Department of Mathematics

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Author: search.filters.author.Kumar, JitenderSubject: search.filters.subject.Graph Theory

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Now showing 1 - 4 of 4
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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).