Degree-Based Graph Entropy in Structure-Property Modeling

Graph entropy plays an essential role in interpreting the structural information and complexity measure of a network. Let G be a graph of order n. Suppose dG(vi) is degree of the vertex vi for each i=1,2,…,n. Now, the k-th degree-based graph entropy for G is defined as Id,k(G)=-∑i=1ndG(vi)k∑j=1ndG(vj)klogdG(vi)k∑j=1ndG(vj)k, where k is real number. The first-degree-based entropy is generated for k=1, which has been well nurtured in last few years. As ∑j=1ndG(vj)k yields the well-known graph invariant first Zagreb index, the Id,k for k=2 is worthy of investigation. We call this graph entropy as the second-degree-based entropy. The present work aims to investigate the role of Id,2 in structure property modeling of molecules.

Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph ΓG of a finite group G is a graph where non-central elements of G are its vertex set, while two different elements are edge connected when they do not commute in G. In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of SL(2,C). We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.

[Root architecture and fine root characteristics of Juglans mandshurica saplings in different habitats in the secondary forest on the west slope of Zhangguangcailing, China]

The whole root excavation method was used to examine root configuration of Juglans mandshurica, with the age of 5-6 years in three habitats (forest edge, gap, and canopy) in a secondary forest on the western part of Zhangguangcailing Mountains. Root structure and fine root function were measured. The root topological index, average joint length, cross-sectional area ratio before and after root branching were calculated and fine root chemical compositions were analyzed. Roots of J. mandshurica at forest edge tended to be dichotomous branch (Topological index:TI=0.68), that under the canopy were herringbone-like branch (TI=0.79), and the gap was between the two (TI=0.72). The average connection length of roots among the three habitats was not significant. The cross-sectional area ratio of roots before and after root branching in three habitats was 1.06, 1.04 and 1.07, respectively, which was not affected by root diameter, in accordance with the Leonardo da Vinci rule. For the same order fine root in different habitats, its length and specific surface area gradually increased from the edge of the forest to the canopy. The N content decreased first and then increased, while the C content and C/N increased first and then decreased. From the forest edge to the gap and to the under canopy, roots tended to move from the dichotomous branch to the herringbone-like branch by reducing the overlap between the secondary branches and roots, increasing specific root length, specific surface area and changing the contents of C and N to cope with environmental change and improve nutrient absorption efficiency.

Genealogy of Conjugated Acyclic Polyenes

Based on the total π-electron energies E_πs of Hückel Molecular Orbital (HMO) method for all the possible isomers of conjugated acyclic polyenes (C_2nH_2n+2) up to n = 7, the structure–stability relation of the possible isomers was analyzed. It was shown that the mean length of conjugation L can roughly predict the ordering of stability among isomers, while the Z-index, or Hosoya-index, can almost perfectly reproduce their stability. Further, the genealogy of the conjugated acyclic polyene family was obtained by drawing systematic diagrams connecting these isomers of different n, and governed by several simple rules. Namely, the stability change of a given isomer in the genealogy connecting n and n + 1 polyenes can be classified into three different modes of vinyl addition (elongation, inner and outer branching) and horn growing, i.e., substitution of –HC=CH– moiety with –HC(=CH₂)–C(=CH₂)H–. By using the Z-index, we can extend this type of discussion to polyene radicals and even to “cross-conjugated” cyclic polyenes containing only one odd-membered cycle, such as radialene and fulvene.

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.