The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

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     

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     

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.     

New vertex invariants and topological indices of chemical graphs based on information on distances

By applying information theory to the set of topological distances from one vertex to all other graph vertices, one obtains four new types of vertex invariants (u _i,v _i,x _i,Y _i) which are real numbers (as opposed to integers). They may be combined in many ways to afford new topological indices. One such type leads to indicesU, V, X andY which show no degeneracy for alkanes with up to 15 vertices.     

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.     

A möbius inversion of the ulam subgraphs conjecture

The generalized Eichinger matrices are defined asE = _j ⁿ ₁( _j S ^T S)^–1, where _j M denotes the matrixM withj th row and column deleted.S is the incidence matrix andM ^T is the transposed matrix. The conjectureS ^T SE = S _K ^TS _K , where SK is the incidence matrix of the complete graph, is proven for trees, simple cycles and complete graphs. The consequence of the conjecture isS _G ^T S _G (E _G -I) = S _G ^TS _G , whereG is the complementary graph ofG. It leads to graphs with imaginary arcs as the complements of graphs with multiple arcs.     

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.     

Theoretical characterization and design of small molecule donor material containing naphthodithiophene central unit for efficient organic solar cells

To seek for high‐performance small molecule donor materials used in heterojunction solar cell, six acceptor–donor–acceptor small molecules based on naphtho[2,3‐b:6,7‐b′]dithiophene ( NDT ) units with different acceptor units were designed and characterized using density functional theory and time‐dependent density functional theory. Their geometries, electronic structures, photophysical, and charge transport properties have been scrutinized comparing with the reported donor material NDT(TDPP)₂ ( TDPP = thiophene‐capped diketopyrrolopyrrole). The open circuit voltage (V_oc), energetic driving force(ΔE_L‐L), and exciton binding energy (E_b) were also provided to give an elementary understanding on their cell performance. The results reveal that the frontier molecular orbitals of 3–7 match well with the acceptor material PC₆₁BM , and compounds 3–5 were found to exhibit the comparable performances to 1 and show promising potential in organic solar cells. In particular, comparing with 1 , system 7 with naphthobisthiadiazole acceptor unit displays broader absorption spectrum, higher V_oc, lower E_b, and similar carrier mobility. An in‐depth insight into the nature of the involved excited states based on transition density matrix and charge density difference indicates that all S₁ states are mainly intramolecular charge transfer states with the charge transfer from central NDT unit to bilateral acceptor units, and also imply that the exciton of 7 can be dissociated easily due to its large extent of the charge transfer. In a word, 7 maybe superior to 1 and may act as a promising donor candidate for organic solar cell. © 2013 Wiley Periodicals, Inc.     

Experimental and theoretical study of the intramolecular C–H···N and C–H···S hydrogen bonding effects in the 1H and 13C NMR spectra of the 2‐(alkylsulfanyl)‐5‐amino‐1‐vinylpyrroles: a particular state of amine nitrogen

In the ¹H NMR spectra of the 1‐vinylpyrroles with amino‐ and alkylsulfanyl groups in 5 and 2 positions, an extraordinarily large difference between resonance positions of the H_A and H_B terminal methylene protons of the vinyl group is discovered. Also, the one‐bond ¹J(C_β,H_B) coupling constant is surprisingly greater than the ¹J(C_β,H_A) coupling constant in pyrroles under investigation, while in all known cases, there was a reverse relationship between these coupling constants. These spectral anomalies are substantiated by quantum chemical calculations. The calculations show that the amine nitrogen lone pair is removed from the conjugation with the π‐system of the pyrrole ring so that it is directed toward the H_B hydrogen. These factors are favorable to the emergence of the intramolecular C–H_B???N hydrogen bonding in the s‐cis(N) conformation. On the other hand, the spatial proximity of the sulfur to the H_B hydrogen provides an opportunity of the intramolecular C–H_B???S hydrogen bonding in the s‐cis(S) conformation. Presence of the hydrogen bond critical points as well as ring critical point for corresponding chelate ring revealed by a quantum theory of atoms in molecules (QTAIM) approach confirms the existence of the weak intramolecular C–H???N and C–H???S hydrogen bonding. Therefore, an unusual high‐frequency shift of the H_B signal and the increase in the ¹J(C_β,H_B) coupling constant can be explained by the effects of hydrogen bonding. Copyright © 2013 John Wiley & Sons, Ltd.     

Another look at the degree‐Kirchhoff index

Let G be an arbitrary graph with vertex set {1,2, …,N} and degrees d_i ≤ D, for fixed D and all i, then for the index R′(G) = ∑_{i < j}d_id_jR_ij we show that We also show that the minimum of R′(G) over all N‐vertex graphs is attained for the star graph and its value is 2N² ? 5N + 3. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Chemical graphs with degenerate topological indices based on information on distances

Recently, four new types of vertex invariants, namelyu, v, x, andy, were defined on the basis of information on graph distances. They were combined to give four highly selective topological indices:U, V, X, andY. The degeneracy, i.e. equal values for nonisomorphic graphs, of the four topological indices is investigated. A structural condition and a graphical method which gives pairs of molecular graphs with identicalU, V, X, andY topological indices are introduced. The smallest pair of 4-trees representing alkanes having degeneratedU, V, X, andY values consists of trees with eighteen vertices.     

Unique description of chemical structures based on hierarchically ordered extended connectivities (HOC procedures). V. New topological indices,ordering of graphs,and recognition of graph similarity

The vertex numbering obtained by application of the HOC algorithm can be converted into two sequences of numbers: If each vertex starting with vertex 1 is only counted once, the sums of numberings of adjacent vertices form sequence S_i (i = 1?N), while the sums of S_i values form sequence M_i (i = 1?N). These two sequences can be used for (i) two new topological indices, ?? and ??, the latter being of extremely low degeneracy, and the former correlating with boiling points of alkanes; (ii) a criterion based on sequence S_i for ordering graphs which possess the same number N of vertices; and (iii) a quantitative measure, also based on sequence S_i, for appreciating the similarity or dissimilarity of pairs of graphs. Comparisons with other topological indices, ordering criteria, and similarity measures for graphs show that the newly devised procedures compare favorably with those known previously.     

Multipolar electrostatics for proteins: Atom–atom electrostatic energies in crambin

Accurate electrostatics necessitates the use of multipole moments centered on nuclei or extra point charges centered away from the nuclei. Here, we follow the former alternative and investigate the convergence behavior of atom‐atom electrostatic interactions in the pilot protein crambin. Amino acids are cut out from a Protein Data Bank structure of crambin, as single amino acids, di, or tripeptides, and are then capped with a peptide bond at each side. The atoms in the amino acids are defined through Quantum Chemical Topology (QCT) as finite volume electron density fragments. Atom‐atom electrostatic energies are computed by means of a multipole expansion with regular spherical harmonics, up to a total interaction rank of L = ?_A+ ?_B + 1 = 10. The minimum internuclear distance in the convergent region of all the 15 possible types of atom‐atom interactions in crambin that were calculated based on single amino acids are close to the values calculated from di and tripeptides. Values obtained at B3LYP/aug‐cc‐pVTZ and MP2/aug‐cc‐pVTZ levels are only slightly larger than those calculated at HF/6‐31G(d,p) level. This convergence behavior is transferable to the well‐known amyloid beta polypeptide Aβ_1–42. Moreover, for a selected central atom, the influence of its neighbors on its multipole moments is investigated, and how far away this influence can be ignored is also determined. Finally, the convergence behavior of AMBER becomes closer to that of QCT with increasing internuclear distance. © 2013 Wiley Periodicals, Inc.     

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     

Topographical distance matrices for porous arrays

The topographical Wiener index is calculated for two-dimensional graphs describing porous arrays, including bee honeycomb. For tiling in the plane, we model hexagonal, triangular, and square arrays and compare with topological formulas for the Wiener index derived from the distance matrix. The normalized Wiener indices of C₄, T₁₃, and O(4), for hexagonal, triangular, and square arrays are 0.993, 0.995, and 0.985, respectively, indicating that the arrays have smaller bond lengths near the center of the array, since these contribute more to the Wiener index. The normalized Perron root (the first eigenvalue, λ ₁), calculated from distance/distance matrices describes an order parameter, f = l₁/n{\phi=\lambda_1/n} , where f = 1{\phi= 1} for a linear graph and n is the order of the matrix. This parameter correlates with the convexity of the tessellations. The distributions of the normalized distances for nearest neighbor coordinates are determined from the porous arrays. The distributions range from normal to skewed to multimodal depending on the array. These results introduce some new calculations for 2D graphs of porous arrays.