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Topological index (TI) is a numerical invariant that helps to understand the natural relationship of the physicochemical properties of a compound in its primary structure. George Polya introduced the idea of counting polynomials in chemical graph theory and Winer made the use of TI in chemical compounds working on the paraffin's boiling point. The literature of the topological indices and counting polynomials of different graphs has grown extremely since that time. Metal-organic network (MON) is a group of different chemical compounds that consist of metal ions and organic ligands to represent unique morphology, excellent chemical stability, large pore volume, and very high surface area. Working on structures, characteristics, and synthesis of various MONs show the importance of these networks with useful applications, such as sensing of different gases, assessment of chemicals, environmental hazard, heterogeneous catalysis, gas and energy storage devices of excellent material, conducting solids, super-capacitors and catalysis for the purification, and separation of different gases. The above-mentioned properties and physical stability of these MONs become a most discussed topic nowadays. In this paper, we calculate the M-polynomials and various TIs based on these polynomials for two different MONs. A comparison among the aforesaid topological indices is also included to represent the better one.

The quantitative structures activity relationships (QSAR) and quantitative structures property relationships (QSPR) between the chemical compounds are studied with the help of topological indices (TI’s) which are the fixed real numbers directly linked with the molecular graphs. Gutman and Trinajstic (1972) defined the first degree based TI to measure the total π-electrone energy of a molecular graph. Recently, Ali and Trinajstic (2018) restudied the connection based TI’s such as first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index to find entropy and accentric factor of the octane isomers. In this paper, we study the modified second Zagreb connection index and modified third Zagreb connection index on the T-sum (molecular) graphs obtained by the operations of subdivision and product on two graphs. At the end, as the applications of the obtained results for the modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes are also included. Mainly, a comparision among the Zagreb indices, Zagreb connection indices and modified Zagreb connection indices of the T-sum graphs of the particular classes of alkanes is performed with the help of numerical tables, 3D plots and line graphs using the statistical tools.

In this paper, some algebraic and combinatorial characterizations of the spanning simplicial complex Δs(𝓙n,m) of the Jahangir’s graph 𝓙n,m are explored. We show that Δs(𝓙n,m) is pure, present the formula for f-vectors associated to it and hence deduce a recipe for computing the Hilbert series of the Face ring k[Δs(𝓙n,m)]. Finally, we show that the face ring of Δs(𝓙n,m) is Cohen-Macaulay and give some open scopes of the current work.

Apart from many branches in mathematical sciences, the main share of applications of graph theory nowadays is in chemistry. The graph-theoretical ideas have been used by many chemists for more than two centuries but over the last 30 years, the great significance of graph theory in chemistry has been extensively acknowledged. In a chemical graph, the vertices can symbolize atoms, orbitals, bonds, collection of atoms, molecules, or group of molecules and the edges symbolize the connection between chemical substances and are used to describe reactions, chemical bonds, kinetic models, reaction mechanisms, or other interactions and alterations of the chemical substances. Benzene is the archetypal aromatic molecule, showing exceptional chemical stability as compared with other unsaturated hydrocarbons. Fullerenes are a unique family of carbon-based cage molecules having remarkable properties. Fullerene has potential applications in medicinal chemistry due to its size, hydrophobicity, three-dimensionality, and electronic configurations. In this paper, we will calculate the fault-tolerant partition dimension of classes of fullerene graphs, n-linear benzene graph and partition dimension of polyphenyl chains. [ABSTRACT FROM AUTHOR]

The k -partition dimension of graphs was introduced as generalization of partition dimension of graphs by Estrado–Moreno in 2020. Its applications can be seen in several areas including robot navigation, network designing, image processing and chemistry. Rotationally symmetric graphs are important in computer network topologies due to uniform rate of data transformation to all nodes. In this paper, we compute the lower bound of the 2 -partition dimension of r -regular graphs for r ≥ 3 and also calculate the 2 -partition dimension of the families of rotationally symmetric graphs, namely, prism graph and n -sunlet graph. [ABSTRACT FROM AUTHOR]

Carbon nanotubes have been actively studied for their applications in filtering of polycyclic aromatic hydrocarbons, sensors, electronics, stretchable devices, and energy storage. The large structures of nanotubes are studied using their induced subgraphs. The partition dimension focusses on the unique distance-based representation of nodes with respect to partition sets such that each node is exclusively identified. The partition and 2 − partition dimension of induced subgraphs of certain nanotubes have been computed in this article. [ABSTRACT FROM AUTHOR]

Let Λ = {B1, B2, . . ., Bl} be an ordered l-partition of a connected graph G(V (G), E(G)). The partition representation of vertex x with respect to Λ is the l-vector, r(x|Λ) = (d(x, B1), d(x, B2), . . ., d(x, Bl)), where d(x, B) = min{d(x, y)|y ∈ B} is the distance between x and B. If the l - vectors r(x|Λ), for all x ∈ V (G) are distinct then l - partition is called a resolving partition. The least value of l for which there is a resolving l - partition is known as the partition dimension of G symbolized as pd(G). In this paper, the partition dimension of circulant graphs Cn(1, 2, 3, 4) is computed for n ≥ 8 as, pd(Cn(1, 2, 3, 4)) = { if 8 ≤ n ≤ 9; { 6, if n = 10; {5, if n ≥ 11. [ABSTRACT FROM AUTHOR]

In this article, we present important combinatorial and algebraic properties of spanning simplicial complex (SSC) of the generalised Jahangir’s graph Jm,k. We describe the relation to find f−vectors associated to ∆s(Jm,k) and determine the Hilbert series for the SR-ring K ∆s(Jm,k) . In the end, we present the associated primes of the facet ideal IF (∆s(Jm,k)) and the Cohen-Macaulay characterization of the SR-ring of ∆s(Jm,k). [ABSTRACT FROM AUTHOR]

The multiple criteria decision making (MCDM) problems indicate the alternatives which have more or less resemblance to each other. An important mathematical tool used by decision-makers (DMs) to quan- tify these resemblances is the similarity measure (SM). SM is a powerful tool that measures the resemblance more accurately. Mostly, fuzzy sets (FSs) and its extensions handle the vague and uncertain information by considering the membership, non-membership, and indeterminacy degrees whose sum always lies in the interval [0; 1]. However, single-valued neutrosophic sets (SVNSs) and interval-valued neutrosophic sets (IVNSs) have information whose sum is bounded in [0; 3]. In the present work, we extended the SM presented by William and Steel for SVNSs and IVNSs by using the concept of Euclidean distance. The weights of criteria indicate much inuence for the selection of the best alternative, sometimes DMs feel hesitation to allocate the weights to the criteria. We applied the linear programming (LP) model to evaluate the weights of the criteria to reduce the hesitancy. Later on, SM is utilized to establish an MCDM model for the selection of the best option. Moreover, the Spearman's rank correlation coe_cient is implemented to analyze the ranking order. Finally, a medical diagnosis example is illustrated for the feasibility and e_ectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a binomial ideal P(Tn) on a simple graph Tn obtained from n-triangulation of a 2-manifols. It is discussed on two particular classes of n-triangulation graphs: the class Tn1 and the class Tn2. The Gröbner bases of the binomial ideals P(Tn1) and P(Tn2) on n1-triangulation and n2-triangulation graphs Tn1 and Tn2 respectively are computed using a systematic way and it is shown that degree of polynomials in the Gröbner bases of these n-triangulation graphs is at most 5. Also, it is shown that the binomial ideals P(Tn1) and Tn2 are regular ideals. [ABSTRACT FROM AUTHOR]

A BCK-module is an action of a BCK-algebra on an abelian group. In this paper, the theory of fuzzy soft sets is applied on BCKmodules and thereby introduced the notion of fuzzy soft BCK-module (fsX--module). In this regard, various algebraic operations between fsX--modules, like fsX-- intersection, fsX--union etc, are studied. Also, fsX---homormorphisms for fsX--modules is defined and discussed. Moreover, fuzzy soft exactness of fsX--modules is introduced and discussed. [ABSTRACT FROM AUTHOR]

In the current work, the Molodtsov's idea of soft sets [14] is applied on the theory of BCK-modules [1]. The aim here, is to introduce the notion of soft BCK-modules and discuss its basic properties. In this regard, three theorems for soft BCK-modules isomorphism are developed. The notion of soft X-exactness of BCK-modules is introduced and its relation with soft X-isomorphism is studied. A transitivity between two soft X-exact sequences is also established. [ABSTRACT FROM AUTHOR]

In this paper, we give some new examples of BCK-modules. We initiate the homology theory of BCK-modules and establish that every short exact sequence of complexes admits an exact homology sequence of BCK-modules. [ABSTRACT FROM AUTHOR]