Department of Mathematics
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Item Lambda number of the enhanced power graph of a finite group(2022-08) Kumar, JitenderThe enhanced power graph of a finite group G is the simple undirected graph whose vertex set is G and two distinct vertices x,y are adjacent if x,y∈⟨z⟩ for some z∈G. An L(2,1)-labeling of graph Γ is an integer labeling of V(Γ) such that adjacent vertices have labels that differ by at least 2 and vertices distance 2 apart have labels that differ by at least 1. The λ-number of Γ, denoted by λ(Γ), is the minimum range over all L(2,1)-labelings. In this article, we study the lambda number of the enhanced power graph PE(G) of the group G. This paper extends the corresponding results, obtained in [22], of the lambda number of power graphs to enhanced power graphs. Moreover, for a non-trivial simple group G of order n, we prove that λ(PE(G))=n if and only if G is not a cyclic group of order n≥3. Finally, we compute the exact value of λ(PE(G)) if G is a finite nilpotent group.Item On the difference graph of power graphs of finite groups(Taylor & Francis, 2023-11) Kumar, JitenderThe power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.Item Affine Near-Semirings Over Brandt Semigroups(Taylor & Francis, 2014) Kumar, JitenderIn order to study the structure of A +(B n )—the affine near-semiring over a Brandt semigroup—this work completely characterizes the Green's classes of its semigroup reducts. In this connection, this work classifies the elements of A +(B n ) and reports the size of A +(B n ). Further, idempotents and regular elements of the semigroup reducts of A +(B n ) have also been characterized and studied some relevant semigroups in A +(B n ).Item The large rank of a finite semigroup using prime subsets(Springer, 2014-03) Kumar, JitenderThe large rank of a finite semigroup , denoted by r5( ), is the least number n such that every subset of with n elements generates . Howie and Ribeiro showed that r5( ) = |V| + 1, where V is a largest proper subsemigroup of . This work considers the complementary concept of subsemigroups, called prime subsets, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro’s result about the large rank of Brandt semigroups. Further, this work obtains the large rank of the semigroup of order-preserving singular selfmapsItem Radicals and Ideals of Affine Near-semirings over Brandt Semigroups(ARXIV, 2015-06) Kumar, JitenderThis work obtains all the right ideals, radicals, congruences and ideals of the affine near-semirings over Brandt semigroups.Item Syntactic semigroup problem for the semigroup reducts of affine near-semirings over Brandt semigroups(World Scientific, 2016) Kumar, JitenderThe syntactic semigroup problem is to decide whether a given finite semigroup is syntactic or not. This work investigates the syntactic semigroup problem for both the semigroup reducts of A+(Bn), the affine near-semiring over a Brandt semigroup Bn. It is ascertained that both the semigroup reducts of A+(Bn) are syntactic semigroups.Item Rank properties of the semigroup of endomorphisms over Brandt semigroup(Springer, 2017-10) Kumar, JitenderSince the work of Marczewski [10], many authors have studied the rank properties in the context of general algebras (cf. [1–3,9,11,12,14]). The concept of rank for general algebras is analogous to the concept of dimension in linear algebra. The dimension of a vector space is the maximum cardinality of an independent subset, or equivalently, it is the minimum cardinality of a generating set of the vector space. A subset U of a semigroup is said to be independent if every element ofU is not in the subsemigroup generated by the remaining elements of U, i.e., ∀a ∈ U, a /∈ U \ {a} . It can be observed that the minimum size of a generating set need not be equal to the maximum size of an independent set in a semigroup. Accordingly, Howie and Ribeiro have considered various concepts of ranks for a finite semigroup (cf. [5,6]). 1. r1( ) = max{k: every subset U of cardinality k in is independent}. 2. r2( ) = min{|U| : U ⊆ , U = }. 3. r3( ) = max{|U| : U ⊆ , U = ,U is independent}. 4. r4( ) = max{|U| : U ⊆ ,U is independent}.Item Maximal subsemigroups of finite transformation and diagram monoids(Elsevier, 2018-06) Kumar, JitenderWe describe and count the maximal subsemigroups of many well-known transformation monoids, and diagram monoids, using a new unified framework that allows the treatment of several classes of monoids simultaneously. The problem of determining the maximal subsemigroups of a finite monoid of transformations has been extensively studied in the literature. To our knowledge, every existing result in the literature is a special case of the approach we present. In particular, our technique can be used to determine the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by I. Dimitrova, V. H. Fernandes, and co-authors. We only present details for the transformation monoids whose maximal subsemigroups were not previously known; and for certain diagram monoids, such as the partition, Brauer, Jones, and Motzkin monoids. The technique we present is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors, and available in the Semigroups package for GAP, an open source computer algebra system. This allows us to concisely present the descriptions of the maximal subsemigroups, and to clearly see their common features.Item Chromatic Number of the Cyclic Graph of Infinite Semigroup(ACM Digital Library, 2020-01) Kumar, JitenderThe cyclic graph Γ(S) of a semigroup S is the simple graph whose vertex set is S, two element being adjacent if the subsemigroup generated by these two elements is monogenic. The purpose of this note is to prove that the chromatic number of Γ(S) is at most countable. The present paper generalizes the results of Shitov (Graphs Comb 33(2):485–487, 2017) and the corresponding results on power graph and enhanced power graph of groups obtained by Aalipour et al. (Electron J Comb 24(3):#P3.16, 2017).Item On enhanced power graphs of certain groups(World Scientific, 2021) Kumar, JitenderThe enhanced power graph Pe(G) of a group G is a simple undirected graph with vertex set G and two distinct vertices x,y are adjacent if both x and y belongs to same cyclic subgroup of G. In this paper, we obtain various graph invariants viz. independence number, minimum degree and matching number of Pe(G), where G is the dicyclic group or a class of groups of order 8n. If G is any of these groups, we prove that Pe(G) is perfect and then obtain its strong metric dimension.
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