Determining the degree of irregularity of a certain molecular structure or a network has been a k... more Determining the degree of irregularity of a certain molecular structure or a network has been a key source of interest for molecular topologists, but it is also important as it provides an insight into the key features used to guess properties of the structures. In this article, we are interested in formulating closed forms of irregularity measures of some popular benzenoid systems, such as hourglass H (m, n), jagged-rectangular J (m, n), and triangular benzenoid T (m, n) systems. We also compared our results graphically and concluded which benzenoid system among the above listed is more irregular than the others.
In this note we classify all homogeneous spaces G/H admitting a G-invariant G 2-structure, assumi... more In this note we classify all homogeneous spaces G/H admitting a G-invariant G 2-structure, assuming that G is a connected compact Lie group and G acts effectively on G/H. They include a subclass of all homogeneous spaces G/H with a G-invariantG 2-structure, where G is a compact Lie group. There are many new examples with nontrivial fundamental group. We study a subclass of homogeneous spaces of high rigidity and low rigidity and show that they admit families of invariant coclosed G 2-structures (resp.G 2-structures). Contents 1. Introduction 1 2. The groupsG 2 and G 2 3 2.1. The groupG 2 and its maximal compact subgroup SO(4) 3 2.2. The group G 2 and its maximal compact subgroups 5 3. Compact homogeneous manifolds admitting invariantG 2-structures 6 3.1. Reduction to a representation problem 6 3.2. Classification theorem 17 4. Compact homogeneous manifolds admitting invariant G 2-structures 18 4.1. Reduction to a representation problem 19 4.2. Classification theorem 20 5. Spaces G/H with high rigidity or with low rigidity 20 5.1. Spaces G/H with d 3 (G/H) = 1 21 5.2. Spaces G/H with d 3 (G/H) = 2 21 5.3. Spaces with d 3 (G/H) = 35 22 References 24
In this note we classify homogeneous spaces $G/H$ admitting a $G$-invariant $G_2$-structure, assu... more In this note we classify homogeneous spaces $G/H$ admitting a $G$-invariant $G_2$-structure, assuming that $G$ is a compact Lie group and $G$ acts effectively on $G/H$. They include a subclass of all homogeneous spaces $G/H$ with a $G$-invariant $\tilde G_2$-structure, where $G$ is a compact Lie group. There are many new examples with nontrivial fundamental group. We study a subclass of homogeneous spaces of high rigidity and show that they admit a family of invariant coclosed $G_2$-structures and invariant coclosed $\tilde G_2$-structures. We use them to construct new metrics of $Spin (3,4)$-holonomy.
Scientific organizations are creating carbon nanotube-based composites like VC 5 C 7 [m, n] and H... more Scientific organizations are creating carbon nanotube-based composites like VC 5 C 7 [m, n] and HC 5 C 7 [m, n], which indicate significant response against voltage. Imbalance-based irregularity indices determine the degree of irregularity of a certain molecular structure and, as a result, determine the properties of a molecular substance. In this article, we aim to compute irregularity indices of two nanotubes, VC 5 C 7 [m, n] and HC 5 C 7 [m, n]. We produce formulas for the irregularity of these two nanotubes, which are functions depending on the parameters of the structure m and n. We compare our results graphically and conclude that VC 5 C 7 [m, n] is more irregular than HC 5 C 7 [m, n]. KEYWORDS irregularity index, VC 5 C 7 [m,n] and HC 5 C 7 [m,n], nanotubes, imbalance-based irregularity measure, molecular computing
Molecular topology provides a basis for the correlation of physical as well as chemical propertie... more Molecular topology provides a basis for the correlation of physical as well as chemical properties of a certain molecule. Irregularity indices are used as functions in the statistical analysis of the topological properties of certain molecular graphs and complex networks, and hence help us to correlate properties like enthalpy, heats of vaporization, and boiling points etc. with the molecular structure. In this article we are interested in formulating closed forms of imbalance-based irregularity measures of boron nanotubes. These tubes are known as-boron nanotube, triangular boron nanotubes, and tri-hexagonal boron nanotubes. We also compare our results graphically and come up with the conclusion that alpha boron tubes are the most irregular with respect to most of the irregularity indices.
The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest ab... more The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.
Journal of Intelligent and Fuzzy Systems, Jan 4, 2022
The largest absolute eigenvalue of a matrix A associated to the graph G is called the A-Spectral ... more The largest absolute eigenvalue of a matrix A associated to the graph G is called the A-Spectral Radius of the graph G, and A-energy of the graph G is defined as the absolute sum of all its eigenvalues. In the present article, we compute Randic energies, Reciprocal Randic energies, Randic spectral radii and Reciprocal Randic radii of s-shadow and s-splitting graph of G. We actually relate these energies and Spectral Radii of new graphs with the energies and Spectral Radii of original graphs.
In this study, we first introduce polygonal cylinder and torus using Cartesian products and topol... more In this study, we first introduce polygonal cylinder and torus using Cartesian products and topologically identifications and then find their Wiener and hyper-Wiener indices using a quick, interesting technique of counting. Our suggested mathematical structures could be of potential interests in representation of computer networks and enhancing lattice hardware security.
The concept of resolving set and metric basis has been very successful because of multi-purpose a... more The concept of resolving set and metric basis has been very successful because of multi-purpose applications both in computer and mathematical sciences. A system in which failure of any single unit, another chain of units not containing the faulty unit can replace the originally used chain is called a fault-tolerant self-stable system. Recent research studies reveal that the problem of finding metric dimension is NP-hard for general graphs and the problem of computing the exact values of fault-tolerant metric dimension seems to be even harder although some bounds can be computed rather easily. In this article, we compute closed formulas for the fault-tolerant metric dimension of lattices of two types of boron nanotubes, namely triangular and alpha boron. These lattices are formed by cutting the tubes vertically. We conclude that both tubes have constant fault tolerance metric dimension 4.
The present article presents some new results relating to Atomic Bond Connectivity energies and S... more The present article presents some new results relating to Atomic Bond Connectivity energies and Spectral radii of generalized splitting and generalized shadow graphs constructed on the basis of some fundamental families of cycle graph Cn, complete graph Kn and complete bipartite graph Kn,n referred as base graphs. In fact we relate the energies and Spectral radii of splitting and shadow graphs with the energies and Spectral radii of original graphs.
Lie symmetry analysis (LSA) is one of the most common, effective, and estimation-free methods to ... more Lie symmetry analysis (LSA) is one of the most common, effective, and estimation-free methods to find the symmetries and solutions of the differential equations (DEs) by following an algorithm. This analysis leads to reduce the order of partial differential equations (PDEs). Many physical problems are converted into non-linear DEs and these DEs or system of DEs are then solved with several methods such as similarity methods, Lie Bäcklund transformation, and Lie group of transformations. LSA is suitable for providing the conservation laws corresponding to Lie point symmetries or Lie Bäcklund symmetries. Short pulse equation (SPE) is a non-linear PDE, used in optical fibers, computer graphics, and physical systems and has been generalized in many directions. We will find the symmetries and a class of solutions depending on one-parameter (ε) obtained from Lie symmetry groups. Then we will construct the optimal system for the Lie algebra and invariant solutions (called similarity soluti...
New results relating to the maximum and minimum degree spectral radii of generalized splitting an... more New results relating to the maximum and minimum degree spectral radii of generalized splitting and shadow graphs have been constructed on the basis of any regular graph, referred as base graph. In particular, we establish the relations of extreme degree spectral radii of generalized splitting and shadow graphs of any regular graph.
We give the general form of the Tutte polynomial of a family of positive-signed connected planar ... more We give the general form of the Tutte polynomial of a family of positive-signed connected planar graphs. Then we give general formulas of the Jones polynomial of some very interesting families of alternating knots and links that correspond to these planar graphs. We also give general forms of the flow, reliability, and chromatic polynomials of these graphs. Finally, we give some useful combinatorial information by evaluating the Tutte polynomial at some special points.
Around 1880, Lie introduced an idea of invariance of the partial differential equation (PDE) unde... more Around 1880, Lie introduced an idea of invariance of the partial differential equation (PDE) under one-parameter Lie group of transformation to find the invariant, similarity, or auto-model solutions. Lie symmetry analysis (LSA) provides us an algorithm to search for point symmetries for solving related linear systems for infinitesimal generators. Actually, point symmetries lead us to one-parameter family of solutions from a known solution. LSA is a program that provides us the exact solutions for the non-linear differential equations (DEs) in analogy of the program designed by Galois for algebraic polynomial equations. In this paper, we have carried out the LSA for computing the similarity solutions (symmetries) of the non-linear short pulse equation (SPE) for the cases when h(u) = e u , k(u) = u xx , h(u) e u n , and k(u) = u xx. In addition, an optimal system of one-dimensional subalgebra has been set up. The reductions and invariant solutions for the generalized SPE are calculated corresponding to this optimal system as well. Reductions reduce the non-linear PDE or system of PDEs into non-linear reduced ordered ODE or system of PDEs. This helps to solve these systems of PDEs to reduced form. Graphical behavior of the transformed points of the 1-parameter solution functions have drawn.
Spectra of network related graphs have numerous applications in computer sciences, electrical net... more Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius.
Spectra of network related graphs have numerous applications in computer sciences, electrical net... more Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius.
Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 el... more Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph's spectral radius represents the adjacency matrix's largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article's main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph.
Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast ... more Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast to reduce the order and non-linearity of the equation. The present article focuses on the solution of Generalized Equal Width wave (GEW) equation using Lie group theory. Over the years, different solution methods have been tried for GEW but Lie symmetry analysis has not been done yet. At first, we obtain the infinitesimal generators, commutation table and adjoint table of Generalized Equal Width wave (GEW) equation. After this, we find the one dimensional optimal system. Then we reduce GEW equation into non-linear ordinary differential equation (ODE) by using the Lie symmetry method. This transformed equation can take us to the solution of GEW equation by different methods. After this, we get the travelling wave solution of GEW equation by using the Sine-cosine method. We also give graphs of some solutions of this equation.
Hilbert series is a simplest way to calculate the dimension and the degree of an algebraic variet... more Hilbert series is a simplest way to calculate the dimension and the degree of an algebraic variety by an explicit polynomial equation. The mixed braid group $ B_{m, n} $ is a subgroup of the Artin braid group $ B_{m+n} $. In this paper we find the ambiguity-free presentation and the Hilbert series of canonical words of mixed braid monoid $ M\!B_{2, 2} $.
Determining the degree of irregularity of a certain molecular structure or a network has been a k... more Determining the degree of irregularity of a certain molecular structure or a network has been a key source of interest for molecular topologists, but it is also important as it provides an insight into the key features used to guess properties of the structures. In this article, we are interested in formulating closed forms of irregularity measures of some popular benzenoid systems, such as hourglass H (m, n), jagged-rectangular J (m, n), and triangular benzenoid T (m, n) systems. We also compared our results graphically and concluded which benzenoid system among the above listed is more irregular than the others.
In this note we classify all homogeneous spaces G/H admitting a G-invariant G 2-structure, assumi... more In this note we classify all homogeneous spaces G/H admitting a G-invariant G 2-structure, assuming that G is a connected compact Lie group and G acts effectively on G/H. They include a subclass of all homogeneous spaces G/H with a G-invariantG 2-structure, where G is a compact Lie group. There are many new examples with nontrivial fundamental group. We study a subclass of homogeneous spaces of high rigidity and low rigidity and show that they admit families of invariant coclosed G 2-structures (resp.G 2-structures). Contents 1. Introduction 1 2. The groupsG 2 and G 2 3 2.1. The groupG 2 and its maximal compact subgroup SO(4) 3 2.2. The group G 2 and its maximal compact subgroups 5 3. Compact homogeneous manifolds admitting invariantG 2-structures 6 3.1. Reduction to a representation problem 6 3.2. Classification theorem 17 4. Compact homogeneous manifolds admitting invariant G 2-structures 18 4.1. Reduction to a representation problem 19 4.2. Classification theorem 20 5. Spaces G/H with high rigidity or with low rigidity 20 5.1. Spaces G/H with d 3 (G/H) = 1 21 5.2. Spaces G/H with d 3 (G/H) = 2 21 5.3. Spaces with d 3 (G/H) = 35 22 References 24
In this note we classify homogeneous spaces $G/H$ admitting a $G$-invariant $G_2$-structure, assu... more In this note we classify homogeneous spaces $G/H$ admitting a $G$-invariant $G_2$-structure, assuming that $G$ is a compact Lie group and $G$ acts effectively on $G/H$. They include a subclass of all homogeneous spaces $G/H$ with a $G$-invariant $\tilde G_2$-structure, where $G$ is a compact Lie group. There are many new examples with nontrivial fundamental group. We study a subclass of homogeneous spaces of high rigidity and show that they admit a family of invariant coclosed $G_2$-structures and invariant coclosed $\tilde G_2$-structures. We use them to construct new metrics of $Spin (3,4)$-holonomy.
Scientific organizations are creating carbon nanotube-based composites like VC 5 C 7 [m, n] and H... more Scientific organizations are creating carbon nanotube-based composites like VC 5 C 7 [m, n] and HC 5 C 7 [m, n], which indicate significant response against voltage. Imbalance-based irregularity indices determine the degree of irregularity of a certain molecular structure and, as a result, determine the properties of a molecular substance. In this article, we aim to compute irregularity indices of two nanotubes, VC 5 C 7 [m, n] and HC 5 C 7 [m, n]. We produce formulas for the irregularity of these two nanotubes, which are functions depending on the parameters of the structure m and n. We compare our results graphically and conclude that VC 5 C 7 [m, n] is more irregular than HC 5 C 7 [m, n]. KEYWORDS irregularity index, VC 5 C 7 [m,n] and HC 5 C 7 [m,n], nanotubes, imbalance-based irregularity measure, molecular computing
Molecular topology provides a basis for the correlation of physical as well as chemical propertie... more Molecular topology provides a basis for the correlation of physical as well as chemical properties of a certain molecule. Irregularity indices are used as functions in the statistical analysis of the topological properties of certain molecular graphs and complex networks, and hence help us to correlate properties like enthalpy, heats of vaporization, and boiling points etc. with the molecular structure. In this article we are interested in formulating closed forms of imbalance-based irregularity measures of boron nanotubes. These tubes are known as-boron nanotube, triangular boron nanotubes, and tri-hexagonal boron nanotubes. We also compare our results graphically and come up with the conclusion that alpha boron tubes are the most irregular with respect to most of the irregularity indices.
The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest ab... more The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.
Journal of Intelligent and Fuzzy Systems, Jan 4, 2022
The largest absolute eigenvalue of a matrix A associated to the graph G is called the A-Spectral ... more The largest absolute eigenvalue of a matrix A associated to the graph G is called the A-Spectral Radius of the graph G, and A-energy of the graph G is defined as the absolute sum of all its eigenvalues. In the present article, we compute Randic energies, Reciprocal Randic energies, Randic spectral radii and Reciprocal Randic radii of s-shadow and s-splitting graph of G. We actually relate these energies and Spectral Radii of new graphs with the energies and Spectral Radii of original graphs.
In this study, we first introduce polygonal cylinder and torus using Cartesian products and topol... more In this study, we first introduce polygonal cylinder and torus using Cartesian products and topologically identifications and then find their Wiener and hyper-Wiener indices using a quick, interesting technique of counting. Our suggested mathematical structures could be of potential interests in representation of computer networks and enhancing lattice hardware security.
The concept of resolving set and metric basis has been very successful because of multi-purpose a... more The concept of resolving set and metric basis has been very successful because of multi-purpose applications both in computer and mathematical sciences. A system in which failure of any single unit, another chain of units not containing the faulty unit can replace the originally used chain is called a fault-tolerant self-stable system. Recent research studies reveal that the problem of finding metric dimension is NP-hard for general graphs and the problem of computing the exact values of fault-tolerant metric dimension seems to be even harder although some bounds can be computed rather easily. In this article, we compute closed formulas for the fault-tolerant metric dimension of lattices of two types of boron nanotubes, namely triangular and alpha boron. These lattices are formed by cutting the tubes vertically. We conclude that both tubes have constant fault tolerance metric dimension 4.
The present article presents some new results relating to Atomic Bond Connectivity energies and S... more The present article presents some new results relating to Atomic Bond Connectivity energies and Spectral radii of generalized splitting and generalized shadow graphs constructed on the basis of some fundamental families of cycle graph Cn, complete graph Kn and complete bipartite graph Kn,n referred as base graphs. In fact we relate the energies and Spectral radii of splitting and shadow graphs with the energies and Spectral radii of original graphs.
Lie symmetry analysis (LSA) is one of the most common, effective, and estimation-free methods to ... more Lie symmetry analysis (LSA) is one of the most common, effective, and estimation-free methods to find the symmetries and solutions of the differential equations (DEs) by following an algorithm. This analysis leads to reduce the order of partial differential equations (PDEs). Many physical problems are converted into non-linear DEs and these DEs or system of DEs are then solved with several methods such as similarity methods, Lie Bäcklund transformation, and Lie group of transformations. LSA is suitable for providing the conservation laws corresponding to Lie point symmetries or Lie Bäcklund symmetries. Short pulse equation (SPE) is a non-linear PDE, used in optical fibers, computer graphics, and physical systems and has been generalized in many directions. We will find the symmetries and a class of solutions depending on one-parameter (ε) obtained from Lie symmetry groups. Then we will construct the optimal system for the Lie algebra and invariant solutions (called similarity soluti...
New results relating to the maximum and minimum degree spectral radii of generalized splitting an... more New results relating to the maximum and minimum degree spectral radii of generalized splitting and shadow graphs have been constructed on the basis of any regular graph, referred as base graph. In particular, we establish the relations of extreme degree spectral radii of generalized splitting and shadow graphs of any regular graph.
We give the general form of the Tutte polynomial of a family of positive-signed connected planar ... more We give the general form of the Tutte polynomial of a family of positive-signed connected planar graphs. Then we give general formulas of the Jones polynomial of some very interesting families of alternating knots and links that correspond to these planar graphs. We also give general forms of the flow, reliability, and chromatic polynomials of these graphs. Finally, we give some useful combinatorial information by evaluating the Tutte polynomial at some special points.
Around 1880, Lie introduced an idea of invariance of the partial differential equation (PDE) unde... more Around 1880, Lie introduced an idea of invariance of the partial differential equation (PDE) under one-parameter Lie group of transformation to find the invariant, similarity, or auto-model solutions. Lie symmetry analysis (LSA) provides us an algorithm to search for point symmetries for solving related linear systems for infinitesimal generators. Actually, point symmetries lead us to one-parameter family of solutions from a known solution. LSA is a program that provides us the exact solutions for the non-linear differential equations (DEs) in analogy of the program designed by Galois for algebraic polynomial equations. In this paper, we have carried out the LSA for computing the similarity solutions (symmetries) of the non-linear short pulse equation (SPE) for the cases when h(u) = e u , k(u) = u xx , h(u) e u n , and k(u) = u xx. In addition, an optimal system of one-dimensional subalgebra has been set up. The reductions and invariant solutions for the generalized SPE are calculated corresponding to this optimal system as well. Reductions reduce the non-linear PDE or system of PDEs into non-linear reduced ordered ODE or system of PDEs. This helps to solve these systems of PDEs to reduced form. Graphical behavior of the transformed points of the 1-parameter solution functions have drawn.
Spectra of network related graphs have numerous applications in computer sciences, electrical net... more Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius.
Spectra of network related graphs have numerous applications in computer sciences, electrical net... more Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number of spanning trees, graph energies and spectral radius.
Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 el... more Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph's spectral radius represents the adjacency matrix's largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article's main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph.
Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast ... more Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast to reduce the order and non-linearity of the equation. The present article focuses on the solution of Generalized Equal Width wave (GEW) equation using Lie group theory. Over the years, different solution methods have been tried for GEW but Lie symmetry analysis has not been done yet. At first, we obtain the infinitesimal generators, commutation table and adjoint table of Generalized Equal Width wave (GEW) equation. After this, we find the one dimensional optimal system. Then we reduce GEW equation into non-linear ordinary differential equation (ODE) by using the Lie symmetry method. This transformed equation can take us to the solution of GEW equation by different methods. After this, we get the travelling wave solution of GEW equation by using the Sine-cosine method. We also give graphs of some solutions of this equation.
Hilbert series is a simplest way to calculate the dimension and the degree of an algebraic variet... more Hilbert series is a simplest way to calculate the dimension and the degree of an algebraic variety by an explicit polynomial equation. The mixed braid group $ B_{m, n} $ is a subgroup of the Artin braid group $ B_{m+n} $. In this paper we find the ambiguity-free presentation and the Hilbert series of canonical words of mixed braid monoid $ M\!B_{2, 2} $.
Uploads
Papers by mobeen munir