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Jitender Kumar

Publications and source records attributed to Jitender Kumar.

At least 37 records · Page 2Linked to original sources
arXiv2023

Line graph characterization of the order supergraph of a finite group

The power graph $\mathcal{P}(G)$ is the simple undirected graph with group elements as a vertex set and two elements are adjacent if one of them is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph $\mathcal{P}(G)$ is the simple undirected graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if $o(x)\vert o(y)$ or $o(y)\vert o(x)$. In this paper, we classify all the finite groups $G$ such that the order supergraph $\mathcal{S}(G)$ is the line graph of some graph. Moreover, we characterize finite groups whose order supergraphs are the complement of line graphs.

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arXiv2023

Characterization of rings with genus two prime ideal sum graphs

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is a simple undirected graph whose vertex set is the set of nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus $2$.

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arXiv2023

Laplacian spectrum of weakly zero-divisor graph of the ring $\mathbb{Z}_{n}$

Let $R$ be a commutative ring with unity. The weakly zero-divisor graph $WΓ(R)$ of the ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$ and two vertices $x$, $y$ are adjacent if and only if there exists $r\in {\rm ann}(x)$ and $s \in {\rm ann}(y)$ such that $rs =0$. The zero-divisor graph of a ring is a spanning subgraph of the weakly zero-divisor graph. It is known that the zero-divisor graph of the ring $\mathbb{Z}_{p^t}$, where $p$ is a prime, is the Laplacian integral. In this paper, we obtain the Laplacian spectrum of the weakly zero-divisor graph $WΓ(\mathbb{Z}_{n})$ of the ring $\mathbb{Z}_{n}$ and show that $WΓ(\mathbb{Z}_{n})$ is Laplacian integral for arbitrary $n$.

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arXiv2023

On rings whose prime ideal sum graphs are line graphs

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$, $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.

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arXiv2023

On Finite groups whose power graphs are line graphs

S. Bera (Line graph characterization of power graphs of finite nilpotent groups, \textit{Communication in Algebra}, 50(11), 4652-4668, 2022) characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian $2$-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose power graphs, proper power graphs, enhanced power graphs and proper enhanced power graphs are the complement of line graphs, respectively.

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arXiv2023

On the idempotent graph of a ring

Let $R$ be a ring with unity. The \emph{idempotent graph} $G_{\text{Id}}(R)$ of a ring $R$ is an undirected simple graph whose vertices are the set of all the elements of ring $R$ and two vertices $x$ and $y$ are adjacent if and only if $x+y$ is an idempotent element of $R$. In this paper, we obtain a necessary and sufficient condition on the ring $R$ such that $G_{\text{Id}}(R)$ is planar. We prove that $G_{\text{Id}}(R)$ cannot be an outerplanar graph. Moreover, we classify all the finite non-local commutative rings $R$ such that $G_{\text{Id}}(R)$ is a cograph, split graph and threshold graph, respectively. We conclude that latter two graph classes of $G_{\text{Id}}(R)$ are equivalent if and only if $R \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$.

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arXiv2023

Strong metric dimension of the prime ideal sum graph of a commutative ring

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we obtain the strong metric dimension of the prime ideal sum graph for various classes of Artinian non-local commutative rings.

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arXiv2023

Characterization of rings with genus two cozero-divisor graphs

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$ 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 $x \notin Ry$ and $y \notin Rx$. The reduced cozero-divisor graph of a ring $R$, is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals of $R$ and two distinct vertices $(a)$ and $(b)$ are adjacent if and only if $(a) \not\subset (b)$ and $(b) \not\subset (a)$. In this paper, we characterize all classes of finite non-local commutative rings for which the cozero-divisor graph and reduced cozero-divisor graph is of genus two.

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arXiv2023

On the Difference Graph of power graphs of finite groups

The power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group $G$ is a simple undirected graph whose vertex set is the group $G$ and two vertices $a$ and $b$ are adjacent if there exists $c \in G$ such that both $a$ and $b$ are powers of $c$. In this paper, we investigate the difference graph $\mathcal{D}(G)$ 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 study the difference graphs of finite groups with forbidden subgraphs among other results. We first characterize an arbitrary finite group $G$ such that $\mathcal{D}(G)$ is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equivalent for $\mathcal{D}(G)$. By applying these results, we classify the nilpotent groups $G$ such that $\mathcal{D}(G)$ belong to the aforementioned five graph classes. This shows that all these graph classes are equivalent for $\mathcal{D}(G)$ 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 $S_n$ and alternating group $A_n$ are cograph, chordal, split, and threshold.

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arXiv2022

Wiener index of the Cozero-divisor graph of a finite commutative ring

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 $x \notin Ry$ and $y \notin Rx$. In this article, we extend some of the results of [24] to an arbitrary ring. In this connection, we derive a closed-form formula of the Wiener index of the cozero-divisor graph of a finite commutative ring $R$. As applications, we compute the Wiener index of $Γ'(R)$, when either $R$ is the product of ring of integers modulo $n$ or a reduced ring. At the final part of this paper, we provide a SageMath code to compute the Wiener index of the cozero-divisor graph of these class of rings including the ring $\mathbb{Z}_{n}$ of integers modulo $n$.

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arXiv2022

Lambda Number of the enhanced power graph of a finite group

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 $x, y \in \langle z \rangle$ for some $z \in G$. 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 $\mathcal{P}_E(G)$ 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 $λ(\mathcal{P}_E(G)) = n$ if and only if $G$ is not a cyclic group of order $n\geq 3$. Finally, we compute the exact value of $λ(\mathcal{P}_E(G))$ if $G$ is a finite nilpotent group.

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arXiv2022

The complement of enhanced power graph of a finite group

The enhanced power graph $\mathcal{P}_E(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we give an affirmative answer of the question posed by Cameron [6] which states that: Is it true that the complement of the enhanced power graph $\bar{\mathcal{P}_E(G)}$ of a non-cyclic group $G$ has only one connected component apart from isolated vertices? We classify all finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G)}$ is bipartite. We show that the graph $\bar{\mathcal{P}_E(G)}$ is weakly perfect. Further, we study the subgraph $\bar{\mathcal{P}_E(G^*)}$ of $\bar{\mathcal{P}_E(G)}$ induced by all the non-isolated vertices of $\bar{\mathcal{P}_E(G)}$. We classify all finite groups $G$ such that the graph is $\bar{\mathcal{P}_E(G^*)}$ is unicyclic and pentacyclic. We prove the non-existence of finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G^*)}$ is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups $G$ such that the graph $\bar{\mathcal{P}_E(G^*)}$ is outerplanar, planar, projective-planar and toroidal, respectively.

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arXiv2022

Certain properties of the enhanced power graph associated with a finite group

The enhanced power graph of a finite group $G$, denoted by $\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of $\mathcal{P}_E(G)$ 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 a lower bound and an upper bound for the Wiener index of $\mathcal{P}_E(G)$, where $G$ is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized.

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arXiv2022

Automorphisms of left Ideal relation graph over full matrix ring

The left-ideal relation graph on a ring $R$, denoted by $\overrightarrow{Γ_{l-i}}(R)$, is a directed graph whose vertex set is all the elements of $R$ and there is a directed edge from $x$ to a distinct $y$ if and only if the left ideal generated by $x$, written as $[x]$, is properly contained in the left ideal generated by $y$. In this paper, the automorphisms of $\overrightarrow{Γ_{l-i}}(R)$ are characterized, where $R$ is the ring of all $n \times n$ matrices over a finite field $F_q$. The undirected left relation graph, denoted by $Γ_{l-i}(M_n(F_q))$, is the simple graph whose vertices are all the elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if either $[x] \subset [y]$ or $[y] \subset [x]$ is considered. Various graph theoretic properties of $Γ_{l-i}(M_n(F_q))$ including connectedness, girth, clique number, etc. are studied.

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arXiv2022

On the intersection ideal graph of semigroups

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 $diam(Γ(S)) \leq 2$. 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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arXiv2021

On the inclusion ideal graph of semigroups

The inclusion ideal graph $\mathcal{I}n(S)$ 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 $I \subset J$ or $J \subset I$. The purpose of this paper is to study algebraic properties of the semigroup $S$ as well as graph theoretic properties of $\mathcal{I}n(S)$. In this paper, we investigate the connectedness of $\mathcal{I}n(S)$. We show that diameter of $\mathcal{I}n(S)$ is at most $3$ if it is connected. We also obtain a necessary and sufficient condition of $S$ such that the clique number of $\mathcal{I}n(S)$ is $n$, where $n$ is the number of minimal left ideals of $S$. Further, various graph invariants of $\mathcal{I}n(S)$ viz. perfectness, planarity, girth etc. are discussed. For a completely simple semigroup $S$, we investigate various properties of $\mathcal{I}n(S)$ including its independence number and matching number. Finally, we obtain the automorphism group of $\mathcal{I}n(S)$.

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arXiv2021

The cyclic graph of a semigroup

The cyclic graph $Γ(S)$ of a semigroup $S$ is the simple 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 classify the semigroup $S$ such that whose cyclic graph $Γ(S)$ is complete, bipartite, tree, regular and a null graph, respectively. Further, we determine the clique number of $Γ(S)$ for an arbitrary semigroup $S$. 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.

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arXiv2021

Enhanced Power Graph of Certain Non-abelian Groups

The enhanced power graph of a group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if they belong to same cyclic subgroup. In this paper, we study distant properties and detour distant properties such as closure, interior, distance degree sequence and eccentric subgraph of the enhanced power graph of semidihedral group. Consequently, we obtained the metric dimension and resolving polynomial of the enhanced power graph of semidihedral group. At the final part of this paper, we obtained the Laplacian spectrum of the enhanced power graph of semidihedral, dihedral and generalized quaternion groups.

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