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Abstract The graph $$G'$$ G ′ obtained from a graph G by identifying two nonadjacent vertices in G having at least one common neighbor is called a 1-fold of G. A sequence $$G_0, G_1, G_2, \ldots , G_k$$ G 0 , G 1 , G 2 , … , G k of graphs such that $$G_0=G$$ G 0 = G and $$G_i$$ G i is a 1-fold of $$G_{i-1}$$ G i - 1 for each $$i=1, 2, \ldots , k$$ i = 1 , 2 , … , k is called a uniform k-folding of G if the graphs in the sequence are all singular or all nonsingular. The fold thickness of G is the largest k for which there is a uniform k-folding of G. We show here that the fold thickness of a singular bipartite graph of order n is $$n-3$$ n - 3 . Furthermore, the fold thickness of a nonsingular bipartite graph is 0, i.e., every 1-fold of a nonsingular bipartite graph is singular. We also determine the fold thickness of some well-known families of graphs such as cycles, fans and some wheels. Moreover, we investigate the fold thickness of graphs obtained by performing operations on these families of graphs. Specifically, we determine the fold thickness of graphs obtained from the cartesian product of two graphs and the fold thickness of a disconnected graph whose components are all isomorphic.

Tuberculosis (TB) is one of the most contagious diseases that has a greater mortality rate than HIV/AIDS and the cases of TB are feared to rise as a repercussion of the COVID-19 pandemic. The pharmaceutical industry is constantly looking for ways to improve drug design processes in order to combat the growth of infections and cure newly identified syndromes or genetically based dysfunctions with the help of QSPR models. QSPR models are mathematical tools that establish relationships between a molecular structure and its physicochemical attributes using structural properties. Topological indices are such properties that are generated from the molecular graph without any empirically derived measurements. This work focuses on developing a QSPR model using distance-based topological indices for anti-tuberculosis medications and their diverse physicochemical features.

A topological index, which is a number, is connected to a graph. It is often used in chemometrics, biomedicine, and bioinformatics to anticipate various physicochemical properties and biological activities of compounds. The purpose of this article is to encourage original research focused on topological graph indices for the drugs azacitidine, decitabine, and guadecitabine as well as an investigation of the genesis of symmetry in actual networks. Symmetry is a universal phenomenon that applies nature’s conservation rules to complicated systems. Although symmetry is a ubiquitous structural characteristic of complex networks, it has only been seldom examined in real-world networks. The M¯-polynomial, one of these polynomials, is used to create a number of degree-based topological coindices. Patients with higher-risk myelodysplastic syndromes, chronic myelomonocytic leukemia, and acute myeloid leukemia who are not candidates for intense regimens, such as induction chemotherapy, are treated with these hypomethylating drugs. Examples of these drugs are decitabine (5-aza-20-deoxycytidine), guadecitabine, and azacitidine. The M¯-polynomial is used in this study to construct a variety of coindices for the three brief medicines that are suggested. New cancer therapies could be developed using indice knowledge, specifically the first Zagreb index, second Zagreb index, F-index, reformulated Zagreb index, modified Zagreb, symmetric division index, inverse sum index, harmonic index, and augmented Zagreb index for the drugs azacitidine, decitabine, and guadecitabine.

The metric dimension has various applications in several fields, such as computer science, image processing, pattern recognition, integer programming problems, drug discovery, and the production of various chemical compounds. The lowest number of vertices in a set with the condition that any vertex can be uniquely identified by the list of distances from other vertices in the set is the metric dimension of a graph. A resolving function of the graph G is a map ϑ:V(G)→[0,1] such that ∑u∈R{v,w}ϑ(u)≥1, for every pair of adjacent distinct vertices v,w∈V(G). The local fractional metric dimension of the graph G is defined as ldimf(G) = min{∑v∈V(G)ϑ(v), where ϑ is a local resolving function of G}. This paper presents a new family of planar networks namely, rotationally heptagonal symmetrical graphs by means of up to four cords in the heptagonal structure, and then find their upper-bound sequences for the local fractional metric dimension. Moreover, the comparison of the upper-bound sequence for the local fractional metric dimension is elaborated both numerically and graphically. Furthermore, the asymptotic behavior of the investigated sequences for the local fractional metric dimension is addressed.

Entropy is a thermodynamic function in physics that measures the randomness and disorder of molecules in a particular system or process based on the diversity of configurations that molecules might take. Distance-based entropy is used to address a wide range of problems in the domains of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines. We explain the basic applications of distance-based entropy to chemical phenomena. These applications include signal processing, structural studies on crystals, molecular ensembles, and quantifying the chemical and electrical structures of molecules. In this study, we examine the characterisation of polyphenylenes and boron (B12) using a line of symmetry. Our ability to quickly ascertain the valences of each atom, and the total number of atom bonds is made possible by the symmetrical chemical structures of polyphenylenes and boron B12. By constructing these structures with degree-based indices, namely the K Banhatti indices, ReZG1-index, ReZG2-index, and ReZG3-index, we are able to determine their respective entropies.

Entropy is a measure of a system’s molecular disorder or unpredictability since work is produced by organized molecular motion. Shannon’s entropy metric is applied to represent a random graph’s variability. Entropy is a thermodynamic function in physics that, based on the variety of possible configurations for molecules to take, describes the randomness and disorder of molecules in a given system or process. Numerous issues in the fields of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines are resolved using distance-based entropy. These applications cover quantifying molecules’ chemical and electrical structures, signal processing, structural investigations on crystals, and molecular ensembles. In this paper, we look at K-Banhatti entropies using K-Banhatti indices for C6H6 embedded in different chemical networks. Our goal is to investigate the valency-based molecular invariants and K-Banhatti entropies for three chemical networks: the circumnaphthalene (CNBn), the honeycomb (HBn), and the pyrene (PYn). In order to reach conclusions, we apply the method of atom-bond partitioning based on valences, which is an application of spectral graph theory. We obtain the precise values of the first K-Banhatti entropy, the second K-Banhatti entropy, the first hyper K-Banhatti entropy, and the second hyper K-Banhatti entropy for the three chemical networks in the main results and conclusion.