On the Randić index

The Randić index of an organic molecule whose molecular graph is 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 Discrete Mathematics 257, 29–38 by 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 [2002]), and then we will show that the result holds when δ(G) ≥ 2.AMS subject classification: 05C18     

On zeroth-order general Randić index of conjugated unicyclic graphs

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as where α is an arbitrary real number. In this paper, we investigate the zeroth-order general Randić index of conjugated unicyclic graphs G (i.e., unicyclic graphs with a perfect matching) and sharp lower and upper bounds are obtained for depending on α in different intervals.     

Randić index and lexicographic order

Let T be a tree and consider the Randi index (T)= ), where v _i –v _j runs over all edges of T and (v _i ) denotes the degree of the vertex v _i . Using counting arguments we show that the Randi index, is monotone increasing over the well (lexicographic order) ordered sequence of trees with unique branched vertex.     

Third order Randić index of phenylenes

Let PH denote a phenylene, whose third order Randić index is denoted by ³χ(P H). The expression of ³χ(P H) in terms of their inlet features is found.     

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.     

On the Randić Index of Quasi-tree Graphs

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 vertex u and v in G. A graph G is called quasi-tree, if there exists such that G − u is a tree. In the paper, we give sharp lower and upper bounds on the Randić index of quasi-tree graphs. Mei Lu: Partially supported by NSFC (No. 10571105).     

On the Randić Index of Unicyclic Conjugated Molecules

The Randić index R(G) of a graph G is the sum of the weights of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given size of matching.     

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)     

The Hosoya index and the Merrifield–Simmons index

In this article, we give sharp bounds on the Hosoya index and the Merrifield–Simmons index for connected graphs of fixed size. As a consequence, we determine all connected graphs of any fixed order and size which maximize the Merrifield–Simmons index. Sharp lower bounds on the Hosoya index are known for graphs of order n and size \(m\in [n-1,2n-3]\cup \left( {n-1\atopwithdelims ()2},{n\atopwithdelims ()2}\right] \); while sharp upper bounds were only known for graphs of order n and size \(m\le n+2\). We give sharp upper bounds on the Hosoya index for dense graphs with \(m\ge {n\atopwithdelims ()2}-2n/3\). Moreover, all extreme graphs are also determined.     

Temperature dependence of the Kováts retention index. The entropy index

According to a novel equation, the temperature dependence of the Kováts retention index, dI/dT is proportional to the difference of the Kováts retention index, I, and the new entropy index, I(S), defined similarly as the retention index, but based on solvation entropy instead of the free energy of solvations. The new relationship was tested with the experimental retention and thermodynamic data published by Kováts and coworkers for 32 compounds on 6 different stationary phases. Very good correlations (r>0.99) were observed for dI/dT versus (I-I(S)) and dI/dT versus deltaDeltaC(p), the molar heat capacity difference of the solute and the hypothetical n-alkane, which has the same retention index as the solute. Deviations in the dI/dT versus deltaDeltaC(p) relationship were observed only for alcohols, suggesting a different solvation mechanism for alcohols as compared with other compounds.     

Defining rules of aromaticity: a unified approach to the Hückel, Clar and Randić concepts

The molecular structure of any system may be unambiguously described by its adjacency matrix, A, in which bonds are assigned entry a(ij) = 1 and non-bonded pairs of atoms entry a(ij) = 0. For π-electron-containing conjugated hydrocarbons, this matrix may be modified in order to represent one of the possible Kekulé structures by assigning entry 1 to double bonds and entry 0 to single bonds, leading to the Kekulé matrix K which can be obtained from the A matrix by subtracting 1 from elements a(pq) that represent single bonds in the Kekulé structure. The A and K matrices are the boundary cases of a general matrix A(ε), named perturbation matrix, in which from elements a(pq) that represent single bonds is subtracted a value ε∈<0,1> representing the magnitude of the perturbation. The determinant of the A(ε) matrix is unambiguously represented by an appropriate polynomial that, in turn, can be written in a form containing terms ±(1-ε)(N/2) that identify types of π-electron conjugated cycles (N is the corresponding number of π-electrons). If the sign before the term is (+), then the contribution is stabilizing, but if it is (-) the contribution is destabilizing. The approach shows why and how the Hückel rule works, how the Randi? conjugated circuits result from the analysis of canonical structures, and also how the Clar rule may be extended to include aromatic cycles larger than six-membered (aromatic sextet).     

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.     

The Graovac–Pisanski index of zig-zag tubulenes and the generalized cut method

The Graovac–Pisanski index, which is also called the modified Wiener index, was introduced in 1991 by Graovac and Pisanski. This variation of the classical Wiener index takes into account the symmetries of a graph. In 2016 Ghorbani and Klav?ar calculated this index by using the cut method, which we generalize in this paper. Moreover, we prove that in some cases the automorphism group of a zig-zag tubulene is isomorphic to the direct product of a dihedral group and a cyclic group. Finally, the closed formulas for the Graovac–Pisanski index of zig-zag tubulenes are calculated.     

Characterization of distribution of pi-electrons amongst benzenoid rings for Randić’s “algebraic” Kekulé structures

For benzenoid hydrocarbons the distribution of pi-electrons amongst rings is characterized in the context of Randis mode of assignment attending to the different Kekulé structures. In particular the mean and mean deviation from the mean are considered, and the benzenoids which achieve maximum deviation are identified.     

The First-Year Chemistry Laboratory in the United States

本文是张树永教授为《大学化学》杂志向Kelter教授特约的专稿,主要介绍美国大学一年级实行分层次化学教学的一些作法,以及大学一年级化学课程教学体系和实验教学体系、教学内容和教学方法。虽然美国大学化学实验课程的内容不像我国那样多,但涉及面广,更强调学生的自主学习和自我训练,同时将仪器分析作为实验的必要内容嵌入日常实验当中,使实验内容的综合性、设计性更强,也更贴近日常生活和科学实验。这些作法对我国的大学化学实验教学改革具有借鉴作用。     

The Transition in the Melt of Fluorinated Ethylene-Propylene Copolymer

The subject of the transition in linear polymer melt has been of particular interest since thediscovery of liquid crystal polymers.The transitions in several polymer melts have been recog-nized in recent years.These polymers are mainly rigid chain in nature and the transitions wereassigned as mesomorphic.Recently we have found a transition in the melt of tetrafluoroethylene-hexafluoropropylene copolymer (FEP copolymer). The authors~([1]) have found that the morphology of FEP copolymer is critically dependent onthe temperature of melt near 310?20?.When the melt was held at the temperature above itsmelting temperature and below 310?,the randomly arranged lamellas morphology was obtained.When the melt was held at the temperature above 320?,the spherulitic morphology was formed.When the copolymer crystallized from the intermediate temperature range,the rodlike morphologywas usually obtained.The variation of morphology with temperature can be easily detected by Small Angle Light Scattering technique.Also the crystallization curves,measured with the Per-