On reciprocal molecular topological index

We report some properties of the reciprocal molecular topological index RMTI of a connected graph, and, in particular, its relationship with the first Zagreb index M₁. We also derive the upper bounds for RMTI in terms of the number of vertices and the number of edges for various classes of graphs, including K _r+1 -free graphs with r ≥ 2, quadrangle-free graphs, and cacti. Additionally, we consider a Nordhaus-Gaddum-type result for RMTI.     

Common vertex matrix: A novel characterization of molecular graphs by counting

We present a novel matrix representation of graphs based on the count of equal‐distance common vertices to each pair of vertices in a graph. The element (i, j) of this matrix is defined as the number of vertices at the same distance from vertices (i, j). As illustrated on smaller alkanes, these novel matrices are very sensitive to molecular branching and the distribution of vertices in a graph. In particular, we show that ordered row sums of these novel matrices can facilitate solving graph isomorphism for acyclic graphs. This has been illustrated on all undecane isomers C₁₁H₂₄ having the same path counts (total of 25 molecules), on pair of graphs on 18 vertices having the same distance degree sequences (Slater's graphs), as well as two graphs on 21 vertices having identical several topological indices derived from information on distances between vertices. © 2013 Wiley Periodicals, Inc.     

QSPR Modeling of Complex Stability by Optimization of Correlation Weights of the Hydrogen Bond Index and the Local Graph Invariants

The hydrogen bond index (HBI) is the global invariant of a molecular graph and equals the number of vertices representing hydrogen and nitrogen atoms. This index was considered a measure of the capability of a complex to form hydrogen bonds. Optimization of the correlation weights of the HBI and local graph invariants was used for the QSPR modeling of the stability of 110 biometal M²⁺ complexes with -amino acids and phosphate derivatives of adenosine. The statistical parameters of the best model are n = 55, r = 0.9921, s = 0.279, and F = 3328 (learning sample) and n = 55, r = 99.35, s = 0.248, and F = 4027 (control sample).     

QSPR Modeling of Complex Stability by Correlation Weighing of the Topological and Chemical Invariants of Molecular Graphs

The concept of modeling of the complex stability based on optimization of the correlation weights of the nearest neighborhood codes (NNC), the hydrogen bond index (HBI), and the cyclicity code (CC) is described. The NNC is a local topochemical invariant of a vertex of the molecular graph whose numerical value is a function of the total number and the composition of vertices adjacent to the given vertex. The HBI is a global chemical invariant of the molecular graph calculated from the number of oxygen and nitrogen atoms. The CC is a global topological invariant of the graph equal to the number of rings present in the ligand structure. The statistical characteristics of the best model of the stability constants of the complexes are as follows: n = 75, r = 0.9738, s = 0.457, F = 1337 (training sample); n = 75, r = 0.9795, s = 0.461, F = 1724 (test sample).     

Van der Waals interaction and hydrogen bond effects on molecular vibrational frequencies

The frequency shift of an AZ two-atomic fragment upon molecule transfer from gas to solution or matrix is investigated. The nearest neighbour ligands are considered to form a cluster. The coupling of the ν_s(AZ) mode with the cluster low-frequency oscillators (ν_Q) is taken into account. One such oscillator can be the ν_σ(AZ…B) intermolecular stretching frequency oscillator in the H-bonded or charge transfer complex. A general expression for the ν_s(AZ) frequency shift (Δω) is obtained, which allows for both the usual van der Waals interaction and the (ν_s, ν_σ) and (ν_s, ν_Q) mode couplings. When the former prevails, the relative frequency shift Δω/ω_f is invariant to the Z → Z* (e.g., H → D) isotopic substitution. When the mode coupling prevails, the quantity Δω/ω²_f is invariant what is typical for H bonds. If the H-bonded or charge transfer complexes are absent, the frequency shift Δω is proportional to α(ϱ_AZ + ϱ_L)⁻⁶ where α is the polarizability of the ligand, ϱ_AZ and ϱ_L are the van der Waals radii of AZ and ligand, respectively. The additional ν_s(AH) frequency shift upon transfer of the AH…B complex from the gas to solution seems to be caused by (ν_s, ν_Q) mode coupling.     

Characterization of molecular complexity

We put forward a novel index of molecular complexity, ξ, taking into account the symmetry of a molecular graph and the specificity of structural components considered. The ξ index is defined as the sum of augmented valences of all mutually nonequivalent vertices in a molecular graph. The augmented valence of a vertex in a graph is the sum of its valence and valences of all neighboring vertices with the weight 1/2^d depending on their distance, d, from the vertex. The ξ index is examined on the set of octane isomers and some special classes of graphs. It is also compared with a certain number of alternative complexity measures considered in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Reorientational and vibrational relaxation in liquid trichloracetonitrile

The first- and second-order reorientational correlation functions were determined from i.r. and Raman bandshapes of ν_s(CN)in liquid trichloracetonitrile and discussed in terms of the J-diffusion model. A consistent fit for G_1r(t) and G_2r(t) was possible for short times only (0–1.5 ps). The vibrational correlation functions, determined from Raman studies of the ν_s(CN) and ν_s(CC) bandshapes, are discussed in terms of recent theories of vibrational dephasing.     

Unicyclic Graphs Possessing Kekulé Structures with Minimal Energy

Unicyclic graphs possessing Kekulé structures with minimal energy are considered. Let n and l be the numbers of vertices of graph and cycle C _l contained in the graph, respectively; r and j positive integers. It is mathematically verified that for and l = 2r + 1 or has the minimal energy in the graphs exclusive of , where is a graph obtained by attaching one pendant edge to each of any two adjacent vertices of C ₄ and then by attaching n/2 − 3 paths of length 2 to one of the two vertices; is a graph obtained by attaching one pendant edge and n/2 − 2 paths of length 2 to one vertex of C ₃. In addition, we claim that for has the minimal energy among all the graphs considered while for has the minimal energy.      

Carbon-13 NMR spectra of a series of para-substituted phenyl isothiocyanates. Linear free energy relationships for carbon-13 substituent chemical shifts

Carbon-13 chemical shifts are reported for 16 para-substituted phenyl isothiocyanates measured at 1 and 10 mol % in chloroform-d solution. Data for the ? N?C?S group were not obtained at 1 mol %, but concentration effects for the other resonances were negligible. Hammett, dual substituent parameter (DSP) and DSP-nonlinear resonance (DSP-NLR) analyses were used to evaluate substituent effects on the substituent chemical shifts (SCS) for the ipso-carbon (C-1), C-2, and the ? N?C?S carbon atoms. A good Hammett correlation was observed for C-1 (ν_p⁺ = 8.1 ppm, r = 0.98 at 1 mol %) but was improved for the higher order correlations with the following results, DSP:ρ _I = 5.4, ρ_R° = 22.2, r = 0.998; DSP-NLR: ρ_I = 5.6, ρ_R° = 20.5, ? = ?0.22, r = 0.999. The 10 mol % data were very similar except the value of ? was ?0.26 and confirms the phenyl-bonded ? N?C?S moiety as a mild electron acceptor substituent. Hammett correlations were unsuccessful for the C-2 data, but fairly good results were obtained from the higher order analyses. For the 1 mol % data, DSP: ν_I = 1.6, ν_R° = ?2.0, r = 0.976; DSP-NLR: ν_I = 1.8, ν_R° = ?2.6, ? = 1.1, r = 0.982. Excellent correlations were obtained for the 10 mol % ? N?C?S carbon data. Hammett: ν_p° = 6.2, r = 0.997; DSP: ν_I = 5.9, ν_R° = 7.0, r = 0.997; DSP-NLR: ν_I = 5.8, ν_R° = 7.6, ? = 0.25, r = 0.997. The positive ν values in these three correlations contrast the negative values usually observed for carbonyl and thiocarbonyl carbons, and more closely parallel results previously reported for the β-carbon of styrenes and benzylidene anilines with para-substituents in the aniline ring.     

Details of the reaction graphs for intramolecular isomerizations of the carboranes

From proposed mechanisms for framework reorganizations of the carboranes C₂B _n-2H _n ,n = 5–12, we present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form. Because they contain both loops and multiple edges, these graphs are actually pseudographs. Loops and multiple edges have chemical significance in several cases. Enantiomeric pairs occur among carborane isomers and among the transition state structures on pathways linking the isomers. For a carborane polyhedral structure withn vertices, each graph hasn(n -1)/2 graph edges. The degree of each graph vertex and the sum of degrees of all graph vertices are independent of the details of the isomerization mechanism. The degree of each vertex is equal to twice the number of rotationally equivalent forms of the corresponding isomer. The total of all vertex degrees is just twice the number of edges orn(n - 1). The degree of each graph vertex is related to the symmetry point group of the structure of the corresponding isomer. Enantiomeric isomer pairs are usually connected in the graph by a single edge and never by more than two edges.     

Infrared spectroscopic studies of hydrogen-bonded complexes in cryogenic sulutions

We have measured the ν_s(OH) band parameters of the IR absorption spectra for a wide variety of hydrogen-bonded (HB) complexes of CH₃OH(D), CF₃CH₂OH, and (CF₃)₃COH(D) with some simplest representatives of various classes of bases in Xe and Kr in the temperature range 120–270 K. The ν_s(OH) absorption bands of the HB complexes in solution in atomic solvents have been demonstrated to be narrower by a factor of 2 to 4 than in molecular solvents at the same temperature. The fact that the ν_s(OH) bandwidths in Xe and in the gas phase at similar temperatures are practically the same indicates that these bandwidths are in both cases governed mainly by the contribution of “hot transitions” from a sequence of excited levels of the ν_β low-frequency bending mode of the hydrogen bond. The other characteristic features revealed for the complexes under study in liquid Xe and Kr at ν_s(OH) frequency shifts up to 500 cm⁻¹ include: (1) slight temperature dependence of the ν_s(OH) bandwidth (0.1–0.3 cm⁻¹/K), (2) almost “normal” isotope frequency ratio ν_s(OH)/ν_s(OD) (1.34–1.36) and (3) low ν_s(OH) temperature shift values (0.1–0.4 cm⁻¹/K).     

Kirchhoff index of linear pentagonal chains

The resistance distance r_ij between two vertices v_i and v_j of a connected graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this article, following the method of Yang and Zhang in the proof of the Kirchhoff index of liner hexagonal chain, we obtain the closed‐form formulae of the Kirchhoff index of liner pentagonal chain P_n in terms of its Laplacian spectrum. Finally, we show that the Kirchhoff index of P_n is approximately one half of its Wiener index. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Graphs corresponding to reference polynomial or to circuit characteristic polynomial

A vertex-weighted graphG ^* is studied which is obtained by deleting edgee _rs in a circuit of a graphG and giving two vertices _r and _s weightsh _r = 1 andh _s = -1, respectively. It is shown that if subgraphG - _r is identical with subgraphG - _s, then the reference polynomial ofG ^* is identical with that ofG and the characteristic polynomial ofG ^* contains the contributions due to only a certain part of the circuits found in the original graphG. This result gives a simple way to find a graph whose characteristic polynomial is equal to the reference polynomial in the topological resonance energy theory or to the circuit characteristic polynomial in the circuit resonance energy theory. This approach can be applied not only to Hilckel graphs but also to Möbius graphs, provided that they satisfy a certain condition. The significances of this new type of reference graph thus obtained are pointed out.     

H-bonds in acetic acid crystals: Influence of temperature and of isotopic dilution on OH(O-D) stretching BANDS

We describe the evolution of the ν_s bands of CH₃COOH, CD₃COOH, CH₃COOD and CD₃COOD crystals when varying temperature from 10 K to 270 K, and also under isotopic dilutions. We discuss the results obtained and we show how we can estimate from these experiments the values of the anharmonic constant coupling ν_s with ν_σ (parameter b) and also how we can evaluate the energy of resonance interactions of ν_s with other modes. The origin of the shift of the centre of gravity of the ν_s, bands with temperature is discussed and we give arguments against an anharmonic one-dimesional potential for the ν_σ mode being at the origin of such a shift in acetic acid crystals. We suggest that this shift might rather be due to a modulation of ν_s or ν_σ by lower frequency modes of the H-band. In the conclusion we summarize our results on acetic acid crystals and stress the interest of performing such quantitative studies.     

Kirchhoff index of linear hexagonal chains

The resistance distance r_ij between vertices i and j of a connected (molecular) graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this work, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of linear hexagonal chain L_n consists of the Laplacian spectrum of path P_2n+1 and eigenvalues of a symmetric tridiagonal matrix of order 2n + 1. By applying the relationship between roots and coefficients of the characteristic polynomial of the above matrix, explicit closed‐form formula for Kirchhoff index of L_n is derived in terms of Laplacian spectrum. To our surprise, the Krichhoff index of L_n is approximately to one half of its Wiener index. Finally, we show that holds for all graphs G in a class of graphs including L_n. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Resistance distance and Laplacian spectrum

The resistance distance r _ij between two vertices v _i and v _j of a (connected, molecular) graph G is equal to the resistance between the respective two points of an electrical network, constructed so as to correspond to G, such that the resistance of any two adjacent points is unity. We show how the matrix elements r _ij can be expressed in terms of the Laplacian eigenvalues and eigenvectors of G. In addition, we determine certain properties of the resistance matrix R=||r _ij ||. AcknowledgementsThis research was supported by the Natural Science Foundation of China and Fujian Province, and by the Ministry of Sciences, Technologies and Development of Serbia, within Project no. 1389. The authors thank Douglas J. Klein (Galveston) for useful comments.