The graph 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 ,. .. , G k of graphs such that G 0 = G and G i is a 1-fold of G i−1 for each 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. 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. Mathematics Subject Classification 05C76 • 05C50 1 Introduction The notion of folding a graph as defined by Gervacio et al. [2] was motivated by the following situation. Consider a finite number of unit bars joined together at the ends where they are free to turn. Some meter sticks are constructed with this kind of structure as shown in bottom drawing in Fig. 1. Note that this particular meter stick can be folded to look like the top drawing in Fig. 1. The meter stick with this structure can be viewed as a physical model of the path graph P n. After a sequence of folding of the meter stick, it becomes a physical model of a complete graph K 2. Based on this observation, they defined the notion of folding a graph as follows. Let x and y be nonadjacent vertices of a graph that have at least one common neighbor. Obtain a new graph G from G by identifying x and y and reducing any resulting multiple edges to simple edges. We say that G is a 1-fold of G. Note that folds of a graph are idempotent homomorphisms and also known as retracts. For more details on the connection between a 1-fold and retracts, please see [6]. Example 1.1 The vertices x and y in the graph G shown in Fig. 2 have two common neighbors, b and c. By identifying x and y, we obtain the graph G , a 1-fold of G. Consider a graph G that is not isomorphic to a complete graph. Suppose that G 1 is a 1-fold of G. If G 1 is not a complete graph, then there should be a pair of non-adjacent vertices in G 1 and a graph G 2 which is a 1-fold of G 1 can be obtained. Thus, we can consider a sequence of graphs
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.
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.
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.
In this study we define a graph operation on a finite simple graph G = (V, E) called the S-splitting graph of G where S is a non-empty subset of vertices of G. If S = V , then it is the splitting graph of G defined by E. Sampathkumar, and H.B. Walikar in the 1980’s. This paper investigates the Wiener and Harary indices of the S-splitting graph of G for some families of graph.
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