Department of Mathematics

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    On the Structure of the Commuting Graph of Brandt Semigroups
    (Springer, 2021) Kumar, Jitender
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    On the Commuting Graph of Semidihedral Group
    (Springer, 2021-04) Kumar, Jitender
    The commuting graph Δ(G) of a finite non-abelian group G is a simple graph with vertex set G, and two distinct vertices x, y are adjacent if xy=yx. In this paper, first we discuss some properties of Δ(G). We determine the edge connectivity and the minimum degree of Δ(G) and prove that both are equal. Then, other graph invariants, namely: matching number, clique number, boundary vertex, of Δ(G) are studied. Also, we give necessary and sufficient condition on the group G such that the interior and center of Δ(G) are equal. Further, we investigate the commuting graph of the semidihedral group SD8n. In this connection, we discuss various graph invariants of Δ(SD8n) including vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of Δ(SD8n).
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    On the inclusion ideal graph of semigroups
    (ARXIV, 2021-10) Kumar, Jitender
    The inclusion ideal graph In(S) of a semigroup S is an undirected simple graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I,J are adjacent if and only if either I⊂J or J⊂I. The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of In(S). In this paper, we investigate the connectedness of In(S). We show that diameter of In(S) is at most 3 if it is connected. We also obtain a necessary and sufficient condition of S such that the clique number of In(S) is n, where n is the number of minimal left ideals of S. Further, various graph invariants of In(S) viz. perfectness, planarity, girth etc. are discussed. For a completely simple semigroup S, we investigate various properties of In(S) including its independence number and matching number. Finally, we obtain the automorphism group of In(S).
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    On the intersection ideal graph of semigroups
    (ARXIV, 2022-01) Kumar, Jitender
    The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I,J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected then diam(Γ(S))≤2. Further we classify the semigroups such that the diameter of their intersection graph is two. Other graph invariants, namely perfectness, planarity, girth, dominance number, clique number, independence number etc. are also discussed. Finally, if S is union of n minimal left ideals then we obtain the automorphism group of Γ(S).
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    A Graph Theoretical Approach for Creating Building Floor Plans
    (Springer, 2019) Shekhawat, Krishnendra
    Existing floor planning algorithms are mostly limited to rectangular room geometries. This restriction is a significant reason why they are not used much in design practice. To address this issue, we propose an algorithm (based on graph theoretic tools) that generates rectangular and, if required, orthogonal floor plans while satisfying the given adjacency requirements. If a floor plan does not exist for the given adjacency requirements, we introduce circulations within a floor plan to have a required floor plan.
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    Construction of architectural floor plans for given adjacency requirements
    (CAADRIA, 2020) Shekhawat, Krishnendra
    For most of the architectural design problems, there are underlying mathematical sub-problems, they may require to consider for generating architectural layouts. One of these sub-problems is to satisfy adjacency constraints for obtaining an initial layout. But in the literature, there does not exist a mathematical procedure that can address any given adjacency requirements, i.e., there does not exist a tool for generating a floor plan corresponding to any given adjacency (planar) graph (there exist algorithms for constructing floor plans for planar triangulated graphs only). In this paper, we are going to present an algorithm that would generate a floor plan corresponding to any given planar graph. The larger aim of this research is to develop a user-friendly tool that can generate a variety of initial layouts corresponding to a given graph, which can be further modified by the architects/designers.
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    A Graph Theoretic Approach for the Automated Generation of Dimensioned Floorplans
    (CAADRIA, 2021) Shekhawat, Krishnendra
    The automated generation of architectural layouts is an intensively studied research area where the aim is to generate a variety of (initial) layouts for the given constraints which can be further modified by designers and architects. From a mathematical perspective, one of the well-known constraints is given in the form of an adjacency graph which represents the adjacency relations of the given rooms and problem is to generate multiple layouts satisfying the adjacency relations. In the literature, the adjacency graph is usually taken as a bi-connected planar triangular graph. In this paper, we present the results of a prototype GPLAN that generates multiple dimensioned layouts for any given planar graph. The larger aim of this work is to develop software that can produce a variety of architecturally acceptable floorplans corresponding to the given constraints.
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    Rectangular floorplans with block symmetry
    (CUMINCAD, 2023) Shekhawat, Krishnendra
    The principle of symmetry has beneficial applications in architecture. Symmetry mainly creates order and equilibrium in complex designs. This study presents a graph theoretic approach for the automatic generation of rectangular floorplans with block symmetry. Existing graph theoretical approaches focus on the floorplan’s outer boundary design, different room shapes, and spatial arrangements. This paper introduces block symmetry as a new concept in floorplan generation. Based on this concept, an algorithm is proposed for generating a rectangular floorplan with rectangular blocks for a given adjacency graph if one exists. Further, suppose two blocks are required to be symmetric, i.e., of equal size. In that case, an optimisation framework is used to equate the widths and heights of the blocks, resulting in the generation of a rectangular floorplan with block symmetry. A GUI is provided for users to perform the automatic generation of floorplans.
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    Enumerating generic rectangular floor plans
    (Elsevier, 2018-08) Shekhawat, Krishnendra
    A rectangular floor plan (RFP) is a floor plan in which plan's boundary and each room is a rectangle. The problem is to construct a RFP for the given adjacency requirements, if it exists. In this paper, we aim to present a generic solution to the above problem by enumerating a set of RFP that topologically contain all possible RFP. This set of RFP is called generic rectangular floor plans (GRFP). Furthermore, the construction of GRFP leads us to the necessary condition for the existence of a RFP corresponding to a given graph.
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    A tool for computer-generated dimensioned floorplans based on given adjacencies
    (Elsevier, 2021-07) Shekhawat, Krishnendra
    In this paper, we present GPLAN, a software aimed at constructing dimensioned floorplan layouts using graph-theoretical and optimization techniques. For GPLAN, the adjacency requirements are given in the following two forms: i. An adjacency graph: It allow users to draw an adjacency graph on a GUI (graphical user interface) corresponding to which GPLAN produces a set of dimensioned floorplans with a rectangular boundary, where each floorplan is topologically distinct from others. ii. A dimensionless layout: Here, users/designers can draw any layout with a rectangular or a non-rectangular boundary on a GUI and GPLAN transforms it into a dimensioned floorplan while preserving the adjacencies, positions, shapes of the rooms. The above approaches represent different ways of inserting adjacencies and GPLAN generate dimensioned floorplans corresponding to the given adjacencies. The larger aim of this work is to provide alternative platforms to users/designers for producing dimensioned floorplans for all given (architectural) constraints, which can be further refined by architects.