Department of Mathematics

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Affine Near-Semirings Over Brandt Semigroups
(Taylor & Francis, 2014) Kumar, Jitender
In 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 ).
Certain properties of the enhanced power graph associated with a finite group
(Springer, 2023-03) Kumar, Jitender
The enhanced power graph of a finite group G, denoted by PE(G), is a simple undirected graph whose vertex set is G and two distinct vertices x, y are adjacent if x,y∈⟨z⟩ for some z∈G. In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of PE(G) are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain the upper and lower bounds of the Wiener index of PE(G), where G is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized.
Chromatic Number of the Cyclic Graph of Infinite Semigroup
(ACM Digital Library, 2020-01) Kumar, Jitender
The 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).
Lambda number of the enhanced power graph of a finite group
(2022-08) Kumar, Jitender
The 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.
The large rank of a finite semigroup using prime subsets
(Springer, 2014-03) Kumar, Jitender
The 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 selfmaps
Maximal subsemigroups of finite transformation and diagram monoids
(Elsevier, 2018-06) Kumar, Jitender
We 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.
On enhanced power graphs of certain groups
(World Scientific, 2021) Kumar, Jitender
The 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.
On the clique number and independence number of the cyclic graph of a semigroup
(World Scientific, 2023-04) Kumar, Jitender
The cyclic graph Γ(S) of a semigroup S is the simple undirected graph whose vertex set is S and two vertices x,y are adjacent if the subsemigroup generated by x and y is monogenic. In this paper, we determine the clique number of Γ(S) for an arbitrary semigroup S. Further, we obtain the independence number of Γ(S) if S is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of Γ(S) if S is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds.
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).
On the Connectivity and Equality of Some Graphs on Finite Semigroups
(Springer, 2022-11) Kumar, Jitender
In this paper, we study various graphs, namely Colon power graph, cyclic graph, enhanced power graph, and commuting graph on a semigroup S. The purpose of this paper is twofold. First, we study the interconnection between the diameters of these graphs on semigroups having one idempotent. Consequently, the results on the connectedness and the diameter of the proper enhanced power graphs (or cyclic graph) of finite groups, viz., symmetric group and alternating group, are obtained. In the other part of this paper, for an arbitrary pair of these four graphs, we classify finite semigroups such that the graphs in this pair are equal. Our results generalize some of the corresponding results of these graphs on groups to semigroups.
On the cozero-divisor graphs associated to rings
(Taylor & Francis, 2022-08) Kumar, Jitender
Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by Γ'(R), is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if x∉Ry and y∉Rx. In this paper, first we study the Laplacian spectrum of Γ'(Zn). We show that the graph Γ'(Zpq) is Laplacian integral. Further, we obtain the Laplacian spectrum of Γ'(Zn) for n=pn1qn2, where n1,n2∈N and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of Γ'(Zn), we characterized the values of n for which the Laplacian spectral radius is equal to the order of Γ'(Zn). Moreover, the values of n for which the algebraic connectivity and vertex connectivity of Γ'(Zn) coincide are also described. At the final part of this paper, we obtain the Wiener index of Γ'(Zn) for arbitrary n.
On the difference graph of power graphs of finite groups
(Taylor & Francis, 2023-11) Kumar, Jitender
The 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.
On the enhanced power graph of a finite group
(Taylor & Francis, 2020-11) Kumar, Jitender
The enhanced power graph Pe(G) of a group G is a graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number, and matching number of enhanced power graphs of finite groups. We first study these graph invariants for Pe(G) when G is any finite group and then determine them when G is a finite abelian p-group, U6n=⟨a,b:a2n=b3=e,ba=ab−1⟩, the dihedral group D2n, or the semidihedral group SD8n. If G is any of these groups, we prove that Pe(G) is perfect and then obtain its strong metric dimension. Additionally, we give an expression for the independence number of Pe(G) for any finite abelian group G. These results along with certain known equalities yield the edge connectivity, vertex covering number, and edge covering number of enhanced power graphs of the respective groups as well.
On the enhanced power graph of a semigroup
(ARXIV, 2021-07) Kumar, Jitender
The enhanced power graph Pe(S) of a semigroup S is a simple graph whose vertex set is S and two vertices x,y∈S are adjacent if and only if x,y∈⟨z⟩ for some z∈S, where ⟨z⟩ is the subsemigroup generated by z. In this paper, first we described the structure of Pe(S) for an arbitrary semigroup S. Consequently, we discussed the connectedness of Pe(S). Further, we characterized the semigroup S such that Pe(S) is complete, bipartite, regular, tree and null graph, respectively. Also, we have investigated the planarity together with the minimum degree and independence number of Pe(S). The chromatic number of a spanning subgraph, viz. the cyclic graph, of Pe(S) is proved to be countable. At the final part of this paper, we construct an example of a semigroup S such that the chromatic number of Pe(S) need not be countable.
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).
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).
On the large rank of certain semigroups of transformations on a finite chain
(World Scientific, 2022) Kumar, Jitender
The large rank of a finite semigroup S is the least number n such that every subset of S with n elements generates S. This paper obtains the large ranks of ODn,PODJn and PODn, the semigroups of singular transformations, injective partial and partial transformations on a finite chain [n]={1,2,…,n}, which preserve or reverse the order, respectively. As a consequence, we obtain the large ranks of POJn and POn, the semigroups of injective order-preserving partial transformations and order-preserving partial transformations on [n], respectively.
On the Structure of the Commuting Graph of Brandt Semigroups
(Springer, 2021) Kumar, Jitender
Radicals and Ideals of Affine Near-semirings over Brandt Semigroups
(ARXIV, 2015-06) Kumar, Jitender
This work obtains all the right ideals, radicals, congruences and ideals of the affine near-semirings over Brandt semigroups.
Rank properties of the semigroup of endomorphisms over Brandt semigroup
(Springer, 2017-10) Kumar, Jitender
Since 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}.