Eigenvector and eigenvalues of some special graphs. IV. Multilevel circulants

A multilevel circulant is defined as a graph whose adjacency matrix has a certain block decomposition into circulant matrices. A general algebraic method for finding the eigenvectors and the eigenvalues of multilevel circulants is given. Several classes of graphs, including regular polyhedra, suns, and cylinders can be analyzed using this scheme.     

Analysis of IR spectra of carbon clusters trapped in noble gas matrices using algorithms based on digital filtering techniques

IR spectra of carbon clusters C_n (n3) trapped in noble gas matrices (Ar and Kr) at temperatures of 13, 30 and 35 K are analyzed using algorithms based on digital filtering techniques and non-linear least-squares fitting. The spectral features at different temperatures and in different matrices are discussed. The majority of the resolved lines are assigned to C_n cluster species based on data obtained via quantum chemical computations and from tunable laser IR spectrometry of C_n species in the gas phase. A complete analysis of the IR spectra is available upon request.     

Symmetry factoring of the characteristic equations of graphs corresponding to polyhedra

A systematic procedure is described which uses two-and three-fold symmetry elements in graphs to reduce their adjacency matrices to lead to corresponding factorings of their characteristic polynomials. A graph splitting algorithm based on this matrix reduction procedure is described. Applications of these methods to the factoring of the characteristic polynomials of 28 polyhedra with nine or less vertices are given. General expressions for the eigenvalues of prisms, pyramids, and bipyramids in terms of the eigenvalues of their basal or equatorial regular polygons are calculated by closely related matrix methods.     

Low-energy vibrations of the group 10 metal monocarbonyl MCO (M = Ni, Pd, and Pt): rotational spectroscopy and force field analysis

The rotational spectra of NiCO and PdCO in the ground and ν(2) excited vibrational states were observed by employing a source-modulated microwave spectrometer. The NiCO and PdCO molecules were generated in a free space cell by the sputtering reaction of nickel and palladium sheets, respectively, lining the inner surface of a stainless steel cathode with a dc glow plasma of CO and Ar. The molecular constants of NiCO and PdCO were determined by least-squares analysis. By force field analysis for the molecular constants of not only NiCO and PdCO but also of PtCO as previously reported, the harmonic force constants were determined for these three group 10 metal monocarbonyls. The vibrational wavenumbers derived for the lower M-C stretching vibrations were in good agreement with those obtained from the IR spectra in noble gas matrices and those predicted by several quantum chemical calculations published in the past. The bending vibrational wavenumbers derived by the force field analysis were also consistent with most quantum chemical calculations previously reported, but showed systematic discrepancies from the matrix IR values by about 40 cm(-1), even after reassignment (ν(2) band → 2ν(2) band) of the matrix IR spectra of PdCO and PtCO.     

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.     

IR and UV spectra of the matrix-isolated peroxy radicals CF3OO, trans-FC(O)OO, and cis-FC(O)OO

The peroxy radicals CF3OO and FC(O)OO are prepared in high yields by vacuum flash pyrolysis of ROONO2 or ROOOR (R=CF3, FC(O)), highly diluted in inert gases, and subsequent isolation in inert-gas matrices by quenching the product mixtures at low temperatures. The IR spectrum of FC(O)OO was observed for the first time and eight fundamentals as well as several combinations were measured and assigned for both cis and trans rotamers of FC(O)OO. Discrepancies in an earlier assignment of the fundamentals of CF3OO have been eliminated and its IR spectrum is reported fully. The matrix UV spectra of both peroxy radicals (X2A"--> 2(2)A" transition) are in agreement with the gas-phase spectra; however, there are differences in the absorption cross-sections, for which possible reasons are discussed. The X2A"--> 1(2)A' transitions in the near IR region are too weak to be detected with our instrumentation.     

Topological analysis of eigenvectors of the adjacency matrices in graph theory: The concept of internal connectivity

The topological properties of eigenvectors of adjacency matrices of a graph have been analyzed. Model systems studied are n-vertex-m-edge (n-V-m-E) graphs where n = 2–4, m = 1–6. The topological information contained in these eigenvectors is described using vertex-signed and edge-signed graphs. Relative ordering of net signs of edge-signed graphs is similar to that of eigenvalues of the adjacency matrix. This simple analysis has also been applied to naphthalene, anthracene and pyrene. It provides a sound basis for the application of graph theory to molecular orbital theory.     

Raman studies of solid hydrogen cyanide (HCN,DCN) and of HCN argon matrices

Raman spectra of single crystal-like oriented solid HCN and DCN, as well as of HCN molecules and clusters in argon matrices, have been recorded and analysed with respect to both so far uninterpreted spectral features reported in the literature and to the applicability of Raman experiments for studying molecular clusters in inert gas matrices. The main results obtained are: (i) splitting of both the intramolecular bending modes and librations of solid HCN has to be interpreted in terms of TO/LO splitting rather than site group splitting; (ii) the nature of the phase transition at 170 K reported as tI6 to oI6 must be doubted; and (iii) the previous results of HCN(DCN)/Ar matrices obtained by IR studies are confirmed.     

A matrix isolation study on Ac-Gly-NHMe and Ac-l-Ala-NHMe, the simplest chiral and achiral building blocks of peptides and proteins

The infrared absorption (IR) spectra of acetyl-N-methyl-glycine and acetyl-N-methyl-alanine have been recorded in dichloromethane and dimethyl sulfoxide-d(6) solution, as well as in Ar and Kr matrices. The spectra were assigned with the help of quantum chemical calculations. Based on the assignments of the matrix-isolation IR spectra, in line with theoretical predictions, two different hydrogen bonded conformers were identified, furthermore a third conformer is likely to be present, which cannot be unambiguously identified. In dichloromethane two conformers could be observed, while in dimethyl sulfoxide a single conformer could be identified. Vibrational circular dichroism (VCD) spectra of acetyl-N-methyl-l-alanine have also been recorded in solutions and matrices. These matrix-isolation VCD spectra not only support the assignments of the matrix-isolation IR spectra, but also demonstrate that these spectra can be interpreted much easier with the help of quantum chemical calculations than the VCD spectra recorded in solutions. It is also shown that the rotatory strength of some vibrational transitions changes rapidly as a function of the backbone torsional coordinates; hence the appearance of some regions in the VCD spectra is extremely dependent on any perturbations, e.g. weak intermolecular interactions.     

Very efficient search for nucleotide alignments

We describe a very efficient search for nucleotide alignments, which is analogous to the novel very efficient search for protein alignment. Just as it has been the case with the alignment of proteins, based on 20 × 20 adjacency matrices for amino acids, obtained from a superposition of labeled amino acids adjacency matrices for the proteins considered, one can construct labeled matrices of size 4 × 4, listing adjacencies of nucleotides in DNA sequence. The matrix elements correspond to 16 pairs of adjacent nucleotides. To obtain DNA alignments, one combines information in the corresponding matrices for a pair of DNA nucleotides. Matrices are obtained by insertion of the sequential labels for pairs of nucleotides in the corresponding cells of the 4 × 4 tables. When two such matrices are superimposed, one can identify all segments in two DNA sequences, which are shifted relative to one another by the same amount in either direction, without using trial‐and‐error displacements of the two sequences one relative to the other to find local nucleotide alignments. © 2012 Wiley Periodicals, Inc.     

An alternative strategy for count and storage of Kekulé and longer range resonance valence bond structures

In this paper, we suggest a simple representation for notation of Kekulé valence bond (KVB) structures and longer range resonance valence bond (RVB) structures, which is called "the adjacency bonding array". In this representation, only an N component one-dimensional array is needed for inscribing each KVB or longer range RVB structure for an N-carbon system. Based on the adjacency bonding arrays, we develop very efficient algorithms for the systematic search of KVB and RVB structures as well as evaluation of the basis set overlap and Hamiltonian matrices.