Browsing by Author "Sohail Zafar"

Now showing 1 - 7 of 7
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    Algebraic properties of the binomial edge ideal of a complete bipartite graph
    (Univ. Ovidius Constant a, 2014) Sohail Zafar; Schenzel, Peter
    Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj 􀀀 xjyi; 1 i < j n; in the polynomial ring S = K[x1; : : : ; xn; y1; : : : ; yn] where fi; jg is an edge of G. We study the arithmetic properties of S=JG for G, the complete bipartite graph. In particular we compute dimen- sions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of de ciencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of S=JG.
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    Algebraic properties of the binomial edge ideal of a complete bipartite graph
    (An. St. Univ. Ovidius Constanta, 2014) Schenzel, Peter; Sohail Zafar
    Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj 􀀀 xjyi; 1 _ i < j _ n; in the polynomial ring S = K[x1; : : : ; xn; y1; : : : ; yn] where fi; jg is an edge of G. We study the arithmetic properties of S=JG for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of decencies, the duals of local co homology modules, and prove the purity of the minimal free resolution of S=JG.
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    Edge version of harmonic index and harmonic polynomial of some classes of graphs.
    (Journal of Applied mathematics and Informatics, Lifecscience Global., 2017) Sohail Zafar; Rabia Nazir; Muhammad Shoaib Sardar; Zohaib Zahid
    In this paper we define the edge version of harmonic index and harmonic polynomial of a graph G. We computed explicit formulas for the edge version of harmonic index and harmonic polynomial of many well known classes of graphs.
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    Fundamental ideas and mathematical basis of ontology learning algorithm
    (Journal of Intelligent & Fuzzy Systems(Preprint), 2018) Zhu, Linli; Hua, Gang; Sohail Zafar; Pan, Yu
    As a data utility and aided tool, ontology has been widely used in many areas of the computer. Owing to its great efficiency, ontologies have also been introduced into various engineering disciplines. In this paper, we present the fundamental ideas of how to deal with similarity measuring problem in ontology learning algorithms. The mathematical basis of ontology learning algorithms is also introduced from a statistical learning theory point of view. Finally, we present two ontology learning algorithms in multi-dividing setting and ontology sparse vector learning setting, respectively.
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    Mixed reality shopping mall
    (UMT.Lahore, 2019) Arham Idrees; Mohammad Lareeb Anas; Sohail Zafar
    Nowadays e-commerce is a very popular platform for buying furniture. But from this platform not everyone is satisfied and happy because of no one can try furniture like how it looks at their place before buying, therefore people have to first purchase the furniture and then they have to check it at their place whether it suits their place or not, from this point of view this problem is very noticeable. The new world is adopting the concept of Augmented Reality to revolutionize the world. So, we came up with an idea that combining Augmented Reality and E-commerce will let the people to try the furniture before buying it, from this experience they will be satisfied from what they purchase and they will not regret after getting it from its suitability factor. The main objective of this is to develop Augmented Reality e-commerce which facilitates the user to purchase only suitable and accurate things to for their home, office etc. The purposed website will provide people a augmented environment to buy furniture online which will help them enhance their satisfaction about the selection of the furniture.
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    On the betti numbers of some classes of binomial edge ideals
    (the electronic journal of combinatorics, 2013) Sohail Zafar; Zohaib Zahid
    We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs depending on induced subgraphs.
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    Vertex version of co-pi index of the polycyclic aromatic hydrocarbon systems pahk.
    (International Journal of Biology, Pharmacy and Allied Sciences, IJBPAS, 2016) Sohail Zafar
    Let G be a simple connected graph having vertex set V and edge set E. The length of the smallest V(G) is called the distance, d(u,v), between the vertices u,v.path between vertices u,v Mathematical chemistry is the area of research engaged in new application of mathematics in chemistry. In mathematics chemistry, we have many topological indices for any molecular graph, that they are invariant on the graph automorphism. The length of the smallest path V(G) is called the distance, d(u,v), between the vertices u,v. For an edgebetween vertices u,v E(G), nu(e|G) represents the number of vertices of G whose distance to u is less than thee=uv distance to v in G and nv(e|G) represents the number of vertices of G whose distance to v is less than the distance to u in G. In 2010, A Iranmanesh et.al introduced the new topological indices. The Co-PIv index is the vertex version of Co-PI topological index and is defined as Co PI G n e G n e G  In this present study, we introduce a closed formula of    this new index of the Polycyclic Aromatic Hydrocarbon systems PAHk