Muhammad Ahsan Banyamin

Government College University Faisalabad | GCUF · Department of Mathematics

In this paper, we introduce the concept of [Formula: see text]-simplicial complexes by generalizing the term of [Formula: see text]-graphs (introduced in [H. Mahmood, I. Anwar and M. K. Zafar, Construction of Cohen–Macaualy [Formula: see text]-graphs, J. Algebra Appl. 13(6) (2014) 1450012]). In particular, we discuss the problem of connectedness of...

In this article we characterize the classi�cation of stably simple curve singularities given by V.I. Arnold, in terms of invariants. On the basis of this characterization we describe an implementation of a classi�er for stably simple curve singularities in the computer algebra system SINGULAR

In this article we characterize the ideal unimodular singularities in terms of their invariants. On the basis of this characterization we give an implementation of a classifier for ideal unimodular singularities in the computer algebra system SINGULAR.

In this article we characterize the classification of uni-modal parametric plane curve singularities given by Ishikawa and Janeczko, in terms of invariants. On the basis of this characterization we present an algorithm to classify the uni-modal parametric plane curve singularities and also give its implementation in computer algebra system SINGULAR...

The H-basis concept allows an investigation of multivariate polynomial spaces degree by degree. In this paper we present the analogue of Hbases for subalgebras in polynomial rings, we call them "SH-bases". We present their connection to the Sagbi basis concept, characterize SH-basis and show how to construct them.

The notion of f-graphs and f-ideals are relatively new and have been studied in many papers. In this paper, we have generalized the idea of f-graphs and f-ideals to quasi f-graphs and quasi f-ideals, respectively. We have characterized all quasi f-graphs and quasi f-ideals of degree 2 and determined all the minimal primes ideals of these ideals. Fu...

In 2011, Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a certain equisingularity class by creating "very short" parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by constructing lists of normal forms. The...

In this paper, we give sharp bounds of the Hankel determinant H2(3)(f) for the coefficients of functions in the class of starlike functions related to a domain that is like three leaves. We also give sharp bounds for the Hankel determinants H3(1)(f) and H2(3)(f) for the coefficients of functions in the class of convex functions related to the three...

The telescopic numerical semigroups are a subclass of symmetric numerical semigroups widely used in algebraic geometric codes. Suer and Ilhan gave the classification of triply generated telescopic numerical semigroups up to multiplicity 10 and by using this classification they computed some important invariants in terms of the minimal system of gen...

The F-index of a graph Q is defined as FQ=∑t∈VQdt3. In this paper, we use edge swapping transformations to find the extremal value of the F-index among the class of trees with given order, pendent vertices, and diameter. We determine the trees with given order, pendent vertices, and diameter having the greatest F-index value. Also, the first five m...

Let $$\Delta$$ be a simplicial complex on the vertex set $$V$$. For $$m=1,2,3,\dots$$, the notion of $$m$$-th $$\mathcal{NF}$$-complex of $$\Delta$$, $$\delta^{(m)}_{\mathcal{NF}}(\Delta)$$, was introduced by Hibi and Mahmood in [5], where $$\delta^{(m)}_{\mathcal{NF}}(\Delta)=\delta_{\mathcal{NF}}(\delta^{(m-1)}_{\mathcal{NF}}(\Delta))$$ with sett...

Let R be a commutative ring with unity 1 ≠ 0 . Recently Bennis et al. defined the concept of extended zero-divisor graph Γ ¯ R by considering the vertex set V Γ ¯ R = Z ∗ R and any two vertices x and y are adjacent if there exist positive integers m and n , such that x m y n = 0 with x m ≠ 0 and y n ≠ 0 . The main objective of this article is to ch...

Let \(\Lambda \) be a numerical semigroup and \(I\subset \Lambda \) be an ideal of \(\Lambda \). The graph \(G_I(\Lambda )\) assigned to an ideal I of \(\Lambda \) is a graph with elements of \((\Lambda {\setminus } I)^*\) as vertices and any two vertices x, y are adjacent if and only if \(x+y \in I\). In this paper we give a complete characterizat...

The classification and the geometry of corank one map germs from $ ({\mathbb{C}}^2, 0) \rightarrow ({\mathbb{C}}^3, 0) $ have been studied by Mond [1,2]. In this paper we characterize the classification of map germs of corank at most $ 1 $, in terms of certain invariants. Moreover, by using this characterization, we develop an algorithm to compute...

Given a finite commutative unital ring S having some non-zero elements x, y such that x.y=0, the elements of S that possess such property are called the zero divisors, denoted by ZS. We can associate a graph to S with the help of zero-divisor set ZS, denoted by ζS (called the zero-divisor graph), to study the algebraic properties of the ring S. In...

Kolgushkin and Sadykov classified the simple singularities of multigerms of curves over the field of complex numbers. We develop a similar classification over algebraically closed fields of characteristic p > 0.

Several chemical and medical experimentation reveals a dependence of physicochemical and biological properties of a compound on its molecular structure. Molecular/topological descriptors/indices retrieve this dependence by employing mathematical/statistical tools to generate quantitative structure property/activity relationship (QSAR/QSPR) models....

Mond gave the classification of simple map germs (C^2,0)------->(C^3, 0) with respect to A-equivalence. The aim of this article is to characterize Mond’s classification in terms of certain invariants. On the basis of this characterization, we present an algorithm to compute the type of the simple map germs (C^2, 0) -------->(C^3, 0) without computi...

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 pro...

The concept of H-bases, introduced long ago by Macauly, has become an important ingredient for the treatment of various problems in computational algebra. The concept of H-bases is for ideals in polynomial rings, which allows an investigation of multivariate polynomial spaces degree by degree. Similarly, we have the analogue of H-bases for subalgeb...

A graph’s entropy is a functional one, based on both the graph itself and the distribution of probability on its vertex set. In the theory of information, graph entropy has its origins. Dominating David derived networks have a variety of important applications in medication store, hardware, and system administration. In this study, we discuss domin...

For a graph , its variable sum exdeg index is defined as , where is a real number other than 1 and is the degree of a vertex . In this paper, we characterize all trees on vertices with first three maximum and first three minimum values of the index. Also, we determine all the trees of order with given diameter and having first three largest values...

Topological indices are quantitative measurements that describe a molecule’s topology and are quantified from the molecule’s graphical representation. The significance of topological indices is linked to their use in QSPR/QSAR modelling as descriptors. Mathematical associations between a particular molecular or biological activity and one or severa...

In this article, we compute the vertex Padmakar-Ivan (PIv) index, vertex Szeged (Szv) index, edge Padmakar-Ivan (PIe) index, edge Szeged (Sze) index, weighted vertex Padmakar-Ivan (wPIv) index, and weighted vertex Szeged (wSzv) index of a graph product called subdivision vertex-edge join of graphs.

A graph’s entropy is a functional one, based on both the graph itself and the distribution of probability on its vertex set. In the theory of information, graph entropy has its origins. Hex-derived networks have a variety of important applications in medication store, hardware, and system administration. In this article, we discuss hex-derived netw...

A complete classification of simple function germs with respect to Lipschitz equivalence over the field of complex numbers was given by Nguyen et al. The aim of this article is to implement a classifier in terms of easy computable invariants to compute the type of the Lipschitz simple function germs without computing the normal form in the computer...

Structure-based topological descriptors of chemical networks enable us the prediction of physico-chemical properties and the bioactivities of compounds through QSAR/QSPR methods. Topological indices are the numerical values to represent a graph which characterises the graph. One of the latest distance-based topological index is the Mostar index. In...

Let be a numerical semigroup and be an irreducible ideal of . The graph assigned to an ideal of is a graph with elements of as vertices, and any two vertices are adjacent if and only if . In this work, we give a complete characterization (up to isomorphism) of the graph having metric dimension 2. 1. Introduction In algebraic combinatorics, the stu...

In this paper, we determine the efficiency of all commonly occurring eigenvalues‐based topological descriptors for measuring the π‐electronic energy of lower polycyclic aromatic hydrocarbons. Results show some favorable outcomes as the spectrum‐based descriptors such as the adjacency energy, the arithmetic‐geometric energy, the geometric‐arithmetic...

In quantitative structure property (QSPR) and quantitative structure activity (QSAR) relationships studies, topological indices are utilized to associate the biological activity of underline structures with their physical properties like strain energy, distortion, melting point, boiling point, and stability, etc. In these studies, the degree based...

The notion of quasi $f$-ideals was first presented in $[14]$ which generalize the idea of $f$-ideals. In this paper, we give the complete characterization of quasi $f$-ideals of degree greater or equal to $2$. Additionally, we show that the property of being quasi $f$-ideals remains the same after taking the Newton complementary dual of a squarefre...

Let $\Lambda$ be a numerical semigroup and $I\subset \Lambda$ be an ideal of $\Lambda$. The graph $G_I(\Lambda)$ assigned to an ideal $I$ of $\Lambda$ is a graph with elements of $(\Lambda \setminus I)^*$ as vertices and any two vertices $x,y$ are adjacent if and only if $x+y \in I$. In this paper we give a complete characterization (up to isomorph...

In this article, we correct the classification of unimodal map germs from the plane to the plane of Boardman symbol (2, 2) given by Dimca and Gibson. Also, we characterize this classification of unimodal map germs in terms of certain invariants. Moreover, on the basis of this characterization we present an algorithm to compute the type of unimodal...

In theoretical chemistry, the numerical parameters that are used to characterize the molecular topology of graphs are called topological indices. Several physical and chemical properties like boiling point, entropy, heat formation, and vaporization enthalpy of chemical compounds can be determined through these topological indices. Graph theory has...

Topological indices are numerical numbers that represent the topology of a molecule and are calculated from the graphical depiction of the molecule. The importance of topological indices is due to their use as descriptors in QSPR/QSAR modeling. QSPRs (quantitative structure-property relationships) and QSARs (quantitative structure-activity relation...

In this paper, we propose the concept of anti-intuitionistic fuzzy sets, anti − intuitionistic fuzzy subgroups and prove some of their algebraic properties. We investigate a necessary and sufficient condition for a-anti intuitionistic fuzzy set to be a-anti intuitionistic fuzzy subgroup. We extend this ideology by defining the notions of anti-intui...

The notion of $f$-ideal is recent and has so far been studied in several papers. In \cite{qfi}, the idea of $f$-ideal is generalized to quasi $f$-ideals, which is much larger class than the class of $f$-ideals. In this paper, we introduce the concept of quasi $f$-simplicial complex and quasi $f$-graph. We give a characterization of quasi $f$-graphs...

The notion off-ideals is recent and has been studied in the papers[1] [2], [5], [10], [11], [12], [13], [14] and [15]. In this paper, we have generalized the idea off-ideals to quasi f-ideals. This extended class of ideals is much bigger than the class of all f-ideals. Apart from giving various characterizations of quasi f-ideals of degree 2, we ha...

Let be an ideal of a numerical semigroup . We define an undirected graph with vertex set and edge set . The aim of this article is to discuss the connectedness, girth, completeness, and some other related properties of the graph . 1. Introduction and Preliminaries In mathematics, graph theory plays an important role in understanding the relationsh...

In this paper, we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated with classes of graphs obtained by the strong product of two graphs. We consider the cases when either both graphs are arbitrary paths or one is an arbitrary path and the other is an arbitrary cycle. We give exact formula for values...

The classification of right unimodal and bimodal hypersurface singularities over a field of positive characteristic was given by H. D. Nguyen. The classification is described in the style of Arnold and not in an algorithmic way. This classification was characterized in [5] for the case when the corank of hypersurface singularities is ≤ 2. The aim o...

Dimca and Gibson gave the classification of all map germs from the plane to the plane of the Boardman symbols (2, 1) and (2, 2) with respect to K-equivalence. The aim of this article is to characterize the map germs from the plane to the plane of the Boardman symbol (2, 1) with respect to K-equivalence by means of easy computable invariants such as...

The notion of [Formula: see text]-ideal was introduced in [G. Q. Abbasi, S. Ahmad, I. Anwar and W. A. Baig, [Formula: see text]-Ideals of degree 2, Algebra Colloq. 19(1) (2012) 921–926] in 2012 and has been studied in many papers after that. In this paper, we have studied those graphs whose Stanley–Reisner ideals turn out to be [Formula: see text]-...

In this paper we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated to classes of graphs obtained by taking the strong product of two graphs. We consider the cases when either both graphs are arbitrary paths or one is an arbitrary path and the other is an arbitrary cycle. We give exact formulae for va...

We use the results of Hefez and Hernandez [ 7 ] and Mehmood and Pfis-ter [ 9, 10 ] to give a classification of bimodal germs of parametrized plane curves singularities of (C, 0) −→ (C 2 , 0) with respect to A-equivalence. The results can be extended over the real numbers in a similar way.

There are numeric numbers that define chemical descriptors that represent the entire structure of a graph, which contain a basic chemical structure. Of these, the main factors of topological indices are such that they are related to different physical chemical properties of primary chemical compounds. The biological activity of chemical compounds c...

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 ) .

In chemical graph theory, a topological index is a numerical representation of a chemical network while a topological descriptor correlates certain physico-chemical characteristics of underlying chemical compounds besides its chemical representation. Graph plays an vital role in modeling and designing any chemical network. F. Simonraj et al. derive...

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical network. Simonraj et al. derive...

Recently, increasing attention has been paid to The Optical Transpose Interconnection System (OTIS) network because of its prospective applications in architectures for parallel as well as distributed systems [27, 28]. Different interconnection networks in the context of topological indices are researched recently in [25, 26]. This article includes...

The present-day trend of the numerical coding of chemical structures with topological indices (TIs) has established quite successful in Medicinal Chemistry and Bioinformatics. This strategy provides the annotation, comparison, rapid collection, mining, and retrieval of chemical structures within large databases. Afterwards, TIs can be used to look...

In this article, we develop the theory of SAGBI bases in G-algebras and create a criterion through which we can check if a set of polynomials in a G-algebra is a SAGBI basis or not. Moreover, we will construct an algorithm to compute SAGBI bases from a subset of polynomials contained in a subalgebra of a G-algebra.

In this paper, we characterize the classification of unimodal maps from the plane to the plane with respect to [Formula: see text]-equivalence given by Rieger in terms of invariants. We recall the classification over an algebraically closed field of characteristic [Formula: see text]. On the basis of this characterization, we present an algorithm t...

The complete classification of right unimodal and bimodal hypersurface singular- ities over a field of positive characteristic was given by H. D. Nguyen in form of a classifier, which allows the concrete classification from the given equation in a step by step procedure. The aim of this article is to characterize right unimodal and bimodal hypersur...

In this article we describe the classification of the resolution graphs of weighted homogeneous plane curve singularities in terms of their weights by using the concepts of graph theory and combinatorics. The classification shows that the resolution graph of a weighted homogeneous plane curve singularity is always a caterpillar.

In this article we characterize the classification of simple maps from the plane to the plane given by Rieger, J.H. in terms of invariants. On the basis of this characterization we present an algorithm to classify the simple maps from the plane to the plane and also give its implementation in computer algebra system SINGULAR.

The aim of the article is to describe the classification of simple iso-lated hypersurface sin-gularities over a field of positive characteristic by certain invariants without computing the normal form. We also give a description of the algorithms to compute the classification which we have implemented in the Singular libraries classifyCeq.lib and c...

In this paper, we give the complete characterization of f-ideals of degree d greater or equal to 2.

In this article we present an algorithm to compute the incidence matrix of the resolution graph, the total multiplicities, the strict multiplicities and the Milnor number of a reduced plane curve singularity and its implemetation in Singular.

In this article we describe our experiences with a parallel SINGULAR-implementation of the signature of a surface singularity defined by z^N+g(x,y)=0.