Current Organic Synthesis - Volume 22, Issue 7, 2025

To investigate the H-eigenvalues and H-energy of various types of graphs, including k-fold graphs, strong k-fold graphs, and extended bipartite double graphs and establish relationships between the H-energy of k-fold and strong k-fold graphs and the H-energy of the original graph G, we explore the connection between the H-energy of extended bipartite double graphs and their ordinary energy and find the graphs that share equienergetic properties with respect to both the ordinary and Harary matrices.

The H-eigenvalues of a graph G are the eigenvalues of its Harary matrix H(G). The H-energy () of a graph, G is the sum of the absolute values of its H-eigenvalues. Two connected graphs are said to be H-equienergetic if they have equal H-energies. They are said to A-equienergetic if they have equal A-energies. Adjacency and Harary matrices have applications in chemistry, such as finding total π-electron energy, quantitative structure-property relationship (QSPR), etc.

We determined the H-spectra of k-fold graphs, strong k-fold graphs and extended bipartite double graphs and established connections between the H-energy of different types of graphs and their original graph G for investigating the relationship between the H-energy of extended bipartite double graphs and their ordinary energy and the graphs that share equienergetic properties with respect to both the adjacency and Harary matrices.

Spectral algebraic techniques are used to calculate the H-eigenvalues and H-energy for each type of graph and compare the H-energies of different graphs to identify the equienergetic properties and derive relationships between the H-energy of extended double cover graphs and their ordinary energy.

We determined the H-spectra of k-fold graphs, strong k-fold graphs and extended bipartite double graphs and established relationships between the H-energy of k-fold and strong k-fold graphs and the H-energy of the original graph G. Then, we explored the connection between the H-energy of extended bipartite double graphs and their ordinary energy and presented graphs demonstrating equienergetic properties concerning both adjacency and Harary matrices.

The study provides insights into the H-eigenvalues, H-energy and equienergetic properties of various types of graphs. The established relationships and connections contribute to a deeper understanding of graph spectra and energy properties and the findings enhance the theoretical framework for analyzing equienergetic graphs and their spectral properties.

Possible extensions of this research could include investigating additional types of graphs and exploring further explicit connections between different graph energies and spectral properties.

Harary matrices are distance-based matrices, which can model distances between atoms in molecular structures and could be useful in organic synthesis to predict how molecular structures behave.

L-type amino acid transporter-1 is a drug that stimulates the functions of the brain’s central nervous system. Membrane transporters have evolved, leading to a distinct approach in L-type amino acid transporter-1 drug delivery. One of the transporters used for transporting drugs across biological membranes is the L-type amino acid transporter-1. It is widely discussed in the medicinal field.

Numerous investigations indicate a close connection between the properties of alkanes and the diversity of central nervous system drugs in the brain, specifically log P and molecular weight. One important study that analyzes structural properties is focused on topological descriptors. Recently, topological indices have found application in the development of quantitative structure-activity relationships. These indices are correlated with the physicochemical properties of BCNS-acting drugs and their biological activity.

The study employs significant methods of calculating topological indices: the edge set partition method and the Djokovi´c-Winkler relation (cut method) are utilized to calculate the values of these descriptors.

The results of distance and degree-based topological descriptors have been derived. The strong correlation between topological descriptors and the physicochemical properties of BCNS-acting drugs has been studied.

This article identifies important topological features for various CNS medications, aiming to support researchers in understanding the properties of molecules and their biological activity. Furthermore, we demonstrate how strongly these behaviors correspond to the physicochemical properties of central nervous system drugs.

Hexagonal fractals are intricate geometric patterns that exhibit self-similarity. They are characterized by their repetitive hexagonal shapes at different scales. Due to their unique properties and potential applications, hexagonal fractals have been studied in various fields, including mathematics, physics, and chemistry.

The primary aim of this research is to provide a comprehensive analysis of hexagonal fractals, focusing on their topological indices, fractal dimensions, and their applications in structure-property modeling. We aim to calculate topological indices to quantify the structural complexity and connectivity of hexagonal fractals. Additionally, we will determine fractal dimensions to characterize their self-similarity and scaling behaviour. Finally, we will explore the relationship between topological indices, fractal dimensions, and relevant properties through structure-property modeling.

A systematic approach was employed to investigate hexagonal fractals. Various topological indices were computed using established mathematical techniques. Fractal dimensions were determined. Structure-property modeling was conducted by establishing relationships between the calculated topological indices and fractal dimensions with experimentally measured properties.

The research yielded significant findings regarding hexagonal fractals. A variety of topological indices were calculated, revealing the intricate connectivity and structural complexity of these fractals. Fractal dimensions were determined, confirming their self-similar nature and scaling behaviour. Structure-property modeling demonstrated strong correlations between the topological indices and fractal dimensions with properties such as conductivity, mechanical strength, and chemical reactivity.

This research provides valuable insights into the topological characteristics, fractal dimensions, and potential applications of hexagonal fractals. The findings contribute to a deeper understanding of these complex structures and their relevance in various scientific domains. The developed structure-property modeling approaches offer a valuable tool for predicting and controlling the properties of materials based on their fractal structure. Future research may explore additional applications and delve into the underlying mechanisms governing the relationship between fractal structure and properties.

Boric acid, a weak monobasic Lewis acid of boron, exists as a white solid that crystallizes from aqueous solutions in triclinic waxy plates. The major role of boric acid in the treatment of Candida vaginitis is through its antifungal properties, which help inhibit the growth of the Candida yeast responsible for the infection.

Treatment for Candida vaginitis involves antifungal medications, which are commonly administered orally or topically to eliminate the Candida yeast causing the infection.

In this study, we employed the Djokovic-Winkler relation to enumerate the distance of each pair of compounds in the molecular graph. Analytical and computational techniques were employed to characterize the physical and chemical properties of boric acid, which help in the treatment of Candida vaginitis.

Moreover, this study introduces the Sadhana index and PI index, derived from specific polynomials (omega, theta, sadhana, and PI), to identify the antifungal properties most conducive to antifungal medications.