Hafiz Muhammad Afzal Siddiqui - Academia.edu (original) (raw)

Papers by Hafiz Muhammad Afzal Siddiqui

Research paper thumbnail of On Degree Based Topological Indices of Transition Metal-Tetra Cyano Polycyclic Benzene Organic Network

Polycyclic Aromatic Compounds

Research paper thumbnail of Some topological properties of uniform subdivision of Sierpiński graphs

Main Group Metal Chemistry

Sierpiński graphs are family of fractal nature graphs having applications in mathematics of Tower... more Sierpiński graphs are family of fractal nature graphs having applications in mathematics of Tower of Hanoi, topology, computer science, and many more diverse areas of science and technology. This family of graphs can be generated by taking certain number of copies of the same basic graph. A topological index is the number which shows some basic properties of the chemical structures. This article deals with degree based topological indices of uniform subdivision of the generalized Sierpiński graphs S(n,G) and Sierpiński gasket Sn . The closed formulae for the computation of different kinds of Zagreb indices, multiple Zagreb indices, reduced Zagreb indices, augmented Zagreb indices, Narumi-Katayama index, forgotten index, and Zagreb polynomials have been presented for the family of graphs.

Research paper thumbnail of Topological Properties of Supramolecular Chain of Different Complexes of N-Salicylidene-L-Valine

Polycyclic Aromatic Compounds

Research paper thumbnail of Theoretical study of energy, inertia and nullity of phenylene and anthracene

Open Chemistry

Energy of a molecule plays an important role in physics, chemistry and biology. In mathematics, t... more Energy of a molecule plays an important role in physics, chemistry and biology. In mathematics, the concept of energy is used in graph theory to help other subjects such as chemistry and physics. In graph theory, nullity is the number of zeros extracted from the characteristic polynomials obtained from the adjacency matrix, and inertia represents the positive and negative eigenvalues of the adjacency matrix. Energy is the sum of the absolute eigenvalues of its adjacency matrix. In this study, the inertia, nullity and signature of the aforementioned structures have been discussed.

Research paper thumbnail of On Degree-Based Topological Indices for Strong Double Graphs

Journal of Chemistry

A topological index is a characteristic value which represents some structural properties of a ch... more 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.

Research paper thumbnail of Comparative Study of Zagreb Indices for Capped, Semi-Capped, and Uncapped Carbon Nanotubes

Polycyclic Aromatic Compounds

Research paper thumbnail of On the zero forcing number and propagation time of oriented graphs

AIMS Mathematics

Zero forcing is a process of coloring in a graph in time steps known as propagation time. These g... more 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 NPhard. 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.

Research paper thumbnail of Topological aspects of metal-organic structure with the help of underlying networks

Arabian Journal of Chemistry

Research paper thumbnail of Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields

Journal of Mathematics

Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset ... more 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.

Research paper thumbnail of Distance Measures for Multiple-Attributes Decision-Making Based on Connection Numbers of Set Pair Analysis with Dual Hesitant Fuzzy Sets

IEEE Access

Dual hesitant fuzzy sets (DHFSs) is the refinement and extension of hesitant fuzzy sets and encom... more 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.

Research paper thumbnail of On Wiener index and Wiener polarity index of some polyomino chains

Journal of Discrete Mathematical Sciences and Cryptography

Research paper thumbnail of On Resolvability Parameters of Some Wheel-Related Graphs

Journal of Chemistry

Let G=V,E be a simple connected graph, w∈V be a vertex, and e=uv∈E be an edge. The distance betwe... more 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.

Research paper thumbnail of Computation of Zagreb indices and Zagreb polynomials of Sierpinski graphs

Hacettepe Journal of Mathematics and Statistics

Research paper thumbnail of Resolvability and fault-tolerant resolvability structures of convex polytopes

Theoretical Computer Science

Research paper thumbnail of Computing Metric Dimension of Certain Families of Toeplitz Graphs

IEEE Access

The position of a moving point in a connected graph can be identified by computing the distance f... more 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 = {q 1 , q 2 ,. .. , q k } 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, q 1), r(a, q 2),. .. , r(a, q k)), 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 T n 1, t and T n 1, 2, t , respectively is discussed and proved that it is constant.

Research paper thumbnail of Reformulated Zagreb Indices of Some Derived Graphs

Mathematics

A topological index is a numeric quantity that is closely related to the chemical constitution to... more 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.

Research paper thumbnail of Characterization of Graphs Associated with Numerical Semigroups

Mathematics

Let Γ be a numerical semigroup. We associate an undirected graph G ( Γ ) with a numerical semigro... more 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 ) .

Research paper thumbnail of Distance Degree Index of Some Derived Graphs

Mathematics

Topological indices are numerical values associated with a graph (structure) that can predict man... more 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.

Research paper thumbnail of Vertex Weighted Wiener Polynomials of the Subdivision Graph and the Line Graph of Subdivision Graph of the Wheel Graph

Advances and Applications in Discrete Mathematics

Research paper thumbnail of Bounds for General Connectivity Indices of Tensor Product of Connected Graphs

Topological indices are numerical parameters of a graph which characterize its topology and are u... more Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In this paper, bounds for the Randić, general Randić, sum-connectivity, the general sum-connectivity and harmonic indices for tensor product of graphs are determined by using the combinatorial inequalities and combinatorial computing.

Research paper thumbnail of On Degree Based Topological Indices of Transition Metal-Tetra Cyano Polycyclic Benzene Organic Network

Polycyclic Aromatic Compounds

Research paper thumbnail of Some topological properties of uniform subdivision of Sierpiński graphs

Main Group Metal Chemistry

Sierpiński graphs are family of fractal nature graphs having applications in mathematics of Tower... more Sierpiński graphs are family of fractal nature graphs having applications in mathematics of Tower of Hanoi, topology, computer science, and many more diverse areas of science and technology. This family of graphs can be generated by taking certain number of copies of the same basic graph. A topological index is the number which shows some basic properties of the chemical structures. This article deals with degree based topological indices of uniform subdivision of the generalized Sierpiński graphs S(n,G) and Sierpiński gasket Sn . The closed formulae for the computation of different kinds of Zagreb indices, multiple Zagreb indices, reduced Zagreb indices, augmented Zagreb indices, Narumi-Katayama index, forgotten index, and Zagreb polynomials have been presented for the family of graphs.

Research paper thumbnail of Topological Properties of Supramolecular Chain of Different Complexes of N-Salicylidene-L-Valine

Polycyclic Aromatic Compounds

Research paper thumbnail of Theoretical study of energy, inertia and nullity of phenylene and anthracene

Open Chemistry

Energy of a molecule plays an important role in physics, chemistry and biology. In mathematics, t... more Energy of a molecule plays an important role in physics, chemistry and biology. In mathematics, the concept of energy is used in graph theory to help other subjects such as chemistry and physics. In graph theory, nullity is the number of zeros extracted from the characteristic polynomials obtained from the adjacency matrix, and inertia represents the positive and negative eigenvalues of the adjacency matrix. Energy is the sum of the absolute eigenvalues of its adjacency matrix. In this study, the inertia, nullity and signature of the aforementioned structures have been discussed.

Research paper thumbnail of On Degree-Based Topological Indices for Strong Double Graphs

Journal of Chemistry

A topological index is a characteristic value which represents some structural properties of a ch... more 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.

Research paper thumbnail of Comparative Study of Zagreb Indices for Capped, Semi-Capped, and Uncapped Carbon Nanotubes

Polycyclic Aromatic Compounds

Research paper thumbnail of On the zero forcing number and propagation time of oriented graphs

AIMS Mathematics

Zero forcing is a process of coloring in a graph in time steps known as propagation time. These g... more 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 NPhard. 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.

Research paper thumbnail of Topological aspects of metal-organic structure with the help of underlying networks

Arabian Journal of Chemistry

Research paper thumbnail of Icosahedral Group and Classification of PSL(2, Z)-Orbits of Real Quadratic Fields

Journal of Mathematics

Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset ... more 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.

Research paper thumbnail of Distance Measures for Multiple-Attributes Decision-Making Based on Connection Numbers of Set Pair Analysis with Dual Hesitant Fuzzy Sets

IEEE Access

Dual hesitant fuzzy sets (DHFSs) is the refinement and extension of hesitant fuzzy sets and encom... more 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.

Research paper thumbnail of On Wiener index and Wiener polarity index of some polyomino chains

Journal of Discrete Mathematical Sciences and Cryptography

Research paper thumbnail of On Resolvability Parameters of Some Wheel-Related Graphs

Journal of Chemistry

Let G=V,E be a simple connected graph, w∈V be a vertex, and e=uv∈E be an edge. The distance betwe... more 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.

Research paper thumbnail of Computation of Zagreb indices and Zagreb polynomials of Sierpinski graphs

Hacettepe Journal of Mathematics and Statistics

Research paper thumbnail of Resolvability and fault-tolerant resolvability structures of convex polytopes

Theoretical Computer Science

Research paper thumbnail of Computing Metric Dimension of Certain Families of Toeplitz Graphs

IEEE Access

The position of a moving point in a connected graph can be identified by computing the distance f... more 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 = {q 1 , q 2 ,. .. , q k } 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, q 1), r(a, q 2),. .. , r(a, q k)), 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 T n 1, t and T n 1, 2, t , respectively is discussed and proved that it is constant.

Research paper thumbnail of Reformulated Zagreb Indices of Some Derived Graphs

Mathematics

A topological index is a numeric quantity that is closely related to the chemical constitution to... more 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.

Research paper thumbnail of Characterization of Graphs Associated with Numerical Semigroups

Mathematics

Let Γ be a numerical semigroup. We associate an undirected graph G ( Γ ) with a numerical semigro... more 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 ) .

Research paper thumbnail of Distance Degree Index of Some Derived Graphs

Mathematics

Topological indices are numerical values associated with a graph (structure) that can predict man... more 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.

Research paper thumbnail of Vertex Weighted Wiener Polynomials of the Subdivision Graph and the Line Graph of Subdivision Graph of the Wheel Graph

Advances and Applications in Discrete Mathematics

Research paper thumbnail of Bounds for General Connectivity Indices of Tensor Product of Connected Graphs

Topological indices are numerical parameters of a graph which characterize its topology and are u... more Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In this paper, bounds for the Randić, general Randić, sum-connectivity, the general sum-connectivity and harmonic indices for tensor product of graphs are determined by using the combinatorial inequalities and combinatorial computing.