‐graphs enables the isolation of 16 atomic L‐graphs. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

The effect of a pendant amine in phosphine ligand on the structure and electrochemical property of diiron dithiolate complexes

Abstract

In order to explore the effect of a pendant amine on a phosphine ligand on the structure and electrochemical properties of diiron dithiolate complexes, this work reports the crystallographic and electrocatalytic comparisons of three diiron monophosphine complexes Fe₂(μ-pdt)(CO)₅{Ph₂P(NHR)} [pdt?=?propanedithiolate (SCH₂CH₂CH₂S); R?=?para-methoxycarbonylphenyl (C₆H₄CO₂Me-p) (1), para-methoxyphenyl (C₆H₄OMe-p) (2) and phenyl (Ph) (3)] with a pendant amine and one reference analogue Fe₂(μ-pdt)(CO)₅{Ph₂P(CH₂Ph)} (4). While the new complex 4 has been characterized by elemental analysis and various spectroscopic techniques, the molecular structures of 3 and 4 were further determined by X-ray crystallography. In addition, the electrochemical properties of 1–4 were studied in acetonitrile (MeCN) in the absence and presence of acetic acid (HOAc) as a mild proton source using cyclic voltammetry (CV). This may demonstrate that they are found to be active electrocatalysts for proton reduction to hydrogen (H₂).     

Construction and utilisation of planar graphs of two series of IPR fullerenes through the use of threefold rotational symmetry

Threefold rotational symmetry has been used to develop an algorithm for the construction of planar graphs of IPR fullerenes and to factorize their characteristic polynomials. Two series of fullerenes of the formula C_60+12n and C_60+18n have thus been obtained. The algorithm has been shown to be useful for predicting the nature of variation of the point groups of the fullerenes with increased n, for counting the number of ¹³C nuclear magnetic resonance (NMR) signals (along with their relative intensities), and also for obtaining a large part of their eigenspectra. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Fibonacci relations. On the computation of some counting polynomials of very large graphs

A definition of a set of Fibonacci graphs is introduced which allows construction of several counting polynomials of very large graphs quite easily using a pencil-and-a-paper approach. These polynomials include matching, sextet, independence, Aihara and Hosoya polynomials. Certain combinatorial properties of Kekulé counts of benzenoid hydrocarbons are given. A relation to a new topological function that counts the cardinality of graph topology [23] is given.Dedicated to Professor Oskar E. Polansky for his enthusiastic support, participation and promotion of chemical graph theory.     

Synthesis of aromatic poly(amide-imide)s from novel diimide-diacid (DIDA) containing sulphone and bulky pendant groups by direct polycondensation with various diamines

A novel diimide-diacid (DIDA) monomer, 4-{4-[(4-methyl phenyl) sulphonyl]}-1,3-bis-trimellitoimido benzene containing sulphone and bulky pendant groups was successfully synthesized and used to synthesize a series of wholly aromatic poly(amide-imide)s (PAIs) by direct polycondensation method. The direct polycondensation of newly synthesized DIDA with different diamines was carried out via Yamazaki’s phosphorylation method using triphenyl phosphite and pyridine system. The resulting poly(amide-imide)s were obtained in quantitative yields with inherent viscosities 0.36-0.47 dl/g in DMAc at 30 ± 0.1 °C. The poly(amide-imide)s were amorphous and were readily soluble in various solvents such as N-methyl-2-pyrrolidinone (NMP), N,N-dimethylacetamide (DMAc), N,N-dimethylformamide (DMF), dimethylsulfoxide (DMSO), and pyridine. Tough and flexible films were obtained by casting their DMAc solution. According to thermogravimetric analysis, the polymers were fairly stable up to temperature around 396 °C, and 10% weight losses in the temperature range of 476-511 °C that showed good thermal stabilities of these polymers.     

Synthesis of aromatic polyesters having pendant carboxyl groups in the side chains and conversion of the carboxyl groups to other reactive groups

Aromatic polyester, copolyester, and poly(ester-amide-thioester) having pendant carboxyl groups are directly synthesized by the organic phase/water phase interfacial polyconden-sation using low-molecular and polymeric phase transfer catalysts. Spectral analysis of the resulting polymers indicates that the nucleophilicity of salts of phenols to diacid chloride is far higher than that of salts of carboxylic acids and chemoselective esterification occurs in a 100% yield. Even if the polymeric catalyst having amino acid moiety as a nucleophilic group is used in the polycondensation, the polymers do not contain anhydride groups. The polyester can be almost quantitatively converted to polymers with different reactive groups by reacting the pendant carboxyl groups with alkyl halides in a DMA_c-H₂O mixture con-taining K₂CO₃. A bifunctional catalytic mechanism is proposed for the chemical modification of the polyesters. © 1995 John Wiley & Sons, Inc.     

Synthesis of novel polybenzimidazoles with pendant amino groups and the formation of their crosslinked membranes for medium temperature fuel cell applications

A series of new polybenzimidazoles (PBIs) with pendant amino groups have been synthesized via condensation polymerization of 5‐aminoisophthalic acid (APTA), isophthalic acid (iPTA), and 3,3′diaminobenzidine (DAB) in polyphosphoric acid at 190 °C for 20 h. The molar ratios between APTA and iPTA were controlled at 1:0, 2:1, 1:1, and 1:2, respectively, and the copolymerization reactions were carried out via both random and sequenced manners. The resulting polymers showed good solubility in some organic solvents such as dimethylsulfoxide (DMSO) and N,N‐dimethylacetamide (DMAc). The pendant amino groups of the PBIs were utilized to react with two kinds of crosslinkers, 1,3‐dibromopropane and ethylene glycol diglycidyl ether, to yield various crosslinked membranes. The crosslinked membranes generally showed good mechanical properties even at high‐phosphoric acid (PA) doping levels, whereas the uncrosslinked membranes highly swelled or even dissolved in PA. Fenton's test revealed that the crosslinked PBI membranes had excellent radical oxidative stability. The proton conductivities of the PA‐doped crosslinked membranes increased with an increase in temperature and high‐proton conductivity up to 0.14 S/cm at 0% relative humidity at 170 °C was achieved. The membranes with high PA‐doping levels, good mechanical properties, and high‐proton conductivities have been successfully developed. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2009