New Nordhaus-Gaddum-type results for the Kirchhoff index

Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index is the sum of resistance distances between all pairs of vertices in G. Zhou and Trinajstić (Chem Phys Lett 455(1–3):120–123, 2008) obtained a Nordhaus-Gaddum-type result for the Kirchhoff index by obtaining lower and upper bounds for the sum of the Kirchhoff index of a graph and its complement. In this paper, by making use of the Cauchy-Schwarz inequality, spectral graph theory and Foster’s formula, we give better lower and upper bounds. In particular, the lower bound turns out to be tight. Furthermore, we establish lower and upper bounds on the product of the Kirchhoff index of a graph and its complement.     

A new approach for devising local graph invariants: Derived topological indices with low degeneracy and good correlation ability

A new approach is presented for obtaining graph invariants which have very high discriminating ability for different vertices within a graph. These invariants are obtained as the solution set (local invariant set, LOIS) of a system of linear equationsQ · X = R, whereQ is a topological matrix derived from the adjacency matrix of the graph, andR is a column vector which encodes either a topological property (vertex degree, number of vertices in the graph, distance sum) or a chemical property (atomic number). Twenty examples of LOOIs are given and their degeneracy and ordering ability of vertices is discussed. Interestingly, in some cases the ordering of vertices obtained by means of these invariants parallels closely the ordering from an entirely different procedure based on Hierarchically Ordered Extended Connectivities which was recently reported. New topological indices are easily constructed from LOISs. Excellent correlations are obtained for the boiling points and vaporization enthalpies of alkanesversus the topological index representing the sum of local vertex invariants. Les spectacular correlations with NMR chemical shifts, liquid phase density, partial molal volumes, motor octane numbers of alkanes or cavity surface areas of alcohols emphasize, however, the potential of this approach, which remains to be developed in the near future.     

Vertex topological indices and tree expressions,generalizations of continued fractions

We expand on the work of Hosoya to describe a generalization of continued fractions called “tree expressions.” Each rooted tree will be shown to correspond to a unique tree expression which can be evaluated as a rational number (not necessarily in lowest terms) whose numerator is equal to the Hosoya index of the entire tree and whose denominator is equal to the tree with the root deleted. In the development, we use Z(G) to define a natural candidate ζ(G, v) for a “vertex topological index” which is a value applied to each vertex of a graph, rather than a value assigned to the graph overall. Finally, we generalize the notion of tree expression to “labeled tree expressions” that correspond to labeled trees and show that such expressions can be evaluated as quotients of determinants of matrices that resemble adjacency matrices.     

Maximum energy trees with two maximum degree vertices

The energy E of a graph G is equal to the sum of the absolute values of the eigenvalues of G. In 2005 Lin et al. determined the trees with a given maximum vertex degree Δ and maximum E, that happen to be trees with a single vertex of degree Δ. We now offer a simple proof of this result and, in addition, characterize the maximum energy trees having two vertices of maximum degree Δ.     

Maximum Randić index on Trees with <Emphasis Type="Italic">k</Emphasis>-pendant Vertices

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. Let T be a tree with n vertices and k pendant vertices. In this paper, we give a sharp upper bound on Randić index of T.     

On the Randić Index

The Randić index of an organic molecule whose molecular graph G is defined as the sum of (d(u)d(v))^−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Delorme et al., Discrete Math. 257 (2002) 29, Delorme et al gave a best-possible lower bound on the Randić index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of Delorme et al., Discrete Math. 257 (2002) 29), and then we will show that the result holds when δ(G)≥ 2.     

Transition states in modern valence-bond theory: application to the Cope rearrangement

Modern valence-bond theory, in its spin-coupled form, is used to study the electronic structure of the transition structures in the Cope rearrangement. It is found that the transition structure described by a “chair” geometry with a “6-in-6” CASSCF/6-31G^* wave function is clearly aromatic while the CASSCF/6-31G^*“boat” transition structure corresponds more closely to two weakly interacting allyl radicals. Moreover, there is a striking resemblance between the CASSCF chair transition structure and the benzene molecule, arising from the modern valence-bond analysis in terms of Rumer spin functions. In agreement with previous works, dynamical correlated wave functions show shorter interallylic distances in the optimized transitions structures. The use of spin-coupled wave functions on the latter geometries results in diradical and aromatic character for the chair and boat transition structures, respectively. Received: 13 October 1998 / Accepted: 30 December 1998 / Published online: 7 June 1999     

The (n,n)-graphs with the first three extremal Wiener indices

Let G = (V, E) be a simple connected graph with vertex set V and edge set E. The Wiener index W(G) of G is the sum of distances between all pairs of vertices in G, i.e., , where d _G (u, v) is the distance between vertices u and v in G. In this paper, we first give a new formula for calculating the Wiener index of an (n,n)-graph according its structure, and then characterize the (n,n)-graphs with the first three smallest and largest Wiener indices by this formula.     

Wiener polynomials of some chemically interesting graphs

The chemist Harold Wiener found ??(G), the sum of distances between all pairs of vertices in a connected graph G, to be useful as a predictor of certain physical and chemical properties. The q‐analogue of ??, called the Wiener polynomial ??(G; q), is also useful, but it has few existing useful formulas. We will evaluate ??(G; q) for certain graphs G of chemical interest. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

The study of alanine oscillating reaction catalyzed by tetraazamacrocyclic nickel(II) complex

A closed oscillation system comprised of alanine, KBrO₃, H₂SO₄ and acetone catalyzed by tetraazamacrocyclic nickel(II) complex is introduced, and quantitatively characterized with kinetic parameters, namely the rate constant (k _in, k _p), the apparent activation energy (E _in, E _p) and pre-exponential constant (A _in, A _p) and thermodynamic functions (ΔH _in, ΔG _in, ΔS _in and ΔH _p, ΔG _p, ΔS _p), where indexes “in” and “p” mean “induction period” and “oscillation period,” respectively. The results indicate that tetraazamacrocyclic nickel(II) complex can catalyze alanine oscillating reaction and the reaction corresponds exactly to the feature of irreversible thermodynamics as the entropy of system is negative.     

The heptakisoctahedral group and its relevance to carbon allotropes with negative curvature

We introduce the pollakispolyhedral groups and describe in detail the representational structure of PSL(2,7) or ⁷ O, the automorphism group of the Klein graph composed of 56 trivalent vertices arranged in 24 heptagonal faces. Leapfrog and quadruple transformations of the graph are described and their eigenvalue spectra derived. Considered as carbon frameworks on the “plumber's nightmare” surface these chiral structures exhibit significant steric strain which prevents the molecular realisation of the Klein symmetry. This revised version was published online in July 2006 with corrections to the Cover Date.     

New upper bounds on Zagreb indices

The first Zagreb index M ₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M ₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper we obtain an upper bound on the first Zagreb index M ₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ₁), second maximum vertex degree (Δ₂) and minimum vertex degree (δ). Using this result we find an upper bound on M ₂(G). Moreover, we present upper bounds on and in terms of n,  m, Δ₁, Δ₂, δ, where denotes the complement of G.     

Second-order energy components in basis-set-superposition-error-free intermolecular perturbation theory

 Recently a basis-set-superposition-error-free second-order perturbation theory was introduced based on the “chemical Hamiltonian approach” providing the full antisymmetry of all wave functions by using second quantization. Subsequently, the “Heitler–London” interaction energy corresponding to the sum of the zero- and first-order perturbational energy terms was decomposed into different physically meaningful components, like electrostatics, exchange and overlap effects. The first-order wave function obtained in the framework of this perturbation theory also consists of terms having clear physical significance: intramolecular correlation, polarization, charge transfer, dispersion and combined polarization–charge transfer excitations. The second-order energy, however, does not represent a simple sum of the respective contributions, owing to the intermolecular overlap. Here we propose an approximate energy decomposition scheme by defining some “partial Hylleraas functionals” corresponding to the different physically meaningful terms of the first-order wave functions. The sample calculations show that at large and intermediate intermolecular distances the total second-order intermolecular interaction energy contribution is practically equal to the sum of these “physical” terms, while at shorter distances the overlap-caused interferences become of increasing importance. Received: 18 June 2001 / Accepted: 28 August 2001 / Published online: 16 November 2001     

On the Randić Index of Acyclic Conjugated Molecules

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. We give a sharp lower bound on the Randić index of conjugated trees (trees with a perfect matching) in terms of the number of vertices. A sharp lower bound on the Randić index of trees with a given size of matching is also given Mei Lu: Partially supported by NNSFC (No. 60172005) Lian-zhu Zhang: Partially supported by NNSFC (No. 10271105) Feng Tian: Partially supported by NNSFC (No. 10431020)     

Extremal (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">n</Emphasis> + 1)-graphs with respected to zeroth- order general Randić index

A (n, n + 1)-graph G is a connected simple graph with n vertices and n + 1 edges. If d _v denotes the degree of the vertex v, then the zeroth-order general Randić index of the graph G is defined as , where α is a real number. We characterize, for any α, the (n,n + 1)-graphs with the smallest and greatest zeroth-order general Randić index.     

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     

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.     

Some upper bounds for the energy of graphs

Let G = (V,E) be a graph with n vertices and e edges. Denote V(G) = {v ₁,v ₂,...,v _n }. The 2-degree of v _i , denoted by t _i , is the sum of degrees of the vertices adjacent to . Let σ _i be the sum of the 2-degree of vertices adjacent to v _i . In this paper, we present two sharp upper bounds for the energy of G in terms of n, e, t _i , and σ _i , from which we can get some known results. Also we give a sharp bound for the energy of a forest, from which we can improve some known results for trees.