Chemical applications of topology and group theory: 19. The even permutations of the ligands in five-coordinate complexes viewed as proper rotations of the regular icosahedron [1]

A unified fixed point approach to study the existence of solutions for a class of fractional boundary value problems arising in a chemical graph theory

A theory of chemical graphs is a part of mathematical chemistry concerned with the effects of connectedness in chemical graphs. Several researchers have studied the solutions of fractional differential equations using the concept of star graphs. They employed star graphs because their technique requires a central node with links to adjacent vertices but no edges between nodes. The purpose of this paper is to extend the method’s range by introducing the concept of an octane graph, which is an essential organic compound having the formula C8H18. In this manner, we analyze a graph with vertices annotated by 0 or 1, which is influenced by the structure of the chemical substance octane, and formulate a fractional boundary value problem on each of the graph’s edges. We use the Schaefer and Krasnoselskii fixed point theorems to investigate the existence of solutions to the presented boundary value problems in the framework of the Caputo fractional derivative. Finally, two examples are provided to highlight the importance of our results in this area of study.

Aromaticity in transition metal oxide structures

The concept of aromaticity is useful for understanding the properties of some polyoxometalates containing transition metals such as vanadium, molybdenum, and tungsten having structures based on metal macropolygons and macropolyhedra with M-O-M edges. Thus, the aromatic macrocuboctahedral Keggin ions readily undergo one-electron reductions to highly colored mixed-valence "blues" (e.g., molybdenum blue), whereas the macroicosahedral Silverton ions, M(IV)Mo12O42(8-) (M(IV) = Ce, Th, U), which, like cyclohexane, do not have vertex valence orbitals available for delocalization, do not undergo analogous reduction reactions. A macrohexagon of d1 vanadium(IV) atoms as V-O-V units has been imbedded into an electronically inactive borate matrix in the ion [V6B20O50H8](8-). The small beta unit for the V-O-V interactions in this V6 macrohexagon leads to an unprecedented example of high spin aromaticity with a paramagnetism corresponding to four unpaired electrons per V6 unit in contrast to benzene, which is diamagnetic and hence exhibits low spin aromaticity. The M-O-M interactions in these aromatic metal oxides are closely related to the Cu-O-Cu interactions in the high critical temperature superconducting copper oxides which are essential to the electron transport in these systems.