arXiv subjects

Angsuman Das

Publications and source records attributed to Angsuman Das.

At least 19 recordsLinked to original sources

arXiv2026

Cyclic Probability of a Finite Group

The cyclic probability, $\theta(G)$, of a finite group $G$, is defined as the probability that two randomly selected elements of $G$ generate a cyclic subgroup of $G$. In this paper, we study various upper and lower bounds of $\theta(G)$ and show that $\theta(G)$ can be used as a criterion for checking nilpotency and solvability of a finite group. We also compare $\theta(G)$ with other group invariants like commuting probability $cp(G)$ and normalized sum of element orders.

math.GR↗

arXiv2026

On Connectivity of Comaximal Subgroup Graph

The co-maximal subgroup graph $\Gamma(G)$ of a finite group $G$ is defined to be a graph with the set of all non-trivial proper subgroups of $G$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. The deleted co-maximal subgroup graph of $G$, denoted by $\Gamma^*(G)$, is defined as the graph obtained by removing the isolated vertices from $\Gamma(G)$. In this paper, we prove that for any finite group $G$, $\Gamma^*(G)$ is connected. Furthermore, we show that $\Gamma^*(G)$ either contains a cycle or is a star. When $\Gamma^*(G)$ contains a cycle, its girth is either $3$ or $4$. Finally, we classify all finite groups $G$ for which $\Gamma^*(G)$ is a star.

math.GR↗

arXiv2026

Solvability of Groups via Cyclic Subgroup Count

In this paper, we provide new criteria for the solvability and supersolvability of a finite group based on its number of cyclic subgroups. A finite group G is called n-cyclic if it contains n cyclic subgroups. This paper also partially extends the classification of n-cyclic groups for n\geq 13.

math.GR↗

arXiv2026

Group Structure from Subgroup and Cyclic Subgroup Counts

For a finite group $G$, let $\sub(G)$ be the number of subgroups of $G$, let $\cyc(G)$ be the number of cyclic subgroups, and let $\pi(G)$ be the number of distinct prime divisors of $|G|$. We study the normalized counts $\lambda(G)=\sub(G)/2^{\pi(G)}$ and $\eta(G)=\cyc(G)/2^{\pi(G)}$. We prove that $\eta(G)<5/4$ or $\lambda(G)<3/2$ implies that $G$ is cyclic of squarefree order. The inequalities $\eta(G)<2$ and $\lambda(G)<5/2$ each force all Sylow subgroups to be cyclic, and hence imply metacyclicity. For the subgroup count, we give an exact arithmetic criterion in the parameters of the corresponding $ZM$-presentation. We determine all values with $1<\eta(G)<2$ and $1<\lambda(G)<5/2$, and prove that $\lambda(G)<59/8$ or $\eta(G)<4$ implies solvability. Both solvability bounds are sharp. We also describe how cyclic direct factors of coprime order affect the two normalized counts.

math.GR↗

arXiv2026

Cayley colour integral groups

A finite group $G$ is said to be Cayley integral if every undirected Cayley graph $\operatorname{Cay}(G,S)$ on $G$ is integral. In this paper, we introduce three natural extensions of this concept; namely as: Cayley colour integral, $\mathfrak{F}$-Cayley colour integral and normal Cayley integral groups. We characterize the first two families in its entirety. The last family of groups is shown to be coinciding with inverse semi-rational groups introduced by Chillag and Dolfi, thereby providing an alternative characterization for the same. We also establish an inclusion hierarchy among these families.

math.CO↗

arXiv2025

An integral family of quasi-strongly regular Cayley graphs

Quasi-strongly regular graphs form a significant generalization of strongly regular graphs. We study the eigenvalues of a family of such graphs, $\Gamma_H(G)$, constructed from a finite group $G$ and a subgroup $H$. Our main results include a sufficient condition for $\Gamma_H(G)$ to be integral and an explicit computation of its entire spectrum when $H$ is normal, revealing that the spectrum in this case depends only on $|G|$ and the index $[G:H]$.

math.CO↗

arXiv2025

The Difference Subgroup Graph of a Finite Group

The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$ but $HK \ne G$. This graph arises naturally as the difference between the join graph $\Delta(G)$ and the comaximal subgroup graph $\Gamma(G)$. In this paper, we initiate a systematic study of $D(G)$ and its reduced version $D^*(G)$, obtained by removing isolated vertices. We establish several fundamental structural properties of these graphs, including conditions for connectivity, forbidden subgraph characterizations, and the relationship between graph parameters - such as independence number, clique number, and girth - and the solvability or nilpotency of the underlying group. The paper concludes with a discussion of open problems and potential directions for future research.

math.GR↗

arXiv2025

Prime Order Element Graph of a Group -- II

In this sequel paper, we continue the analysis of the prime order element graph $\Gamma(G)$ of a finite group $G$, where vertices are elements of $G$ and edges connect distinct elements $x, y$ satisfying $\circ(xy) = p$ for some prime $p$. Our investigation focuses on the adjacency and Laplacian spectra, planarity, and clique number of this graph. We conclude by outlining open issues and potential directions for future investigations.

math.GR↗

arXiv2025

Non-isomorphic $d$-integral circulant graphs

The algebraic degree $Deg(G)$ of a graph $G$ is the dimension of the splitting field of the adjacency polynomial of $G$ over the field $\mathbb{Q}$. It can be shown that for every positive integer $d$, there exists a circulant graph with algebraic degree $d$. Let $C(d)$ be the least positive integer such that there exists a circulant graph of order $C(d)$ having algebraic degree $d$. A graph $G$ is called $d$-integral if $Deg(G)=d$. We call a $d$-integral circulant graph \textit{minimal} if order of that graph equals $C(d)$. Let $\mathcal{F}_{n,d}$ denote the collection of isomorphism classes of connected, $d$-integral circulant graphs of some given possible order $n$. In this paper we compute the exact value of $C(d)$ and provide some bounds on $|\mathcal{F}_{n,d}|$, thereby showing that the minimal $d$-integral circulant graph is not unique. Moreover, we find the exact value of $|\mathcal{F}_{p,d}|$ where both $p$ and $d$ are prime.

math.CO↗

arXiv2025

On Independence Number of Comaximal Subgroup Graph

In this paper, we establish sharp thresholds on the independence number of the comaximal subgroup graph $\Gamma(G)$ that guarantee solvability, supersolvability, and nilpotency of the underlying group $G$. Specifically: \begin{itemize} \item For solvability, we prove that any group $G$ with independence number $\alpha(\Gamma(G))\leq 51$ must be solvable, and show that the alternating group $A_5$ is uniquely determined by its graph. \item For supersolvability, we show that $\alpha(\Gamma(G))\leq 14$ implies $G$ is supersolvable, except for three explicit exceptions. \item For nilpotency, we prove that $\alpha(\Gamma(G))\leq 6$ ensures nilpotency, except for five groups. \end{itemize} Finally, we conclude with some open issues involving domination parameters.

math.GR↗

arXiv2024

Forbidden Subgraphs of Prime Order Element Graph

In this paper, we study different forbidden subgraph characterizations of the prime-order element graph $\Gamma(G)$ defined on a finite group $G$. Its set of vertices is the group $G$ and two vertices $x,y \in G$ are adjacent if the order of $xy$ is prime. More specifically, we investigate the conditions when $\Gamma(G)$ is perfect, cograph, chordal, claw-free, and interval graph.

math.CO↗

arXiv2024

Automorphism group of a family of distance regular graphs which are not distance transitive

Let $G_n=\mathbb{Z}_n\times \mathbb{Z}_n$ for $n\geq 4$ and $S=\{(i,0),(0,i),(i,i): 1\leq i \leq n-1\}\subset G_n$. Define $\Gamma(n)$ to be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is known that $\Gamma(n)$ is a strongly regular graph with the parameters $(n^2,3n-3,n,6)$ \cite{19}. Hence $\Gamma(n)$ is a distance regular graph. It is known that every distance transitive graph is distance regular, but the converse is not true. In this paper, we study some algebraic properties of the graph $\Gamma(n)$. Then by determining the automorphism group of this family of graphs, we show that the graphs under study are not distance transitive.

math.CO↗

arXiv2024

Solvability of a group based on its number of subgroups

In this paper, we provide some conditions of (super)-solvability and nilpotency of a finite group $G$ based on its number of subgroups $Sub(G)$. Our results generalize the classification of finite groups with less than $20$ subgroups by Betz and Nash. We also provide an application of our results in studying comaximal subgroup graph of a group. Finally, we conclude with some open issues.

math.GR↗

arXiv2024

On a family of quasi-strongly regular Cayley graphs

In this paper, we construct a family of quasi-strongly regular Cayley graphs $\Gamma_H(G)$ which is defined on a finite group $G$ with respect to a subgroup $H$ of $G$. We also compute its full automorphism group and characterize various transitivity properties of it.

math.GR↗

arXiv2023

A Family of Iterated Maps on Natural Numbers

In this paper, we introduce and study the iterates of the following family of functions $\varphi_k$ defined on natural numbers which exhibits nice properties. $$\varphi_k(x)=\left\lbrace \begin{array}{ll} x+k, & \mbox{ if $x$ is prime;}\\ \mbox{largest prime divisor of $x$,} & \mbox{ if $x$ is composite;} \end{array} \right.$$ In particular, we study the periodic behaviour of the trajectories of these iterated functions. In some cases, we provide proofs of these properties and in some other cases we pose some open problems based on numerical evidences supported by heuristic arguments.

math.NT↗

arXiv2023

On Some Intersection Properties of Finite Groups

In this article, we introduce the study of a class of finite groups $G$ which admits a subgroup which intersects all non-trivial subgroups of $G$. We also explore a subclass of it consisting of all groups $G$ in which the prime order elements commute. In particular, we discuss the relationship between these class of groups with other known classes of finite groups, like simple groups, perfect groups etc. Moreover, we also prove some results on the possible orders of such groups. Finally, we conclude with some open issues.

math.GR↗

arXiv2022

On Difference of Enhanced Power Graph and Power Graph of a Finite Group

The difference graph $D(G)$ of a finite group $G$ is the difference of enhanced power graph of $G$ and power graph of $G$, with all isolated vertices are removed. In this paper we study the connectedness and perfectness of $D(G)$ with respect to various properties of the underlying group $G$. We also find several connection between the difference graph of $G$ and the Gruenberg-Kegel graph of $G$.

math.CO↗

arXiv2021

On Perfectness of Annihilating-Ideal Graph of $\mathbb{Z}_n$

The annihilating-ideal graph of a commutative ring $R$ with unity is defined as the graph $\mathbb{AG}(R)$ with the vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ = 0$. Nikandish {\it et.al.} proved that $\mathbb{AG}(\mathbb{Z}_n)$ is weakly perfect. In this short paper, we characterize $n$ for which $\mathbb{AG}(\mathbb{Z}_n)$ is perfect.

math.CO↗