Maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix

We report some properties of the maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix of a connected graph, in particular, various lower and upper bounds, and the Nordhaus–Gaddum‐type results for them. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Connections between Wiener index and matchings

Let T be an acyclic molecule with n vertices, and let S(T) be the acyclic molecule obtained from T by replacing each edge of T by a path of length two. In this work, we show that the Wiener index of T can be explained as the number of matchings with n−2 edges in S(T). Furthermore, some related results are also obtained MSC: 05C12 Weigen Yan: This work is supported by FMSTF (2004J024) and NSFF(E0540007) Yeong-Nan Yeh: Partially supported by NSC94-2115-M001-017     

A formula for the Kirchhoff index

We show here that the Kirchhoff index of a network is the average of the Wiener capacities of its vertices. Moreover, we obtain a closed‐form formula for the effective resistance between any pair of vertices when the considered network has some symmetries, which allows us to give the corresponding formulas for the Kirchhoff index. In addition, we find the expression for the Foster's n‐th formula.     

Terminal Wiener index

Motivated by some recent research on the terminal (reduced) distance matrix, we consider the terminal Wiener index (TW) of trees, equal to the sum of distances between all pairs of pendent vertices. A simple formula for computing TW is obtained and the trees with minimum and maximum TW are characterized.     

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.     

On resistance-distance and Kirchhoff index

We provide some properties of the resistance-distance and the Kirchhoff index of a connected (molecular) graph, especially those related to its normalized Laplacian eigenvalues.     

The extremal structures of k-uniform unicyclic hypergraphs on Wiener index

The Wiener index of a connected k-uniform hypergraph is defined as the summation of distances between all pairs of vertices. We determine the unique k-uniform unicyclic hypergraphs with maximum and second maximum, minimum and second minimum Wiener indices, respectively.     

On Wiener‐type polynomials of thorn graphs

We derive the expressions of the ordinary, the vertex‐weighted and the doubly vertex‐weighted Wiener polynomials of a type of thorn graph, for which the number of pendant edges attached to any vertex of the underlying parent graph is a linear function of its degree. We also define variable vertex‐weighted Wiener polynomials and calculate them for the same type of thorn graphs. Copyright © 2009 John Wiley & Sons, Ltd.     

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     

Ordering chemical unicyclic graphs by Wiener polarity index

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d _G (u, v) = 3}, where d _G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W _p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given.     

Graphs with maximum connectivity index

Let G be a graph and d_v the degree (=number of first neighbors) of its vertex v. The connectivity index of G is χ=∑ (d_ud_v)^−1/2, with the summation ranging over all pairs of adjacent vertices of G. In a previous paper (Comput. Chem. 23 (1999) 469), by applying a heuristic combinatorial optimization algorithm, the structure of chemical trees possessing extremal (maximum and minimum) values of χ was determined. It was demonstrated that the path P_n is the n-vertex tree with maximum χ-value. We now offer an alternative approach to finding (molecular) graphs with maximum χ, from which the extremality of P_n follows as a special case. By eliminating a flaw in the earlier proof, we demonstrate that there exist cases when χ does not provide a correct measure of branching: we find pairs of trees T, T′, such that T is more branched than T′, but χ(T)>χ(T′). The smallest such examples are trees with 36 vertices in which one vertex has degree 31.     

On the path-Zagreb matrix

The definition of the path-Zagreb matrix for (chemical) trees PZ and its generalization to any (molecular) graph is presented. Additionally, the upper bound of , where G _n is a graph with n vertices is given.     

Trees with the minimum Wiener number

The Wiener number (𝒲) of a connected graph is the sum of distances for all pairs of vertices. As a graphical invariant, it has been found extensive application in chemistry. Considering the family of trees with n vertices and a fixed maximum vertex degree, we derive some methods that can strictly reduce 𝒲 by shifting leaves. And then, by a process, we prove that the dendrimer on n vertices is the unique graph reaching the minimum Wiener number. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 331–340, 2000     

Modification of the Wiener index 4

A novel topological index W(F) is defined by the matrices X, W, and L as W(F) = XWL. Where L is a column vector expressing the characteristic of vertices in the molecule; X is a row vector expressing the bonding characteristics between adjacent atoms; W is a reciprocal distance matrix. The topological index W(F), based on the distance-related matrix of a molecular graph, is used to code the structural environment of each atom type in a molecular graph. The good QSPR/QSAR models have been obtained for the properties such as standard formation enthalpy of inorganic compounds and methyl halides, retention indices of gas chromatography of multiple bond-containing hydrocarbons, aqueous solubility, and octanol/water partition of benzene halides. These models indicate that the idea of using multiple matrices to define the modified Wiener index is valid and successful.     

Chemical trees minimizing energy and Hosoya index

The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph’s eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. It is quite a natural problem to minimize the energy of trees with bounded maximum degree—clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimum energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.     

Novel topological index F based on incidence matrix

Based on incidence matrix W, the novel topological index F is defined by the matrices L, W, X as F = LWX. The properties, the chemical environments, and the interaction of the vertexes in a molecule are taken into account in this definition. Several good QSPR models in the hetero-atom-containing organic compounds and inorganic compounds are obtained. Moreover, based on the definition of F and the values C(i) of the vertexes or the values (m)F(ij) of the chemical bonds, we have obtained serial indices, (m)F(v), (m)F(b), and F(w). They are successfully applied to QSPR models and good correlation results have been obtained as well.