Combinatorial Chemistry & High Throughput Screening - Volume 25, Issue 3, 2022

Introduction: In this paper, we present a novel hybrid model m-polar Diophantine fuzzy N-soft set and define its operations. Methods: We generalize the concepts of fuzzy sets, soft sets, N-soft sets, fuzzy soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft sets, Pythagorean fuzzy sets, Pythagorean fuzzy soft sets and Pythagorean fuzzy N-soft sets by incorporating our proposed model. Additionally, we define three different sorts of complements for Pythagorean fuzzy N-soft sets and examine few outcomes, which do not hold in Pythagorean fuzzy N-soft sets complements unlike to crisp set. We further discuss (α, β, γ) -cut of m-polar Diophantine fuzzy N-soft sets and their properties. Lastly, we prove our claim that the defined model is a generalization of the soft set, N-soft set, fuzzy Nsoft set, intuitionistic fuzzy N soft set, and Pythagorean fuzzy N-soft set. Results: m-polar Diophantine fuzzy N-soft set is more efficient and an adaptable model to manage uncertainties as it also overcomes drawbacks of existing models, which are to be generalized. Conclusion: We introduced the novel concept of m-polar Diophantine fuzzy N-soft sets (MPDFNS sets).

Aims and Objective: The idea of partition and resolving sets play an important role in various areas of engineering, chemistry and computer science such as robot navigation, facility location, pharmaceutical chemistry, combinatorial optimization, networking, and mastermind game. Methods: In a graph, to obtain the exact location of a required vertex, which is unique from all the vertices, several vertices are selected; this is called resolving set, and its generalization is called resolving partition, where selected vertices are in the form of subsets. A minimum number of partitions of the vertices into sets is called partition dimension. Results: It was proved that determining the partition dimension of a graph is a nondeterministic polynomial time (NP) problem. In this article, we find the partition dimension of convex polytopes and provide their bounds. Conclusion: The major contribution of this article is that due to the complexity of computing the exact partition dimension, we provide the bounds and show that all the graphs discussed in the results have partition dimensions either less or equals to 4, but not greater than 4.

Background: Computing Hosoya polynomial for a graph associated with a chemical compound plays a vital role in the field of chemistry. From Hosoya polynomial, it is easy to compute the Weiner index(Weiner number) and Hyper Weiner index of the underlying molecular structure. The Wiener number enables the identifying of three basic features of molecular topology: branching, cyclicity, and centricity (or centrality) and their specific patterns, which are well reflected by the physicochemical properties of chemical compounds. Caterpillar trees are used in chemical graph theory to represent the structure of benzenoid hydrocarbons molecules. In this representation, one forms a caterpillar in which each edge corresponds to a 6-carbon ring in the molecular structure, and two edges are incident at a vertex whenever the corresponding rings belong to a sequence of rings connected end-to-end in the structure. Due to the importance of Caterpillar trees, it is interesting to compute the Hosoya polynomial and the related indices. Methods:The Hosoya polynomial of a graph G is defined as H(G;x) = Σ^d(G)K=0 d(G.k)x^k. In order to compute the Hosoya polynomial, we need to find its coefficient d(G.k) which is the number of pairs of vertices of G which are at distance k. We classify the ordered pair of vertices which are at distance , 2 ≤ m ≤ (n + 1)k in the form of sets. Then finding the cardinality of these sets and adding them up will give us the value of coefficient d(G.m) . Finally, using the values of coefficients in the definition, we get the Hosoya polynomial of uniform subdivision of caterpillar graph. Result: In this work, we compute the closed formula of Hosoya polynomial for subdivided caterpillar trees. This helps us to compute the Weiner index and hyper-Weiner index of uniform subdivision of caterpillar graph. Conclusion: Caterpillar trees are among the important and general classes of trees. Thorn rods and thorn stars are the important subclasses of caterpillar trees. The idea of the present research article is to provide a road map to those researchers who are interested in studying the Hosoya polynomial for different trees.

Background: Topological indices have numerous implementations in chemistry, biology and a lot of other areas. It is a real number associated with a graph, which provides information about its physical and chemical properties and their correlations. For a connected graph H, the degree distance DD index is defined as DD(H) = Σ{h1,h2}⊆V(H) [degH(h1)+degH (h2)]dh (h1,h2), where degH (h1)is the degree of vertex h1 and dH (h1,h2) is the distance between h1 and h2 in the graph H. Aim and Objective: In this article, we characterize some extremal trees with respect to degree distance index which has a lot of applications in theoretical and computational chemistry. Materials and Methods: A novel method of edge-grafting transformations is used. We discuss the behavior of DD index under four edge-grafting transformations Results: With the help of those transformations, we derive some extremal trees under certain parameters, including pendant vertices, diameter, matching and domination numbers. Some extremal trees for this graph invariant are also characterized Conclusion: It is shown that balanced spider approaches to the smallest DD index among trees having given fixed leaves. The tree Cn,d has the smallest DD index, among all trees of diameter d. It is also proved that the matching number and domination numbers are equal for trees having a minimum DD index.

Background: Sierpinski graphs S(n, k) are largely studied because of their fractal nature with applications in topology, chemistry, mathematics of Tower of Hanoi and computer sciences. Applications of molecular structure descriptors are a standard procedure which are used to correlate the biological activity of molecules with their chemical structures, and thus can be helpful in the field of pharmacology. Objective: The aim of this article is to establish analytically closed computing formulae for eccentricity-based descriptors of Sierpinski networks and their regularizations. These computing formulae are useful to determine a large number of properties like thermodynamic properties, physicochemical properties, chemical and biological activity of chemical graphs Methods: At first, vertex sets have been partitioned on the basis of their degrees, eccentricities and frequencies of occurrence. Then these partitions are used to compute the eccentricity-based indices with the aid of some combinatorics. Results: The total eccentric index and eccentric-connectivity index have been computed. We also compute some eccentricity-based Zagreb indices of the Sierpinski networks. Moreover, a comparison has also been presented in the form of graphs. Conclusion: These computations will help the readers to estimate the thermodynamic properties and physicochemical properties of chemical structure which are of fractal nature and can not be dealt with easily. A 3D graphical representation is also presented to understand the dynamics of the aforementioned topological descriptors.

Background and Objective: Blood pressure is vital evidence for clinicians to predict diseases and check the curative effect of diagnosis and treatment. To further improve the prediction accuracy of blood pressure, this paper proposes a combined prediction model of blood pressure based on coritivity theory and photoplethysmography. Method: First of all, we extract eight features of photoplethysmogram, followed by using eight machine learning prediction algorithms, such as K-nearest neighbor, classification and regression trees, and random forest, to predict systolic blood pressure. Secondly, aiming at the problem of sub-model selection of combination forecasting model, from the point of graph theory, we construct an undirected network graph G, the results of each single prediction model constitute a vertex set. If the maximum mutual information coefficient between vertices is greater than or equal to 0.69, the vertices are connected by edges. The maximum core of graph G is a submodel of the combinatorial model. Results: According to the definition of core and coritivity, the maximum core of G is random forest regression and Gaussian kernel support vector regression model. The results show that the SDP estimation error of the combined prediction model based on random forest regression and Gaussian kernel support vector regression is 3.56 ±5.28mmhg, which is better than other single models and meets the AAMI standards. Conclusion: The combined model determined by core and coritivity has higher prediction performance for blood pressure.