On the Edge Metric Dimension of Certain Polyphenyl Chains

The most productive application of graph theory in chemistry is the representation of molecules by the graphs, where vertices and edges of graphs are the atoms and valence bonds between a pair of atoms, respectively. For a vertex w and an edge f = c 1 c 2 of a connected graph G , the minimum number from distances of w with c 1 and c 2 is called the distance between w and f . If for every two distinct edges f 1 , f 2 ∈ E G , there always exists w 1 ∈ W E ⊆ V G such that d f 1 , w 1 ≠ d f 2 , w 1 , then W E is named as an edge metric generator. The minimum number of vertices in W E is known as the edge metric dimension of G . In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph O n , meta-polyphenyl chain graph M n , and the linear [n]-tetracene graph T n and also find the edge metric dimension of para-polyphenyl chain graph L n . It has been proved that the edge metric dimension of O n , M n , and T n is bounded, while L n is unbounded.

Uncompleted Fullerenelike Structures: Synthesis and Chracteristics

The literature is considering completely formed fullerenes as usually. Without attentions have been left uncompleted ones, and their properties are practically unknown. Such structures may be received at arc discharge synthesis in the limited space with short time of synthesis. Here synthesis was made at arc discharge chamber with little radius and less gas flow. The time-of-flight mass spectrum synthesized product on m/z size showed containing of hydrated uncompleted-composite fullerene. They were like with structures: C54H4, C56H4, C56H8, C68H4 and other. A presence at mass spectrum of abnormal structures was manifested in form of doublet many-peaks lines. They also confirmed termperature-formating of new bonds with successive griping valence bonds with poliin structures at arc discharge. From mass spectrum, on base of Mendileive’s chemical plate was supposed the way forming conjectural structure. The MS-picture was confirmed with estimation for forming many-peaks abnormal poliin like structures.

Chains of Metallic Clusters With Ligands

This chapter geometrically investigated the structure of clusters, the core of which represent the metal chains (linear or curved) of both identical and different elements. It was shown that the dimension of the structures of these clusters is more than three. To create a model of these chains in a higher dimension space, a new geometric approach has been developed that allows us to construct convex, closed polytopes of these chains. It consists of removing part of the octahedron edges necessary for constructing the octahedron and adding the same number of new edges necessary to build a closed polytope chain while maintaining the number of metal atoms and ligands and their valence bonds. As a result, it was found that metal chain polytopes consist of polytopes of higher dimension, adjacent to each other along flat sections.

Valence bonds in planar and quasi-planar boron disks

Valence bonds within the perimeter of disk-like boron clusters with a concentric topology follow simple 4n and 8n electron counting rules.