In the field of chemical and medical sciences, topological indices are used to study the chemical... more In the field of chemical and medical sciences, topological indices are used to study the chemical, biological, clinical, and therapeutic aspects of pharmaceuticals. The COVID-19 pandemic is largely recognized as the most life-threatening crisis confronting medical advances. Scientists have tested various antiviral drugs and discovered that they help people recover from viral infections like COVID-19. Antiviral medications, such as Arbidol, Chloroquine, Hydroxy-Chloroquine, Lopinavir, Remdesivir, Ritonavir, Thalidomide and Theaflavin, are often used to treat COVID-19. In this paper, we define Diameter Eccentricity Based vertex degree and employ it to introduce a new polynomial called $ D\varepsilon- $ Polynomial. Using the newly introduced polynomial, we derive new topological indices, namely, diameter eccentricity based and hyper diameter eccentricity based indices. In order to check the efficacy of our indices, we derive the $ D\varepsilon- $ polynomials for the eight COVID-19 drug...
Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique Kc (the ... more Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique Kc (the core) and η copies of Ks (the satellites) meeting in a common clique. They belong to the class of graphs of diameter two. Agave graphs Θ(2, 1, η) ∼= K2 ▽ (ηK1) belong to the general class of complete split graphs, where the graphs consist of a central clique K2 and η copies of K1 which are connected to all the nodes of the clique. They are the subclass of Core-satellite graphs. Let μ(G) be the spectral radius of the signless Laplacian matrix Q(G). In this paper, we have obtained the greatest lower bound and the least upper bound of signless Laplacian spectral radius of Agave graphs. These bounds have been expressed in terms of graph invariants like m the number of edges, n the number of vertices, δ the minimum degree, ∆ the maximum degree, and η copies of the satellite. We have made use of the approximation technique to derive these bounds. This unique approach can be utilized to determi...
The permeable materials known as metal–organic frameworks (MOFs) have a large porosity volume, ex... more The permeable materials known as metal–organic frameworks (MOFs) have a large porosity volume, excellent chemical stability, and a unique structure that results from the potent interactions between metal ions and organic ligands. Work on the synthesis, architectures, and properties of various MOFs reveals their utility in a variety of applications, including energy storage devices with suitable electrode materials, gas storage, heterogeneous catalysis, and chemical assessment. A topological index, which is a numerical invariant, predicts the physicochemical properties of chemical entities based on the underlying molecular graph or framework. In this article, we consider two different zinc-based MOFs, namely zinc oxide and zinc silicate MOFs. We compute 14 neighbourhood degree sum-based topological indices for these frameworks, and the numerical and graphical representations of all the aforementioned 14 indices are made.
A new class of graphs called dumbbell graphs, denoted by DB(Wm,n) is the graph obtained from two ... more A new class of graphs called dumbbell graphs, denoted by DB(Wm,n) is the graph obtained from two copies of generalized wheel graph Wm,n, m ≥ 2, n ≥ 3. It is a graph on 2 (m + n) vertices obtained by connecting m-vertices in one copy with the corresponding vertices in the other copy. The resistance distance between two vertices vi and vj, denoted by rij , is defined as the effective electrical resistance between them if each edge of G is replaced by 1 ohm resistor. The Kirchhoff index is the sum of the resistance distances between all pairs of vertices in the graph. In this paper, we formulate the resistance distance of Wm,n and DB(Wm,n) using Symmetric {1}-inverse of Laplacian matrices. We provide examples to illustrate the proposed method and also obtain the Kirchhoff indices for these examples.
Network equilibrium models are significantly distinct in supply chain networks, traffic networks,... more Network equilibrium models are significantly distinct in supply chain networks, traffic networks, and e-waste flow networks. The idea of network equilibrium is strongly perceived while determining the tuner sets of a graph (network). Tuner sets are subsets of vertices of the graph G whose degrees are lower than the average degree of G, d(G) that can compensate or balance the presence of vertices whose degrees are greater than d(G). Generalised core-satellite graph Θ c, ¯ S, ¯η = Kc ▽ (¯ηK¯ S) comprises ¯η copies of K¯ S (the satellites) meeting in Kc (the core) and it belongs to the family of graphs of diameter two. It has a central core of vertices connected to a few satellites, where all satellite cliques need not be identical and can be of different sizes. Properties like hierarchical structure of large real-world networks, are competently modeled using core-satellite graphs [1, 2, 5]. This family of graphs exhibits the properties similar to scale-free network as they possess anomalous vertex connectivity, where a small fraction of vertices (the core) are densely connected. Since these graphs possess such a structural property, interesting results are obtained for these graphs when tuner sets are determined. In this paper, we have considered the graph G = (ηKq + γKp) ▽ K1, with p > q, a subclass of the generalized core-satellite graph which is a join of η copies of the clique Kq and γ copies of the clique Kp with the core K1. We have obtained the tuner set for this subclass and established the relation between the Top T(G) and the cardinality of the tuner set |Ψ| through necessary and sufficient conditions. We analyze and characterize these graphs and obtain some interesting results while simultaneously examining the existence of tuner sets.
The properties and activities of chemical compounds can be understood by computing topological de... more The properties and activities of chemical compounds can be understood by computing topological descriptors of molecular compounds. We investigate the physical and topological aspects of crystal structure of metal-insulator transition superlattice (GST-SL) in this study. Recently, researchers have turned their attention to modifying this substance into a form that is useful for human life. Metal-insulator transition superlattices (GST-SL) are also useful as two-dimensional (2D) transition metal dichalcogenides (TMDs) in the form of thin films. For this Superlattice Network SLη, we calculate open and closed neighbourhood degree sum based on topological indices.
International Journal of Recent Technology and Engineering (IJRTE), 2019
Parallelism is a process by which a sequential string is broken down into a number of alphabets a... more Parallelism is a process by which a sequential string is broken down into a number of alphabets and used to speed up the acceptance of a string. To identify the parallelisable string, we have used parallel operator || and defined the language as parallel series languages. Algebraic and recognition properties of series parallel posets have been studied by Lodaya in [8]. In this paper, we have introduced finite and infinite parallel series language (parallel strings are connected sequentially). We have considered the set of all parallel series language as topological space and prefix order relation (poset relation) has been used to relate two parallel series strings. Topological concepts like limit, sequence, open set, closed set and basis for parallel series languages and their properties have been derived.
Topological descriptors defined on chemical structures enable understanding the properties and ac... more Topological descriptors defined on chemical structures enable understanding the properties and activities of chemical molecules. In this paper, we compute closed neighborhood degree sum-based indices for four different Graphene structures. The cardinality of closed neighborhood degree-based edge partitions for four different Graphene structures is used to compute the closed neighborhood degree sum-based indices.
Metal-organic frameworks (MOFs) are permeable substances with a high porosity volume, excellent c... more Metal-organic frameworks (MOFs) are permeable substances with a high porosity volume, excellent chemical stability, and a distinctive shape created by strong interactions between metal ions and organic ligands. Work on the synthesis, structures, and properties of numerous MOFs demonstrates their usefulness in a variety of applications, including energy storage devices with good electrode materials, gas storage, heterogeneous catalysis, and chemical assessment. The physico-chemical characteristics of the chemical compounds in the underlying molecular graph or structure are predicted by a topological index, which is a numerical invariant. In this article, we look at two different metal-organic frameworks in terms of the number of layers, as well as metal and organic ligands. We compute the reduced reverse degree-based topological indices and some closed neighbourhood degree sum-based topological indices for these frameworks.
A new class of graphs called dumbbell graphs denoted as DB (Wm,n) on 2(m+n) vertices is obtained... more A new class of graphs called dumbbell graphs denoted as DB (Wm,n) on 2(m+n) vertices is obtained by connecting m - vertices at the centres of the two generalized wheel graphs Wm,n, m ≥2, n ≥3 through m - edges. In this paper, we have extended this class of graphs to form Hyper-Dumbbell graph and also obtained its the distance spectrum, distance Laplacian spectrum and distance signless Laplacian spectrum.
Journal of Informatics and Mathematical Sciences, 2017
In this paper we give a brief overview of the adjacency matrix based page rank algorithm and eige... more In this paper we give a brief overview of the adjacency matrix based page rank algorithm and eigen vector based page rank that are used in the Google search engine. In this paper a new approach has been introduced by considering the web as a mixed graph rather than a simple graph. We propose an improved method for the computation of page rank on the basis of penalty assigned to web pages which are accessed through Advertisement links/pages. Consequently, we have applied the concept of column-stochastic Penalty Matrix to web page ranking. This approach does not involve any iterative technique. This method is based only on the concept of Eigen values and Eigen vectors of the Penalty matrix.
In the field of chemical and medical sciences, topological indices are used to study the chemical... more In the field of chemical and medical sciences, topological indices are used to study the chemical, biological, clinical, and therapeutic aspects of pharmaceuticals. The COVID-19 pandemic is largely recognized as the most life-threatening crisis confronting medical advances. Scientists have tested various antiviral drugs and discovered that they help people recover from viral infections like COVID-19. Antiviral medications, such as Arbidol, Chloroquine, Hydroxy-Chloroquine, Lopinavir, Remdesivir, Ritonavir, Thalidomide and Theaflavin, are often used to treat COVID-19. In this paper, we define Diameter Eccentricity Based vertex degree and employ it to introduce a new polynomial called $ D\varepsilon- $ Polynomial. Using the newly introduced polynomial, we derive new topological indices, namely, diameter eccentricity based and hyper diameter eccentricity based indices. In order to check the efficacy of our indices, we derive the $ D\varepsilon- $ polynomials for the eight COVID-19 drug...
Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique Kc (the ... more Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique Kc (the core) and η copies of Ks (the satellites) meeting in a common clique. They belong to the class of graphs of diameter two. Agave graphs Θ(2, 1, η) ∼= K2 ▽ (ηK1) belong to the general class of complete split graphs, where the graphs consist of a central clique K2 and η copies of K1 which are connected to all the nodes of the clique. They are the subclass of Core-satellite graphs. Let μ(G) be the spectral radius of the signless Laplacian matrix Q(G). In this paper, we have obtained the greatest lower bound and the least upper bound of signless Laplacian spectral radius of Agave graphs. These bounds have been expressed in terms of graph invariants like m the number of edges, n the number of vertices, δ the minimum degree, ∆ the maximum degree, and η copies of the satellite. We have made use of the approximation technique to derive these bounds. This unique approach can be utilized to determi...
The permeable materials known as metal–organic frameworks (MOFs) have a large porosity volume, ex... more The permeable materials known as metal–organic frameworks (MOFs) have a large porosity volume, excellent chemical stability, and a unique structure that results from the potent interactions between metal ions and organic ligands. Work on the synthesis, architectures, and properties of various MOFs reveals their utility in a variety of applications, including energy storage devices with suitable electrode materials, gas storage, heterogeneous catalysis, and chemical assessment. A topological index, which is a numerical invariant, predicts the physicochemical properties of chemical entities based on the underlying molecular graph or framework. In this article, we consider two different zinc-based MOFs, namely zinc oxide and zinc silicate MOFs. We compute 14 neighbourhood degree sum-based topological indices for these frameworks, and the numerical and graphical representations of all the aforementioned 14 indices are made.
A new class of graphs called dumbbell graphs, denoted by DB(Wm,n) is the graph obtained from two ... more A new class of graphs called dumbbell graphs, denoted by DB(Wm,n) is the graph obtained from two copies of generalized wheel graph Wm,n, m ≥ 2, n ≥ 3. It is a graph on 2 (m + n) vertices obtained by connecting m-vertices in one copy with the corresponding vertices in the other copy. The resistance distance between two vertices vi and vj, denoted by rij , is defined as the effective electrical resistance between them if each edge of G is replaced by 1 ohm resistor. The Kirchhoff index is the sum of the resistance distances between all pairs of vertices in the graph. In this paper, we formulate the resistance distance of Wm,n and DB(Wm,n) using Symmetric {1}-inverse of Laplacian matrices. We provide examples to illustrate the proposed method and also obtain the Kirchhoff indices for these examples.
Network equilibrium models are significantly distinct in supply chain networks, traffic networks,... more Network equilibrium models are significantly distinct in supply chain networks, traffic networks, and e-waste flow networks. The idea of network equilibrium is strongly perceived while determining the tuner sets of a graph (network). Tuner sets are subsets of vertices of the graph G whose degrees are lower than the average degree of G, d(G) that can compensate or balance the presence of vertices whose degrees are greater than d(G). Generalised core-satellite graph Θ c, ¯ S, ¯η = Kc ▽ (¯ηK¯ S) comprises ¯η copies of K¯ S (the satellites) meeting in Kc (the core) and it belongs to the family of graphs of diameter two. It has a central core of vertices connected to a few satellites, where all satellite cliques need not be identical and can be of different sizes. Properties like hierarchical structure of large real-world networks, are competently modeled using core-satellite graphs [1, 2, 5]. This family of graphs exhibits the properties similar to scale-free network as they possess anomalous vertex connectivity, where a small fraction of vertices (the core) are densely connected. Since these graphs possess such a structural property, interesting results are obtained for these graphs when tuner sets are determined. In this paper, we have considered the graph G = (ηKq + γKp) ▽ K1, with p > q, a subclass of the generalized core-satellite graph which is a join of η copies of the clique Kq and γ copies of the clique Kp with the core K1. We have obtained the tuner set for this subclass and established the relation between the Top T(G) and the cardinality of the tuner set |Ψ| through necessary and sufficient conditions. We analyze and characterize these graphs and obtain some interesting results while simultaneously examining the existence of tuner sets.
The properties and activities of chemical compounds can be understood by computing topological de... more The properties and activities of chemical compounds can be understood by computing topological descriptors of molecular compounds. We investigate the physical and topological aspects of crystal structure of metal-insulator transition superlattice (GST-SL) in this study. Recently, researchers have turned their attention to modifying this substance into a form that is useful for human life. Metal-insulator transition superlattices (GST-SL) are also useful as two-dimensional (2D) transition metal dichalcogenides (TMDs) in the form of thin films. For this Superlattice Network SLη, we calculate open and closed neighbourhood degree sum based on topological indices.
International Journal of Recent Technology and Engineering (IJRTE), 2019
Parallelism is a process by which a sequential string is broken down into a number of alphabets a... more Parallelism is a process by which a sequential string is broken down into a number of alphabets and used to speed up the acceptance of a string. To identify the parallelisable string, we have used parallel operator || and defined the language as parallel series languages. Algebraic and recognition properties of series parallel posets have been studied by Lodaya in [8]. In this paper, we have introduced finite and infinite parallel series language (parallel strings are connected sequentially). We have considered the set of all parallel series language as topological space and prefix order relation (poset relation) has been used to relate two parallel series strings. Topological concepts like limit, sequence, open set, closed set and basis for parallel series languages and their properties have been derived.
Topological descriptors defined on chemical structures enable understanding the properties and ac... more Topological descriptors defined on chemical structures enable understanding the properties and activities of chemical molecules. In this paper, we compute closed neighborhood degree sum-based indices for four different Graphene structures. The cardinality of closed neighborhood degree-based edge partitions for four different Graphene structures is used to compute the closed neighborhood degree sum-based indices.
Metal-organic frameworks (MOFs) are permeable substances with a high porosity volume, excellent c... more Metal-organic frameworks (MOFs) are permeable substances with a high porosity volume, excellent chemical stability, and a distinctive shape created by strong interactions between metal ions and organic ligands. Work on the synthesis, structures, and properties of numerous MOFs demonstrates their usefulness in a variety of applications, including energy storage devices with good electrode materials, gas storage, heterogeneous catalysis, and chemical assessment. The physico-chemical characteristics of the chemical compounds in the underlying molecular graph or structure are predicted by a topological index, which is a numerical invariant. In this article, we look at two different metal-organic frameworks in terms of the number of layers, as well as metal and organic ligands. We compute the reduced reverse degree-based topological indices and some closed neighbourhood degree sum-based topological indices for these frameworks.
A new class of graphs called dumbbell graphs denoted as DB (Wm,n) on 2(m+n) vertices is obtained... more A new class of graphs called dumbbell graphs denoted as DB (Wm,n) on 2(m+n) vertices is obtained by connecting m - vertices at the centres of the two generalized wheel graphs Wm,n, m ≥2, n ≥3 through m - edges. In this paper, we have extended this class of graphs to form Hyper-Dumbbell graph and also obtained its the distance spectrum, distance Laplacian spectrum and distance signless Laplacian spectrum.
Journal of Informatics and Mathematical Sciences, 2017
In this paper we give a brief overview of the adjacency matrix based page rank algorithm and eige... more In this paper we give a brief overview of the adjacency matrix based page rank algorithm and eigen vector based page rank that are used in the Google search engine. In this paper a new approach has been introduced by considering the web as a mixed graph rather than a simple graph. We propose an improved method for the computation of page rank on the basis of penalty assigned to web pages which are accessed through Advertisement links/pages. Consequently, we have applied the concept of column-stochastic Penalty Matrix to web page ranking. This approach does not involve any iterative technique. This method is based only on the concept of Eigen values and Eigen vectors of the Penalty matrix.
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