Entropy Measures of Distance Matrix

Bonchev and Trinajstic defined two distance based entropy measures to measure the molecular branching of molecular graphs in 1977 [Information theory, distance matrix, and molecular branching, J. Chem. Phys., 38 (1977), 4517–4533]. In this paper we use these entropy measures which are based on distance matrices of graphs. The first one is based on distribution of distances in distance matrix and the second one is based on distribution of distances in upper triangular submatrix. We obtain the two entropy measures of paths, stars, complete graphs, cycles and complete bipartite graphs. Finally we obtain the minimal trees with respect to these entropy measures with fixed diameter.

Non-Equilibrium Molecular Dynamics Study of the Influence of Branching on the Soret Coefficient of Binary Mixtures of Heptane Isomers

Equimolar mixtures composed of isomers were firstly used to investigate the molecular branching effect on thermal diffusion behavior, which was not disturbed by factors of molecular mass and composition in this work. Eight heptane isomers, including n-heptane, 2-methylhexane, 3-methylhexane, 2,2-dimethylpentane, 2,3-dimethylpentane, 2,4-dimethylpentane, 3,3-dimethylpentane and 3-ethylpentane, were chosen as the researched mixtures. A non-equilibrium molecular dynamics (NEMD) simulation with enhanced heat exchange (eHEX) algorithm was applied to calculate the Soret coefficient at T = 303.15 T=303.15 K and P = 1.0 atm P=1.0\hspace{0.1667em}\text{atm} . An empirical correlation based on an acentric factor was proposed and its calculation coincides with the simulated results, which showed the validity of the NEMD simulation. It is demonstrated that the isomer with higher acentric factor has a stronger thermophilic property and tends to migrate to the hot region in the heptane isomer mixture, and the extent of thermal diffusion is proportional to the difference between the acentric factors of the isomers.

Hosoya and Harary Polynomials of Hourglass and Rhombic Benzenoid Systems

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Harry Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we compute the Hosoya polynomials for hourglass and rhombic benzenoid systems and recover Wiener and hyper-Wiener indices from them.

Hosoya and Harary Polynomials of TOX(n),RTOX(n),TSL(n) and RTSL(n)

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials for TOX(n),RTOX(n),TSL(n), and RTSL(n) networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.

Zagreb Connection Indices of Some Nanostructures

The first Zagreb index (occurred in an approximate formula of total π-electron energy, communicated in 1972) and the second Zagreb index (appeared in 1975, within the study of molecular branching) are among the most studied topological indices. Recently, three modified versions of the Zagreb indices were proposed independently in [A. Ali, N. Trinajstić, A novel/old modification of the first Zagreb index, arXiv:1705.10430 [math.CO], 2017] and [A. M. Naji, N. D. Soner, I. Gutman, On leap Zagreb indices of graphs, Commun. Comb. Optim., 2017, 2, 99–117], which were named as the Zagreb connection indices and the leap Zagreb indices, respectively. In this paper, we derive formulas for calculating these modified versions of the Zagreb indices of four well known nanostructures.