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Abstract In this paper, we introduce the concept of fuzzy zero divisor graph (FZDG) for a commutative ring $$R$$ R denoted by $${\Gamma }_{f}\left(\text{R}\right)$$ Γ f R . We explore the multiset dimension (Mdim), a new variant of the metric dimension (MD), specifically in the context of FZDGs. To illustrate our findings, we analyze the FZDG for the ring $${\mathbb{Z}}_{n}$$ Z n of integers modulo $$n$$ n of integers modulo $$n$$ n , denoted by $${\Gamma }_{f}\left({\mathbb{Z}}_{n}\right).$$ Γ f Z n . We compute the multiset dimension for all possible values of $$n$$ n for the FZDG $${\Gamma }_{f}\left({\mathbb{Z}}_{n}\right)$$ Γ f Z n , providing significant theoretical insights into its structure. Our results not only advance the understanding of FZDGs and their multiset dimensions but also have practical implications across various fields, including cryptography, coding theory, and network analysis. This study lays the groundwork for future research on the application of fuzzy concepts in graph theory and algebraic structures.

In the contemporary technological landscape, ensuring confidentiality is a paramount concern addressed through various skillsets. Cryptography stands out as a scientific methodology for safeguarding communication against unauthorized access. Within the realm of cryptography, numerous encryption algorithms have been developed to enhance data security. Recognizing the imperative for nonstandard encryption algorithms to counter traditional attacks, this paper puts forth novel encryption techniques. These methods leverage special corona graphs, star graphs, and complete bipartite graphs, incorporating certain algebraic properties to bolster the secure transmission of messages. The introduction of these proposed encryption schemes aims to elevate the level of security in confidential communication, some of the applications of these schemes are given in the later section.

This article investigates the concept of dominant metric dimensions in zero divisor graphs (ZD-graphs) associated with rings. Consider a finite commutative ring with unity, denoted as R, where nonzero elements x and y are identified as zero divisors if their product results in zero (x.y=0). The set of zero divisors in ring R is referred to as L(R). To analyze various algebraic properties of R, a graph known as the zero-divisor graph is constructed using L(R). This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of R. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo m, denoted as Zm[i], the ring of integers modulo n, denoted as Zn, and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.

This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set Z(RE)\{[0]}=RE\{[0],[1]}, where RE={[x] :x∈R} and [x]={y∈R : ann(x)=ann(y)} is called a compressed zero-divisor graph, denoted by ΓER. An edge is formed between two vertices [x] and [y] of Z(RE) if and only if [x][y]=[xy]=[0], that is, iff xy=0. For a ring R, graph G is said to be realizable as ΓER if G is isomorphic to ΓER. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance.

Zero forcing is a process of coloring in a graph in time steps known as propagation time. These graph-theoretic parameters have diverse applications in computer science, electrical engineering and mathematics itself. The problem of evaluating these parameters for a network is known to be NP-hard. Therefore, it is interesting to study these parameters for special families of networks. Perila et al. (2017) studied properties of these parameters for some basic oriented graph families such as cycles, stars and caterpillar networks. In this paper, we extend their study to more non-trivial structures such as oriented wheel graphs, fan graphs, friendship graphs, helm graphs and generalized comb graphs. We also investigate the change in propagation time when the orientation of one edge is flipped.

Dual hesitant fuzzy sets (DHFSs) is the refinement and extension of hesitant fuzzy sets and encompasses fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. DHFSs have two parts, that is, the membership function and the non-membership function, in which each function is defined by two sets of some feasible values. Therefore, according to the practical demand, DHFSs are more adjustable than the existing ones and provide the information regarding different objects in much better way. The set pair analysis (SPA) illustrates unsureness in three angles, called “identity”, “discrepancy” and “contrary”, and the connection number (CN) is one of its main features. In the present article, the axiom definition of distance measure between DHFSs and CN is introduced. The distance measures are established on the basis of Hamming distance, Hausdorff distance and Euclidean distance. The previous identities and relationship between them are discussed in detail. On the basis of the geometric distance model, the set-theoretic approach, and the matching functions several novel distance formulas of CN are introduced. The novel distance formulas are then applied to multiple-attribute decision making for dual hesitant fuzzy environments. Finally, to demonstrate the validity of the introduced measures, a practical example of decision-making is presented. The benefits of the new measures over the past measures are additionally talked about.

Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, B⊆V(Γ). If the representation of two different edges is unequal, then B is an edge-multiset resolving a set of Γ. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of Γ. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions.

The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q1, q2, ... , qk} be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q1), r(a, q2), ... , r(a, qk)), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by Tn〈1, t〉 and Tn〈1, 2, t〉, respectively is discussed and proved that it is constant.

Metal–organic structures/Networks (MONs) are useful in modern chemistry. It is common to use as a storehouse for the storage of gases, it plays a role in the separation of gases and purification. The most important feature of MONs is that it acts as a predecessor for the development of a large number of nanostructures. Furthermore, MONs reflect very useful chemical-physical properties, changing organic ligands, exchanging of ions, etc. The method used to forecast the natural behaviors among the chemical-physical specifications of the chemical compounds in their primitive network is known as topological indices or TIs. This numerical quantity is used in the method of forecasting. TIs of MONs shows a key role in the environmental and theoretical pharmacology and chemistry. Line graphs also have powerful applications in chemistry and predicting the boiling point of cycloalkanes. In this paper, we study Randić, atom bound connectivity, geometric arithmetic, Zagreb, Multiplicative Zagreb, redefined Zagreb indices and Zagreb coindices for line graph of first organic network L(MON1(χ)) and second organic network L(MON2(χ)),χ⩾2.

Graph theory is one of those subjects that is a vital part of the digital world. It is used to monitor the movement of robots on a network, to debug computer networks, to develop algorithms, and to analyze the structural properties of chemical structures, among other things. It is also useful in airplane scheduling and the study of diffusion mechanisms. The parameters computed in this article are very useful in pattern recognition and image processing. A number df,w=mindw,t,dw,s is referred as distance between f=ts an edge and w a vertex. dw,f1≠dw,f2 implies that two edges f1,f2∈E are resolved by node w∈V. A set of nodes A is referred to as an edge metric generator if every two links/edges of Γ are resolved by some nodes of A and least cardinality of such sets is termed as edge metric dimension, edimΓ for a graph Γ. A set B of some nodes of Γ is a mixed metric generator if any two members of V∪E are resolved by some members of B. Such a set B with least cardinality is termed as mixed metric dimension, mdimΓ. In this paper, the metric dimension, edge metric dimension, and mixed metric dimension of dragon graph Tn,m, line graph of dragon graph LTn,m, paraline graph of dragon graph LSTn,m, and line graph of line graph of dragon graph LLTn,m have been computed. It is shown that these parameters are constant, and a comparative analysis is also given for the said families of graphs.

A topological index is a characteristic value which represents some structural properties of a chemical graph. We study strong double graphs and their generalization to compute Zagreb indices and Zagreb coindices. We provide their explicit computing formulas along with an algorithm to generate and verify the results. We also find the relation between these indices. A 3D graphical representation and graphs are also presented to understand the dynamics of the aforementioned topological indices.

Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset of Qm/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields. In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪k∈NQ∗k2m containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. This study also helps us in classifying reduced numbers lying in the PSL2,ℤ-orbits.

Abstract Topological indices are the mathematical tools that correlate the chemical structure with various physical properties, chemical reactivity or biological activity numerically. A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. In QSAR/QSPR study, a prediction about the bioactivity of chemical compounds is made on the basis of physico-chemical properties and topological indices such as Zagreb, Randić and multiple Zagreb indices. In this paper, we determine the lower and upper bounds of Zagreb indices, the atom-bond connectivity (ABC) index, multiple Zagreb indices, the geometric-arithmetic (GA) index, the forgotten topological index and the Narumi-Katayama index for the Cartesian product of F-sum of connected graphs by using combinatorial inequalities.

Let G=V,E be a simple connected graph, w∈V be a vertex, and e=uv∈E be an edge. The distance between the vertex w and edge e is given by de,w=mindw,u,dw,v, A vertex w distinguishes two edges e1, e2∈E if dw,e1≠dw,e2. A set S is said to be resolving if every pair of edges of G is distinguished by some vertices of S. A resolving set with minimum cardinality is the basis for G, and this cardinality is the edge metric dimension of G, denoted by edimG. It has already been proved that the edge metric dimension is an NP-hard problem. The main objective of this article is to study the edge metric dimension of some families of wheel-related graphs and prove that these families have unbounded edge metric dimension. Moreover, the results are compared with the metric dimension of these graphs.

Let Γ be a numerical semigroup. We associate an undirected graph G ( Γ ) with a numerical semigroup Γ with vertex set { v i : i ∈ N \ Γ } and edge set { v i v j ⇔ i + j ∈ Γ } . In this article, we discuss the connectedness, diameter, girth, and some other related properties of the graph G ( Γ ) .

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.

Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree (DD) index was introduced by Dobrynin and Kochetova in 1994 for characterizing alkanes by an integer. In this paper, we have determined expressions for a DD index of some derived graphs in terms of the parameters of the parent graph. Specifically, we establish expressions for the DD index of a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph, and paraline graph.