s-四嗪-水簇复合物的理论研究

用量子化学B3LYP方法和6-31＋＋G**基函数研究了s-四嗪-水簇复合物基态分子间相互作用, 并进行了构型优化和频率计算, 分别得到无虚频稳定的s-四嗪-(水)₂复合物、s-四嗪-(水)₃复合物和s-四嗪-(水)₄复合物6个、9个和12个. 复合物存在较强的氢键作用, 复合物结构中形成一个N…H—O氢键并终止于O…H—C氢键的氢键水链构型最稳定. 经基组重叠误差和零点振动能校正后, 最稳定的1∶2, 1∶3和1∶4(摩尔比)复合物的结合能分别是41.35, 70.9和 94.61 kJ/mol. 振动分析显示氢键的形成使复合物中水分子H—O键对称伸缩振动频率减小(红移). 研究表明N…H键越短, N…H—O键角越接近直线, 稳定化能越大, 氢键作用越强. 同时, 用含时密度泛函理论方法在TD-B3LYP/6-31＋＋G**水平计算了s-四嗪单体及其氢键复合物的第一¹(n, p*)激发态的垂直激发能.     

The Merrifield–Simmons Indices and Hosoya Indices of Trees with <Emphasis Type="Italic">k</Emphasis> Pendant Vertices

The Merrifield–Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we characterize the trees with maximal Merrifield–Simmons indices and minimal Hosoya indices, respectively, among the trees with k pendant vertices.     

Extremal Wiener and Kirchhoff indices of globular caterpillars

The Wiener and Kirchhoff indices of a graph G are two of the most important topological indices in mathematical chemistry. A graph G is called to be a globular caterpillar if G is obtained from a complete graph K _s with vertex set {v₁,v₂,…, v _s} by attaching n _i pendent edges to each vertex v _i of K _s for some positive integers s and n₁,n₂,…,n _s, denoted by . Let be the set of globular caterpillars with n vertices (). In this article, we characterize the globular caterpillars with the minimal and maximal Wiener and Kirchhoff indices among , respectively.     

Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

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     

Hosoya polynomials of TUC4C8(R) nanotubes

For a connected graph G, the Hosoya polynomial of G is defined as H(G, x) = ∑_{u,v}?V(G)x^{d(u, v)}, where V(G) is the set of all vertices of G and d(u,v) is the distance between vertices u and v. In this article, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(R) nanotubes. Furthermore, the Wiener index and the hyper‐Wiener index can be calculated. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

The exponent in the general Randić index

We determine conditions for the parameters n and δ, for which the general Randić index R _δ is not an acceptable index of branching of n-vertex trees, i.e., for which the n-vertex star and the n-vertex path have not extremal R _δ-values among all n-vertex trees. Analogous results are established also in the case of n-vertex chemical trees. Numerous other results for the general Randić index of trees and chemical trees are obtained.      

On the difference between atom‐bond connectivity index and Randić index of binary and chemical trees

This article is devoted to establishing some extremal results with respect to the difference of two well‐known bond incident degree indices [atom‐bond connectivity (ABC ) index and Randi? (R ) index] for the chemical graphs representing alkanes. More precisely, the first three extremal trees with respect to ABC – R are characterized among all n‐vertex binary trees (the trees with maximum degree at most 3). The n‐vertex chemical trees (the trees with maximum degree at most 4) having the first three maximum ABC – R values are also determined.     

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.     

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.     

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.     

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.     

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.     

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     

On the connectivity index of trees

The connectivity index χ₁(G) of a graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. Let T(n, r) be the set of trees on n vertices with diameter r. In this paper, we determine all trees in T(n, r) with the largest and the second largest connectivity index. Also, the trees in T(n, r) with the largest and the second largest connectivity index are characterized. Mei Lu is partially supported by NNSFC (No. 10571105).     

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     

Trees with the same Topological Index JJ

Abstract

In the present paper we investigate the trees with the same JJ index (called JJ-equivalent trees). The topological index JJ is derived from the so called Wiener matrix introduced by Randic et al., in 1994. The Wiener matrix is constructed by generalizing the procedure of Wiener for evaluation of Wiener numbers in alkanes. From such matrices several novel structural invariants of potential interest in structure-property studies were obtained. These include higher Wiener numbers, Wiener sequences, and hyper-Wiener number, etc. The topological index JJ is constructed by considering the row sums of the Wiener matrix. A construction method for a class of JJ-equivalent trees is given. By using this method we construct the smallest pairs of non-isomorphic JJ-equivalent trees which have 18 vertices. In addition we report on groups of 3,4, and 6 non-isomorphic JJ-equivalent trees. The smallest such trees are of size 28 for triples and quadruples, and 34 for the group of 6 elements.     

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).