Browsing by Author "Farahani, M.R."

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Analyzing the Boron Triangular Nanotube Through Topological Indices Via M-Polynomial
(Taylor and Francis Ltd., 2021) Hussain, S.; Afzal, F.; Afzal, D.; Cancan, M.; Ediz, S.; Farahani, M.R.
he current discovery of different types of nanostructures has inspired the researcher to study the applications of these structures in different fields. In this study, we have analyzed the boron triangular nanotube through topological indices. M-polynomial of a boron triangular nanotube has the capability to recover the topological indices which are dependent on the degree of the vertex. We have presented the results in graphical form. © 2021 Taru Publications.
Choosing Friends in Everyday Life by Using Graph Theory
(Charles Babbage Research Centre, 2024) Mughal, A.A.; Jamil, R.N.; Farahani, M.R.; Alaeiyan, M.; Cancan, M.
Graph theory is playing vital role in almost every field of our routine life. You make a conference call with your friends by using vertices (yourself and your friends) and edges (network connection). You construct a printed grid floor with different faces in your home by the help of graph theory. Authors in this study are using labelling of graphs and applying it in choosing best friends around you. The helping graphs in this article will be plane graphs which will be labelling under k−labelling M of kind (λ, µ, ν). This study can be applied in many fields of everyday life. © 2024 the Author(s), licensee Combinatorial Press.
Computing Metric Dimension of Two Types of Claw-Free Cubic Graphs With Applications
(Charles Babbage Research Centre, 2024) Sardar, M.S.; Xu, S.-J.; Cancan, M.; Farahani, M.R.; Alaeiyan, M.; Patil, S.V.
Consider the simple connected graph G with vertex set V(G) and edge set E(G). A graph G can be resolved by R if each vertex’s representation of distances to the other vertices in R uniquely identifies it. The minimum cardinality of the set R is the metric dimension of G. The length of the shortest path between any two vertices, x, y in V(G), is signified by the distance symbol d(x, y). An ordered k-tuple r(x/R) = (d(x, z1), d(x, z2), ..., d(x, zk)) represents representation of x with respect to R for an ordered subset R = {z1, z2, z3..., zk} of vertices and vertex x in a connected graph. Metric dimension is used in a wide range of contexts where connection, distance, and connectedness are essential factors. It facilitates understanding the structure and dynamics of complex networks and problems relating to robotics network design, navigation, optimization, and facility location. Robots can optimize their localization and navigation methods using a small number of reference sites due to the pertinent idea of metric dimension. As a result, many robotic applications, such as collaborative robotics, autonomous navigation, and environment mapping, are more accurate, efficient, and resilient. A claw-free cubic graph (CCG) is one in which no induced subgraph is a claw. CCG proves helpful in various fields, including optimization, network design, and algorithm development. They offer intriguing structural and algorithmic properties. Developing algorithms and results for claw-free graphs frequently has applications in solving of challenging real-world situations. The metric dimension of a couple of claw-free cubic graphs (CCG), a string of diamonds (SOD), and a ring of diamonds (ROD) will be determined in this work. © 2024 the Author(s), licensee Combinatorial Press.
Distance and Degree Based Topological Polynomial and Indices of X-Level Wheel Graph
(Abdus Salam School of mathematical Sciences, 2021) Hasan, A.; Qasmi, M.H.A.; Alsinai, A.; Alaeiyan, M.; Farahani, M.R.; Cancan, M.
In this paper we discussed the partitioning of the wheel graph and we calculate the M-polynomial, Hosoya polynomial, Harary polynomial, Schultz polynomial, Modified Schultz polynomial, Eccentric connectivity polynomial, Modified Wiener index, Modified Hyper Wiener index, Generalized Harary index, Multiplicative Wiener index, Schultz index, Modified Schultz index, Eccentric connectivity index and also derived the Randic index, Generalized Randic index, First Zagreb, Second Zagreb, Second Modified Zagreb, General Randic and Inverse General Randic, Harmonic, Symmetric Division and Inverse Sum index of generalized wheel networks Wx,y. © 2021
Fault Tolerant Metric Dimension of Arithmetic Graphs
(Charles Babbage Research Centre, 2024) Sardar, M.S.; Rasheed, K.; Cancan, M.; Farahani, M.R.; Alaeiyan, M.; Patil, S.V.
For a graph G, two vertices x, y ∈ G are said to be resolved by a vertex s ∈ G if d(x|s), d(y|s), where d(x|s) denotes the distance between x and s. The minimum cardinality of such a resolving set R in G is called the metric dimension. A resolving set R is said to be fault-tolerant if, for every p ∈ R, the set R − p preserves the property of being a resolving set. The fault-tolerant metric dimension of G is the minimal possible order of a fault-tolerant resolving set. The concept of metric dimension has wide applications in areas where connection, distance, and network connectivity are critical. This includes understanding the structure and dynamics of complex networks, as well as addressing problems in robotic network design, navigation, optimization, and facility placement. By utilizing the concept of metric dimension, robots can optimize their methods for localization and navigation using a limited number of reference points. As a result, various applications in robotics, such as collaborative robotics, autonomous navigation, and environment mapping, have become more precise, efficient, and resilient. The arithmetic graph Al is defined as the graph where the vertex set is the set of all divisors of a composite number l, where l = pγ11 pη22 · · · pαnn and the pi’s are distinct primes with pi ≥ 2. Two distinct divisors x and y of l are said to have the same parity if they share the same prime factors (e.g., x = p1p2 and y = p21p32 have the same parity). Furthermore, two distinct vertices x, y ∈ Al are adjacent if and only if they have different parity and gcd(x, y) = pi (greatest common divisor) for some i ∈ {1, 2, . . ., t}. This article focuses on the investigation of the arithmetic graph of a composite number l, referred to throughout as Al. In this study, we compute the fault-tolerant resolving set and the fault-tolerant metric dimension of the arithmetic graph Al, where l is a composite number. © 2024 the Author(s), licensee Combinatorial Press.
Foreword
(Taru Publications, 2025) Farahani, M.R.; Alaeiyan, M.; Ameen, H.B.; Zhang, X.; Cancan, M.; Afzal, F.
On Exploring the Topological Aspects of the Chemical Structure of the Nanotube Hac5c7
(Utilitas Mathematica Publishing Inc., 2024) Zakir, M.S.; Naseer, M.K.; Farahani, M.R.; Ahmad, I.; Kanwal, Z.; Alaeiyan, M.; Cancan, M.
Graph theory has experienced notable growth due to its foundational role in applied mathematics and computer science, influencing fields like combinatorial optimization, biochemistry, physics, electrical engineering (particularly in communication networks and coding theory), and operational research (with scheduling applications). This paper focuses on computing topological properties, especially in molecular structures, with a specific emphasis on the nanotube HAC5C7[w, t]. © 2024 Utilitas Mathematica Publishing Inc.. All rights reserved.
On Reformulated Narumi-katayama İndex
(Universidad Catolica del Norte, 2020) Cancan, M.; De, N.; Alaeiyan, M.; Farahani, M.R.
A graph is a mathematical model form by set of dots for vertices some of which are connected bylines named as edges. A topological index is a numeric value obtained from a graph mathematically which characterize its topology. The reformulated Narumi-Katayama index of a graph G is defined as the product of edge degrees of all the vertices of G which is introduced in 1984, to used the carbon skeleton of a saturated hydrocarbons. The degree of an edge is given by the sum of degrees of the end vertices of the edge minus 2. In this paper, we compute the reformulated Narumi-Katayama index for different graph operations. copyright © Murat Cancan, Nilanjan De, Mehdi Alaeiyan, and Mohammad Reza Farahani.
On Some Degree Based Topological Indices of Mk-Graph
(Taylor and Francis Ltd., 2020) De, N.; Cancan, M.; Alaeiyan, M.; Farahani, M.R.
A topological index is a real number which is same under graph isomorphism and it is derived from a graph by mathematically. In chemical graph theory, a molecular graph is a simple graph having no loops and multiple edges in which atoms and chemical bonds are represented by vertices and edges respectively. Topological indices defined on these chemical molecular structures can help researchers better understand the physical features, chemical reactivity, and biological activity. In this paper, we compute general expressions of some degree based topological indices of a special graph named as mk-graph for some positive integer k. © 2020 Taru Publications.
On Ve-Degree and Ev-Degree Based Topological Invariants of Chemical Structures
(Utilitas Mathematica Publishing Inc., 2024) Nigar, N.; Alam, S.M.; Rasheed, M.W.; Farahani, M.R.; Alaeiyan, M.; Cancan, M.
In the realm of graph theory, recent developments have introduced novel concepts, notably the νε-degree and εν-degree, offering expedited computations compared to traditional degree-based topological indices (TIs). These TIs serve as indispensable molecular descriptors for assessing chemical compound characteristics. This manuscript aims to meticulously compute a spectrum of TIs for silicon carbide SiC4-I[r, s], with a specific focus on the εν-degree Zagreb index, the νε-degree Geometric-Arithmetic index, the εν-degree Randić index, the νε-degree Atom-bond connectivity index, the νε-degree Harmonic index, and the νε-degree Sum connectivity index. This study contributes to the ongoing advancement of graph theory applications in chemical compound analysis, elucidating the nuanced structural properties inherent in silicon carbide molecules. © 2024 Utilitas Mathematica Publishing Inc.. All rights reserved.
Partition Dimension of Generalized Peterson and Harary Graphs
(Abdus Salam School of mathematical Sciences, 2021) Khalaf, A.J.M.; Nadeem, M.F.; Azeem, M.; Farahani, M.R.; Cancan, M.
The distance of a connected, simple graph (Formula presented) is denoted by d(α1, α2), which is the length of a shortest path between the vertices α1,α2 (Formula presented) V((Formula presented)), where V((Formula presented)) is the vertex set of (Formula presented). The l-ordered partition of V((Formula presented)) is K = {K1, K2,..., Kl}. A vertex α (Formula presented) V((Formula presented)), and r(α|K) = {d(α, K1), d(α, K2),..., d(α, Kl)} be a l-tuple distances, where r(α|K) is the representation of a vertex a with respect to set K. If r(a|K) of a is unique, for every pair of vertices, then K is the resolving partition set of V((Formula presented)). The minimum number l in the resolving partition set K is known as partition dimension (pd(P)). In this paper, we studied the generalized families of Peterson graph, Pλx and proved that these families have bounded partition dimension. © 2021. All Rights Reserved.
Special Issue on Applied Discrete Mathematics, Combinatorics, Cryptography, Computer Science and Computation
(Taru Publications, 2025) Farahani, M.R.; Alaeiyan, M.; Ameen, H.B.; Zhang, X.; Cancan, M.