Two new graph-theoretical methods for generation of eigenvectors of chemical graphs

Two new graph-theoretical methods, (A) and (B), have been devised for generation of eigenvectors of weighted and unweighted chemical graphs. Both the methods show that not only eigenvalues but also eigenvectors have full combinatorial (graph-theoretical) content. Method (A) expresses eigenvector components in terms of Ulam’s subgraphs of the graph. For degenerate eigenvalues this method fails, but still the expressions developed yield a method for predicting the multiplicities of degenerate eigenvalues in the graph-spectrum. Some well-known results about complete graphs (K _n) and annulenes (C _n ), viz. (i)K _n has an eigenvalue −1 with (n−1)-fold degeneracy and (ii) C _n cannot show more than two-fold degeneracy, can be proved very easily by employing the eigenvector expression developed in method (A). Method (B) expresses the eigenvectors as analytic functions of the eigenvalues using the cofactor approach. This method also fails in the case of degenerate eigenvalues but can be utilised successfully in case of accidental degeneracies by using symmetry-adapted linear combinations. Method (B) has been applied to analyse the trend in charge-transfer absorption maxima of the some molecular complexes and the hyperconjugative HMO parameters of the methyl group have been obtained from this trend.     

Construction of characteristic polynomial of reciprocal graphs from the number of pendant vertices

A method for construction of the characteristic polynomial (CP) coefficients of the three classes of reciprocal graphs, viz., L_n + n(p), C_n + n(p), and K_1,n?1 + n(p), has been developed that requires only the value of n. The working formulas have been expressed in matrix product form, computer programs for which can easily be developed. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

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.      

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     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

Eigenvalues of large even annulenes in terms ofK 3,S 4 andS 5

Squares of the adjacency matrices of bipartite cycles (C_v) can be block-factored into matrices which correspond to vertex-weighted complete graphs forv = 6, vertex-weighted strongly regular graphs forv = 8 and 10, and vertex-weighted metrically regular graphs forv > 10. Using this fact and some properties of strongly and metrically regular graphs, it is shown that eigenvalues of large bipartite C _v graphs (i.e. large even annulenes) can be expressed by the general formula ± (2 ± (2 ± (... ± (2 +r _p)) ...), wherev = 2 ⁿ ×p,n is the number of surd () signs required andp = 3, 4 and 5. Here,r ₃,r ₄, andr ₅, are the eigenvalues of the complete graphK ₃ and the strongly regular graphsS ₄ andS ₅ respectively. The procedure does not require construction of characteristic polynomials for the determination of eigenvalues, and brings out a common topological origin for the two-fold degeneracies observed in the eigenvalue spectra of all even cycles and many odd cycles.     

Generalized graphs and the Sinanoğlu graphical rules

A comparison of Sinano?lu's VIF (Ref. 1) and generalized graph is presented. Generalized graphs have vertex and edge weights. An abridged history of generalized graphs in theoretical chemistry is given. VIF 's are generalized graphs and therefore have adjacency matrices. The “graphical” rules of Sinano?lu can be represented by congruent transformations on the adjacency matrix. Thus the method of Sinano?lu is incorporated into the broad scheme of graph spectral theory. If the signature of a graph is defined as the collection of the number of positive, zero, and negative eigenvalues of the graph's adjacency matrix, then it is identical to the all-important {n₊, n₀, n_?}, the {number of positive, zero, and negative loops of a reduced graph} or the {number of bonding, nonbonding, and antibonding MO s}. A special case of the Sinano?lu rules is the “multiplication of a vertex” by (?1). In matrix language, this multiplication is an orthogonal transformation of the adjacency matrix. Thus, one can multiply any vertex of a generalized graph by ?1 without changing its eigenvalues.     

Mapping the Metal Positions inside Spherical C80 Cages: Crystallographic and Theoretical Studies of Ce2@D5h‐C80 and Ce2@Ih‐C80

The dynamic positions of the dimetallic cluster inside the mid‐sized spherical cages of C₈₀–C₈₂ have been seldom studied, despite the high abundance of M₂@C_2n (2n=80, 82) species among various endohedral metallofullerenes. Herein, using crystallographic methods, we first unambiguously map the metal positions for both Ce₂@D_5h‐C₈₀ and Ce₂@I_h‐C₈₀, showing how the symmetry or geometrical change in cage structure can influence the motional behavior of the cluster. Inside the D_5h cage, the primary cerium sites have been identified along a cage belt of the contiguous hexagons, which suggests the significant influence of such a cage motif on endohedral cluster motion. Further analysis revealed a distorted D_5h cage owing to the “punch‐out” effect of cerium atoms. The consequence is the presence of two localized electrostatic potential minima inside the cage of (D_5h‐C₈₀)^6?, thus reflecting the primary ionic cerium–cage interaction. In contrast, a different motional behavior of Ce₂ cluster was observed inside the I_h cage. With the major cerium sites, the molecule of Ce₂@I_h‐C₈₀ presented an approximate D_2h configuration. With the combined theoretical study, we propose that the additional unidentified influence of Ni^II(OEP) (OEP=octaethylporphyrin) might be also relevant for the location of cerium sites inside the I_h cage.     

Relativistic symmetry orbitals for the double groups C2v,C∞v,D∞h,and Oh

Relativistic symmetry orbitals are given for molecular LCAO calculations for the double groups: C_2v, C_∞v, D_∞h, and O_h (6-fold) coordination. The atomic orbitals used in the LCAO are of the four component form. A discussion of the comparison between nonre ativistic and relativistic molecular eigenvalues is presented.     

Zur Konformation des Bicyclo[2.2.2]octan-Systems

Crystals of bicyclo [2.2.2] octane-1, 4-dicarboxylic acid are monoclinic, a = 6.01 Å, b = 16.73 Å, c = 10.29 Å, β = 115.08°, space group P2₁/c, with 4 molecules in the unit cell. The structure was solved with the help of direct methods and refined by full-matrix least-squares analysis of the three-dimensional intensity data. Within experimental error the bicyclo [2.2.2]-octane (BCO) skeleton has apparent D_3h-symmetry, corresponding to the totally eclipsed conformation. Analysis of the thermal ellipsoids in terms of the translational and rotational motion of the BCO skeleton leads to an r.m.s. amplitude of 5.9 ± 0.2° for rotation about the threefold axis. On the assumption that the bond lengths remain effectively constant during a torsional vibration of BCO, the potential energy surface has been calculated for a range of semi-empirical potential functions. These calculations show that the energy minimum may be slightly displaced from D_3h symmetry, but if so the barrier between the two such equivalent minima is only about 0.1 kcal mole^?1. The energy eigenvalues and eigenfunctions for a typical variation of potential energy vs torsion angle have been calculated. From the form of the eigenfunction of the ground vibrational state we conclude that BCO has effective D_3h symmetry at all temperatures as far as diffraction methods are concerned.     

Vibration spectra and NMR spectra of octahedral (Oh and O) fullerenes

On analyzing the topological structures of the three big types of octahedral fullerenes: (1) C_n(O_h, n=24h²; h=1, 2,…); (2) C_n(O_h, n=8h²; h=1, 2,…), and (3) C_n(O, n=8(h²+hk+k²); h>k, h, k=1, 2,…), we have obtained theoretically the infrared and Raman active modes by means of the derived formulas for the decomposition of their nuclear motions into irreducible representations and the NMR spectra by using the distribution functions for all of the octahedral (O_h and O) fullerenes, respectively. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 199–205, 1999     

Structures and thermodynamic properties of aluminum oxyhalides: a computational study

The molecular geometries and vibrational frequencies of aluminum oxyhalides, AlOX, and their dimers, (AlOX)₂, have been calculated by MP2 and density functional methods. The monomeric molecules are linear while the dimers have a D _2h -symmetry geometry with a four-membered ring, in which the oxygen atoms serve as bridging ligands. Different thermodynamic properties have been calculated; for AlOBr and AlOI for the first time.     

Graph factorization: A new mode of application of a vertex-alternation scheme

Linear chains where the vertex weights change sign alternantly but are equal in magnitude were able to be reduced to smaller chains by a procedure analogous to that given by Coulson and Rushbrooke. The algorithm for constructing the reduced chains has been stated and proved. The results have been utilized, in conjunction with McClelland's graph-factorization method using reflection (σ) planes, to reduce the HMO secular determinants of some chemical graphs to an extent beyond the ability of group theory. McClelland's σ-plane algorithm, used repetitively where possible, produces factors whose sizes (n_M) are equal to those (n_G) of the group-theoretic factor blocks. For linear polyacenes (LP ), however, a new observation has been made: If the LP has an even number of fused rings, n_M = n_G; but when the LP has an odd number of fused rings, McClelland's process is effective in further reduction, i.e., n_M < n_G. In any case, however, the vertex alternation procedure reported in the present paper brings about further reduction. To demonstrate the utility of the present method, a sample calculation of the LUMO eigenvector graph theoretically has been shown for p-benzoquinone and the result has been utilized to obtain an inductive effect HMO parameter of the methyl group from the charge-transfer bands of some molecular complexes of methylated p-benzoquinones.© 1993 John Wiley & Sons, Inc.     

Two mononuclear cobalt(III) complexes exhibiting phenoxazinone synthase activity

Two mononuclear cobalt(III) complexes, namely [LCo(tmtp)(H₂O)]ClO₄?MeOH ( 1 ) (tmtp = tri(m‐tolyl)phosphine) and [LCo(PPh₃)(H₂O)]PF₆ ( 2 ), have been prepared from a polydentate ligand, N,N′‐bis(3‐methoxysalicylidehydene)cyclohexane‐1,2‐diamine ( H ₂ L ). Standard analytical techniques such as elemental analysis and UV–visible and Fourier transform infrared spectroscopies were used to characterize both complexes. The solid‐state molecular structures of both complexes were confirmed from single‐crystal X‐ray diffraction analysis. Structural analyses show that the Co(III) ion occupies the centre of a distorted octahedron in a complex cation: [LCo(tmtp)(H₂O)]⁺ and [LCo(PPh₃)(H₂O)]⁺ for 1 and 2 , respectively. Phenoxazinone synthase activities of both complexes were screened. Kinetic studies and other experimental observations reveal that the reaction follows rate saturation kinetics and proceeds through the formation of a catalyst (complex)–substrate adduct. The turnover number (K_cat) of complex 2 is 54.07 h^?1, exhibiting better catalytic activity compared to 1 (K_cat = 45.11 h^?1).     

Diffusion Coefficients for Binary, Ternary, and Polydisperse Solutions from Peak-Width Analysis of Taylor Dispersion Profiles

Binary mutual diffusion coefficients D can be estimated from the width at half height W _1/2 of Taylor dispersion profiles using D=(ln 2)r ² t _R/(3W ² h) and values of the retention time t _R and dispersion tube radius r. The generalized expression D _h=−(ln h)r ² t _R/(3W ² _h ) is derived to evaluate diffusion coefficients from peak widths W _h measured at other fractional heights (e.g., (h = 0.1, 0.2,…,0.9). Tests show that averaging the D _h values from binary profiles gives mutual diffusion coefficients that are as accurate and precise as those obtained by more elaborate nonlinear least-squares analysis. Dispersion profiles for ternary solutions usually consist of two superimposed pseudo-binary profiles. Consequently, D _h values for ternary profiles generally vary with the fractional peak height h. Ternary profiles with constant D _h values can however be constructed by taking appropriate linear combinations of profiles generated using different initial concentration differences. The invariant D _h values and corresponding initial concentration differences give the eigenvalues and eigenvectors for the evaluation of the ternary diffusion coefficient matrix. Dispersion profiles for polymer samples of N i-mers consist of N superimposed pseudo-binary profiles. The edges of these profiles are enriched in the heavier polymers owing to the decrease in polymer diffusion coefficients with increasing polymer molecular weight. The resulting drop in D _h with decreasing fractional peak height provides a signature of the polymer molecular weight distribution. These features are illustrated by measuring the dispersion of mixed polyethylene glycols.     

Graphical Models of Characters of Groups

A graphical method of generating one- and (some) two-dimensional characters () has been developed on the basis of a reduced homomer set, which has been derived from a new concept of negative graphs. Thus, a homomer set H[G(/G _i )]={h₁,...,h _d–1,h _d } (d=|G|/|G _i |) has been generated from a regular body of G so that it has been governed by the coset representation G(/G _i ). The homomer set has been reduced into a reduced homomer set []={h₁,...,h _d–1}, where we have placed h _d –(h₁++h _d–1) in terms of negative graphs. The action of the symmetry operations of G on the reduced homomer set [] has graphically generated a one- or (some) two-dimensional character (). The versatility of the graphical method has been tested by using C _3v , D _2h , C _2h , C _2v , D _3h , and C _3h as examples. The graphical method has been compared with an alternative algebraic generation using marks (or markaracters), i.e., =G(/G _i )–G(/G).     

Vibrational spectra of icosahedral (Ih and I) fullerenes

On the bases of the topological structures of the three big classes of icosahedral fullerenes: (1) C_n(I_h, n=60h²; h=1, 2,…), (2) C_n(I_h, n=20h²; h=1, 2,…), and (3) C_n(I, n=20(h²+hk+k²), h>k; h, k=1, 2,…), we derived formulas for the decomposition of their nuclear motions into irreducible representations. Hence, we obtained the infrared and Raman active modes for all of the icosahedral (I_h and I) fullerenes theoretically. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 113–117, 1998     

Cyclical Edge-Connectivity of Fullerene Graphs and (k, 6)-Cages

It is shown that every fullerene graph G is cyclically 5-edge-connected, i.e., that G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges. The result is then generalized to the case of (k,6)-cages, i.e., polyhedral cubic graphs whose faces are only k-gons and hexagons. Certain linear and exponential lower bounds on the number of perfect matchings in such graphs are also established.     

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.