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A. Satyanarayana Reddy

Publications and source records attributed to A. Satyanarayana Reddy.

At least 19 recordsLinked to original sources
arXiv2026

Cyclic Subgroup Graph of a Group

A cyclic subgroup graph of a group is a graph whose vertices are cyclic subgroups of the group, and two distinct vertices are adjacent if one is a maximal subgroup of the other. In this paper, we examine several graph-theoretic properties of cyclic subgroup graphs, including bipartiteness, regularity, girth, diameter, and vertex degrees. We also classify all groups whose cyclic subgroup graphs are connected, complete, star graphs, trees, or paths. Further, we examine the cyclic subgroup graph for specific families of groups such as cyclic, dihedral, dicyclic, generalized quaternion, minimal non-cyclic and nilpotent groups. We also discuss the relationship of cyclic subgroup graph with the power graph and the enhanced power graph. Additionally, we give a relationship between a cyclic subgroup graph and a poset.

math.GR↗
arXiv2026

On the Generating Graph of Finite Abelian Groups

The generating graph $Γ(G)$ of a group $G$ is the graph whose vertex set is $G$, where two distinct vertices are adjacent if and only if they generate $G$. In this paper, we systematically study the structure of generating graphs of finite abelian groups (non-cyclic) and determine the set of all generating pairs. Moreover, we give some structural characterizations, in particular, we determine conditions under which $Γ(G)$ is regular, characterize when the isolated vertices form a subgroup, and establish necessary and sufficient conditions for two non-isomorphic finite abelian groups $G$ and $H$ to satisfy $Γ(G)\cong Γ(H)$. Furthermore, we compute the spectra of the adjacency and Laplacian matrices of these graphs.

math.CO↗
arXiv2025

CLT-groups with cyclic or abelian subgroups

A finite group is called a CLT-group if it contains a subgroup corresponding to every divisor of the order of the group. It is said to be a Cyclic (Abelian) CLT group if it contains a cyclic (abelian) subgroup corresponding to every proper divisor of the order of the group. A natural number is said to be a CCLT (ACLT) number if every group of that order is a cyclic (abelian) CLT group. In this work, we classify all CCLT and ACLT numbers and study various properties of Cyclic (Abelian) CLT groups. We also show that the classes of CCLT and ACLT groups are contained in the class of supersolvable groups. Moreover, we introduce the function CCLT-degree on the set of non-cyclic finite groups and study the properties of this function.

math.GR↗
arXiv2025

Groups having 12 cyclic subgroups

A finite group is said to be $n$-cyclic if it contains $n$ cyclic subgroups. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group $G$ and is denoted by $cdeg (G)$. In this paper, we classify all $12$-cyclic groups. We also prove that the set of cyclicity degrees for all the finite groups is dense in $[0,1]$, which gives a solution to the problem asked by Tărnăuceanu and Tóth in [20] "For every $a\in [0, 1]$, does there exist a sequence $(G_n)$ of finite groups such that $\lim_{n\to\infty} cdeg(G_n)=a$ "?

math.CO↗
arXiv2025

Spectral Bounds of the Generating Graph of $\mathbb{Z}_n.$

Let $G$ be a group. A group is said to be $k$-generated if it can be generated by its $k$ elements. A generating set of $G$ is called a minimal generating set if no proper subset of it generates $G.$ A minimal generating set of a group can have different sizes. The generating graph $Γ(G)$ of a group $G$ is defined as a graph with the vertex set $G$, where two distinct vertices are adjacent if they together generate $G.$ This graph is particularly useful when studying 2-generated groups. In this context, consider the group $G = \mathbb{Z}_n$, the integers modulo $n.$ In this paper, we explore various graph-theoretic properties of the generating graph $Γ(\mathbb{Z}_n)$ and investigate the spectra of its adjacency and Laplacian matrices. Additionally, we explicitly determine the set of all possible minimal generating sets of $\mathbb{Z}_n$ of size $k.$

math.CO↗
arXiv2024

The Generating graph of Dicyclic Groups

For a group $G,$ the generating graph of $G,$ denoted by $Γ(G).$ We define $Q_n=\langle x,y: x^{2n}=y^4=1, x^n=y^2,y^{-1}xy=x^{-1}\rangle,$ the dicyclic group of order $4n.$ This paper primarily delves into exploring the graph characteristics and spectral properties of various matrices associated with $Γ(Q_n)$. Specifically, we determine the complete spectrum of the adjacency, Laplacian, distance, and eccentricity matrices. Additionally, we completely determine the spectrum pertaining to the distance and eccentricity matrices of the dihedral group of order $2n$, denoted as $D_n$.

math.CO↗
arXiv2023

Spectral properties of edge Laplacian matrix

Let $N(X)$ be the Laplacian matrix of a directed graph obtained from the edge adjacency matrix of a graph $X.$ In this work, we study the bipartiteness property of the graph with the help of $N(X).$ We computed the spectrum of the edge Laplacian matrix for the regular graphs, the complete bipartite graphs, and the trees. Further, it is proved that given a graph $X,$ the characteristic polynomial of $N(X)$ divides the characteristic polynomial of $N(X^{\prime\prime}),$ where $X^{\prime\prime}$ denote the Kronecker double cover of $X.$

math.CO↗
arXiv2023

Algebraic connectivity of Kronecker products of line graphs

Let $X$ be a tree with $n$ vertices and $L(X)$ be its line graph. In this work, we completely characterize the trees for which the algebraic connectivity of $L(X)\times K_m$ is equal to $m-1$, where $\times$ denotes the Kronecker product. We provide a few necessary and sufficient conditions for $L(X)\times K_m$ to be Laplacian integral. The algebraic connectivity of $L(X)\times K_m$, where $X$ is a tree of diameter $4$ and $k$-book graph is discussed.

math.CO↗
arXiv2023

Generating Graphs of Finite Dihedral Groups

For a group $G$, the generating graph $Γ(G)$ is defined as the graph with the vertex set $G$, and any two distinct vertices of $Γ(G)$ are adjacent if they generate $G$. In this paper, we study the generating graph of $D_n,$ where $D_n$ is a Dihedral group of order $2n$. We explore various graph theoretic properties, and determine complete spectrum of the adjacency and the Laplacian matrix of $Γ(D_n)$. Moreover, we compute some distance and degree based topological indices of $Γ(D_n)$.

math.CO↗
arXiv2023

On the double covers of a line graph

Let $L(X)$ be the line graph of graph $X$. Let $X^{\prime\prime}$ be the Kronecker product of $X$ by $K_2$. In this paper, we see that $L(X^{\prime\prime})$ is a double cover of $L(X)$. We define the symmetric edge graph of $X$, denoted as $γ(X)$ which is also a double cover of $L(X)$. We study various properties of $γ(X)$ in relation to $X$ and the relationship amongst the three double covers of $L(X)$ that are $L(X^{\prime\prime}),γ(X)$ and $L(X)^{\prime\prime}$. With the help of these double covers, we show that for any integer $k\geq 5$, there exist two equienergetic graphs of order $2k$ that are not cospectral.

math.CO↗
arXiv2023

Achiral words

A word $w$ in a free group is {\em achiral} if for every group $G,$ $G_w=G_{w^{-1}},$ where $G_w$ is the image of the word map $w$ on $G.$ We will give few classes of examples of achiral words. Cocke and Ho asked whether Engel words are achiral or not. We will prove that it is enough to apply Whitehead's algorithm to check the same.

math.GR↗
arXiv2022

Positivity of Hadamard powers of a few band matrices

Let $\mathbb{P}_G([0,\infty))$ and $\mathbb{P}_G^{'}([0,\infty))$ be the sets of positive semidefinite and positive definite matrices of order $n$, respectively, with nonnegative entries, where some positions of zero entries are restricted by a simple graph $G$ with $n$ vertices. It is proved that for a connected simple graph $G$ of order $n\geq 3$, the set of powers preserving positive semidefiniteness on $\mathbb{P}_G([0,\infty))$ is precisely the same as the set of powers preserving positive definiteness on $\mathbb{P}_G^{'}([0,\infty))$. In particular, this provides an explicit combinatorial description of the critical exponent for positive definiteness, for all chordal graphs. Using chain sequences, it is proved that the Hadamard powers preserving the positive (semi) definiteness of every tridiagonal matrix with nonnegative entries are precisely $r\geq 1$. The infinite divisibility of tridiagonal matrices is studied. The same results are proved for a special family of pentadiagonal matrices.

math.CO↗
arXiv2020

Survey on irreducibility of trinomials

Let $a,b,c$ be non-zero integers and $f(x)=ax^n+bx^m+c$ be a trinomial of degree $n$. We surveyed the irreducibility criteria of $f(x)$ over rational numbers.

math.HO↗
arXiv2020

Irreducibility of $x^n-a$

A. Capelli gave a necessary and sufficient condition for the reducibility of $x^n-a$ over $\Q$. In this article, we are providing an alternate elementary proof for the same.

math.NT↗
arXiv2020

An irreducible class of polynomials over integers

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We will show that under the strict inequality these polynomials are irreducible for certain values of $n_1$. In the case of equality, apart from its cyclotomic factors, they have exactly one irreducible non-reciprocal factor.

math.NT↗
arXiv2020

Calkin-Wilf tree

In this article, we prove various properties of Calkin-Wilf tree. We also see how the Minkowski question mark function will act on Calkin-Wilf tree and its diagonals.

math.CO↗