Binod Kumar Sahoo

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Laplacian eigenvalues of the zero divisor graph of the ring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>

2019-08-20

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

Vertex connectivity of the power graph of a finite cyclic group

2018-06-25

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

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Algebraic Connectivity of Connected Graphs with Fixed Number of Pendant Vertices

2010-08-20

A. K. Lal , Kamal Lochan Patra , Binod Kumar Sahoo

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Minimal cut-sets in the power graphs of certain finite non-cyclic groups

2020-10-16

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

We study minimal cut-sets of the power graph of a finite non-cyclic nilpotent group which are associated with its maximal cyclic subgroups. As a result, we find the vertex connectivity … We study minimal cut-sets of the power graph of a finite non-cyclic nilpotent group which are associated with its maximal cyclic subgroups. As a result, we find the vertex connectivity of the power graphs of certain finite non-cyclic abelian groups whose order is divisible by at most three distinct primes.

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Bounds for the Laplacian spectral radius of graphs

2017-10-30

Kamal Lochan Patra , Binod Kumar Sahoo

This paper is a survey on the upper and lower bounds for the largest eigenvalue of the Laplacian matrix, known as the Laplacian spectral radius, of a graph. The bounds … This paper is a survey on the upper and lower bounds for the largest eigenvalue of the Laplacian matrix, known as the Laplacian spectral radius, of a graph. The bounds are given as functions of graph parameters like the number of vertices, the number of edges, degree sequence, average 2-degrees, diameter, covering number, domination number, independence number and other parameters.

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Blocking sets of certain line sets related to a hyperbolic quadric in PG(3, q)

2018-07-20

Binod Kumar Sahoo , Bikramaditya Sahu

Abstract For a fixed hyperbolic quadric 𝓗 in PG(3, q ), let 𝔼 (respectively 𝕋, 𝕊) denote the set of all lines of PG(3, q ) which are external (respectively … Abstract For a fixed hyperbolic quadric 𝓗 in PG(3, q ), let 𝔼 (respectively 𝕋, 𝕊) denote the set of all lines of PG(3, q ) which are external (respectively tangent, secant) with respect to 𝓗. We characterize the minimum size blocking sets of PG(3, q ) with respect to each of the line sets 𝕊, 𝕋 ∪ 𝕊 and 𝔼 ∪ 𝕊.

Binary codes of the symplectic generalized quadrangle of even order

2015-01-28

Binod Kumar Sahoo , N. S. Narasimha Sastry

A characterization of finite symplectic polar spaces of odd prime order

2006-04-19

Binod Kumar Sahoo , N. S. Narasimha Sastry

Minimum size blocking sets of certain line sets related to a conic in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>P</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

2016-02-23

Kamal Lochan Patra , Binod Kumar Sahoo , Bikramaditya Sahu

Blocking sets of tangent and external lines to a hyperbolic quadric in $$\varvec{PG(3,q)}$$ P G ( 3 , q ) , $$\varvec{q}$$ q even

2018-12-17

Binod Kumar Sahoo , Bikramaditya Sahu

Vertex connectivity of the power graph of a finite cyclic group II

2019-02-08

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which … The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.

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Blocking sets of tangent lines to a hyperbolic quadric in PG(3, 3)

2019-01-21

Bart De Bruyn , Binod Kumar Sahoo , Bikramaditya Sahu

Blocking sets of tangent and external lines to a hyperbolic quadric in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>P</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

2018-07-14

Bart De Bruyn , Binod Kumar Sahoo , Bikramaditya Sahu

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On the order of a non-abelian representation group of a slim dense near hexagon

2008-03-03

Binod Kumar Sahoo , N. S. Narasimha Sastry

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A non-abelian representation of the dual polar space DQ(2n,2)

2009-01-01

Kamal Lochan Patra , Binod Kumar Sahoo

We prove that the dual polar space DQ(2n, 2), n ≥ 3, of rank n associated with a non-singular parabolic quadric in PG(2n, 2) admits a faithful non-abelian representation in … We prove that the dual polar space DQ(2n, 2), n ≥ 3, of rank n associated with a non-singular parabolic quadric in PG(2n, 2) admits a faithful non-abelian representation in the extraspecial 2-group 2 1+2 n + .The near 2n-gon I n (section 2.4) is a geometric hyperplane of DQ(2n, 2).For n ≥ 3, we first construct a faithful non-abelian representation of I n in 2 1+2 n + and subsequently extend it to a faithful non-abelian representation of DQ(2n, 2) in 2 1+2 n + .

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On the minimum degree of the power graph of a finite cyclic group

2019-12-18

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices … The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

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Blocking sets of tangent and external lines to an elliptic quadric in PG(3, q)

2021-10-01

Bart De Bruyn , Puspendu Pradhan , Binod Kumar Sahoo

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Non-abelian representations of the slim dense near hexagons on 81 and 243 points

2010-06-02

B. De Bruyn , Binod Kumar Sahoo , N. S. Narasimha Sastry

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Minimal cut-sets in the power graph of certain finite non-cyclic groups

2018-02-21

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph of a group is the simple graph with vertices as the group elements, in which two distinct vertices are adjacent if and only if one of them … The power graph of a group is the simple graph with vertices as the group elements, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. We study (minimal) cut-sets of the power graph of a (finite) non-cyclic (nilpotent) group which are associated with its maximal cyclic subgroups. Let $G$ be a finite non-cyclic nilpotent group whose order is divisible by at least two distinct primes. If $G$ has a Sylow subgroup which is neither cyclic nor a generalized quaternion $2$-group and all other Sylow subgroups of $G$ are cyclic, then under some conditions we prove that there is only one minimum cut-set of the power graph of $G$. We apply this result to find the vertex connectivity of the power graphs of certain finite non-cyclic abelian groups whose order is divisible by at most three distinct primes.

Minimum size blocking sets of certain line sets with respect to an elliptic quadric in $PG(3,q)$

2020-07-30

Bart De Bruyn , Puspendu Pradhan , Binod Kumar Sahoo

For a given nonempty subset $\mathcal{L}$ of the line set of $\PG(3,q)$, a set $X$ of points of $\PG(3,q)$ is called an $\mathcal{L}$-blocking set if each line in $\mathcal{L}$ contains … For a given nonempty subset $\mathcal{L}$ of the line set of $\PG(3,q)$, a set $X$ of points of $\PG(3,q)$ is called an $\mathcal{L}$-blocking set if each line in $\mathcal{L}$ contains at least one point of $X$. Consider an elliptic quadric $Q^-(3,q)$ in $\PG(3,q)$. Let $\mathcal{E}$ (respectively, $\mathcal{T}, \mathcal{S}$) denote the set of all lines of $\PG(3,q)$ which meet $Q^-(3,q)$ in $0$ (respectively, $1,2$) points. In this paper, we characterize the minimum size $\mathcal{L}$-blocking sets in $\PG(3,q)$, where $\mathcal{L}$ is one of the line sets $\mathcal{S}$, $\mathcal{E}\cup \mathcal{S}$, and $\mathcal{T}\cup \mathcal{S}$.

On Blocking Sets of the Tangent Lines to a Nonsingular Quadric in $\mathrm{PG}(3,q)$, $q$ Prime

2024-12-26

Bart De Bruyn , Francesco Pavese , Puspendu Pradhan +1 more

Let $Q^{-}(3,q)$ be an elliptic quadric and $Q^{+}(3,q)$ a hyperbolic quadric in $\mathrm{PG}(3,q)$. For $\epsilon\in\{-,+\}$, let $\mathcal{T}^{\epsilon}$ denote the set of all tangent lines of $\mathrm{PG}(3,q)$ with respect to $Q^{\epsilon}(3,q)$. … Let $Q^{-}(3,q)$ be an elliptic quadric and $Q^{+}(3,q)$ a hyperbolic quadric in $\mathrm{PG}(3,q)$. For $\epsilon\in\{-,+\}$, let $\mathcal{T}^{\epsilon}$ denote the set of all tangent lines of $\mathrm{PG}(3,q)$ with respect to $Q^{\epsilon}(3,q)$. If $k$ is the minimum size of a $\mathcal{T}^{\epsilon}$-blocking set in $\mathrm{PG}(3,q)$, then it is known that $q^2+1 \leq k \leq q^2+q$. For an odd prime $q$, we prove that there are no $\mathcal{T}^+$-blocking sets of size $q^2+1$ and that the quadric $Q^-(3,q)$ is the only $\mathcal{T}^-$-blocking set of size $q^2 +1$ in $\mathrm{PG}(3,q)$. When $q=3$, we show with the aid of a computer that there are no minimal $\mathcal{T}^-$-blocking sets of size $11$ and that, up to isomorphism, there are eight minimal $\mathcal{T}^-$-blocking sets of size $12$ in $\mathrm{PG}(3,3)$. We also provide geometrical constructions for these eight mutually nonisomorphic minimal $\mathcal{T}^-$-blocking sets of size $12$.

New constructions of two slim dense near hexagons

2007-05-09

Binod Kumar Sahoo

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Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

2013-12-01

Kamal Lochan Patra , Binod Kumar Sahoo

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Revisiting Eisenstein-type criterion over integers

2016-01-01

Akash Jena , Binod Kumar Sahoo

The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer … The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive integer relatively prime to the degree of $f(x)$. Suppose that there exists a prime number $p$ such that the leading coefficient of $f(x)$ is not divisible by $p$, all the remaining coefficients are divisible by $p^k$, and the constant term of $f(x)$ is not divisible by $p^{k+1}$. Then $f(x)$ is irreducible over $\mathbb{Z}$. For $k=1$, this is precisely the Eisenstein criterion. The aim of this article is to give an alternate proof, accessible to the undergraduate students, of this result for $k\in \{2,3,4\}$ using basic divisibility properties of integers.

On minimum size blocking sets of the outer tangents to a hyperbolic quadric in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="normal">PG</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

2018-11-19

Bart De Bruyn , Binod Kumar Sahoo

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Erratum: Vertex connectivity of the power graph of a finite cyclic group II

2019-11-01

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

We retract [1, Lemma 3.4] as the statement is incorrect. In consequence, we correct the statements of Theorems 1.2 and 4.5 and their proofs. We retract [1, Lemma 3.4] as the statement is incorrect. In consequence, we correct the statements of Theorems 1.2 and 4.5 and their proofs.

Proper divisor graph of a positive integer

2020-01-01

Hitesh Kumar , Kamal Lochan Patra , Binod Kumar Sahoo

The proper divisor graph $Υ_n$ of a positive integer $n$ is the simple graph whose vertices are the proper divisors of $n$, and in which two distinct vertices $u, v$ … The proper divisor graph $Υ_n$ of a positive integer $n$ is the simple graph whose vertices are the proper divisors of $n$, and in which two distinct vertices $u, v$ are adjacent if and only if $n$ divides $uv$. The graph $Υ_n$ plays an important role in the study of the zero divisor graph of the ring $\mathbb{Z}_n$. In this paper, we study some graph theoretic properties of $Υ_n$ and determine the graph parameters such as clique number, chromatic number, chromatic index, independence number, matching number, domination number, vertex and edge covering numbers of $Υ_n$. We also determine the automorphism group of $Υ_n$.

Characterizing finite nilpotent groups associated with a graph theoretic equality

2021-05-27

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of … The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. We characterize the finite nilpotent groups whose power graphs have equal vertex connectivity and minimum degree.

Characterizing finite nilpotent groups associated with a graph theoretic equality

2021-09-28

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

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A characterization of the family of secant lines to a hyperbolic quadric in PG(3,q), q odd, Part II

2022-11-16

Bart De Bruyn , Puspendu Pradhan , Binod Kumar Sahoo +1 more

On the minimum cut-sets of the power graph of a finite cyclic group

2023-04-11

Sanjay Mukherjee , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph [Formula: see text] of a finite group [Formula: see text] is the simple graph with vertex set [Formula: see text], in which two distinct vertices are adjacent … The power graph [Formula: see text] of a finite group [Formula: see text] is the simple graph with vertex set [Formula: see text], in which two distinct vertices are adjacent if one of them is a power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. For [Formula: see text], the minimum cut-sets of [Formula: see text] are characterized in [S. Chattopadhyay, K. L. Patra and B. K. Sahoo, Vertex connectivity of the power graph of a finite cyclic group, Discrete Appl. Math. 266 (2019) 259–271]. In this paper, for [Formula: see text], we identify certain cut-sets of [Formula: see text] such that any minimum cut-set of [Formula: see text] must be one of them.

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On the existence of non-abelian representations of slim dense near hexagons having big quads

2006-01-01

Binod Kumar Sahoo , N. S. Narasimha Sastry

This paper has been withdrawn as the statements in Proposition 4.4 and Theorem 1.4(i) are not correct. This paper has been withdrawn as the statements in Proposition 4.4 and Theorem 1.4(i) are not correct.

Polarized non-abelian representations of slim near-polar spaces

2015-12-10

Bart De Bruyn , Binod Kumar Sahoo

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Existence of non-abelian representations of the near hexagon Q(5,2)⊗Q(5,2)

2016-03-17

Binod Kumar Sahoo

Laplacian eigenvalues of the zero divisor graph of the ring $\mathbb{Z}_{n}$

2019-03-19

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

We study the Laplacian eigenvalues of the zero divisor graph $\Gamma\left(\mathbb{Z}_{n}\right)$ of the ring $\mathbb{Z}_{n}$ and prove that $\Gamma\left(\mathbb{Z}_{p^t}\right)$ is Laplacian integral for every prime $p$ and positive integer $t\geq … We study the Laplacian eigenvalues of the zero divisor graph $\Gamma\left(\mathbb{Z}_{n}\right)$ of the ring $\mathbb{Z}_{n}$ and prove that $\Gamma\left(\mathbb{Z}_{p^t}\right)$ is Laplacian integral for every prime $p$ and positive integer $t\geq 2$. We also prove that the Laplacian spectral radius and the algebraic connectivity of $\Gamma\left(\mathbb{Z}_{n}\right)$ for most of the values of $n$ are, respectively, the largest and the second smallest eigenvalues of the vertex weighted Laplacian matrix of a graph which is defined on the set of proper divisors of $n$. The values of $n$ for which algebraic connectivity and vertex connectivity of $\Gamma\left(\mathbb{Z}_{n}\right)$ coincide are also characterized.

On the minimum degree of the power graph of a finite cyclic group

2019-05-26

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them … The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum degree $\delta(\mathcal{P}(C_n))$ of $\mathcal{P}(C_n)$ is known for $r\in\{1,2\}$, see [18]. For $r\geq 3$, under certain conditions involving the prime divisors of $n$, we identify at most $r-1$ vertices such that $\delta(\mathcal{P}(C_n))$ is equal to the degree of at least one of these vertices. If $r=3$ or if $n$ is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of $n$.

Non-abelian representations of the slim dense near hexagons on 81 and 243 points

2010-03-24

Bart De Bruyn , Binod Kumar Sahoo , N. S. Narasimha Sastry

We prove that the near hexagon $Q(5,2) \times \mathbb{L}_3$ has a non-abelian representation in the extra-special 2-group $2^{1+12}_+$ and that the near hexagon $Q(5,2) \otimes Q(5,2)$ has a non-abelian representation … We prove that the near hexagon $Q(5,2) \times \mathbb{L}_3$ has a non-abelian representation in the extra-special 2-group $2^{1+12}_+$ and that the near hexagon $Q(5,2) \otimes Q(5,2)$ has a non-abelian representation in the extra-special 2-group $2^{1+18}_-$. The description of the non-abelian representation of $Q(5,2) \otimes Q(5,2)$ makes use of a new combinatorial construction of this near hexagon.

Algebraic connectivity of connected graphs with fixed number of pendant vertices

2010-03-24

A. K. Lal , Kamal Lochan Patra , Binod Kumar Sahoo

In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, … In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

2010-12-04

Kamal Lochan Patra , Binod Kumar Sahoo

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on $n$ vertices with girth $g$ ($n,g$ being fixed), which graph minimizes the … In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on $n$ vertices with girth $g$ ($n,g$ being fixed), which graph minimizes the Laplacian spectral radius? We prove that the graph $U_{n,g}$ (defined in Section 1) uniquely minimizes the Laplacian spectral radius for $n\geq 2g-1$ when $g$ is even and for $n\geq 3g-1$ when $g$ is odd.

On the minimum degree of power graphs of finite nilpotent groups

2022-07-18

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph P(G) of a group G is the simple graph with vertex set G and two distinct vertices are adjacent if one of them is a positive power … The power graph P(G) of a group G is the simple graph with vertex set G and two distinct vertices are adjacent if one of them is a positive power of the other. For a finite noncyclic nilpotent group G, we study the minimum degree δ(P(G)) of P(G). Under some conditions involving the prime divisors of |G| and the Sylow subgroups of G, we identify certain vertices associated with the generators of maximal cyclic subgroups of G such that δ(P(G)) is the degree of one of them. As an application, we obtain δ(P(G)) for some classes of nilpotent groups G.

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On the minimum degree of power graphs of finite nilpotent groups

2021-01-01

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph $\mathcal{P}(G)$ of a group $G$ is the simple graph with vertex set $G$ and two vertices are adjacent whenever one of them is a positive power of … The power graph $\mathcal{P}(G)$ of a group $G$ is the simple graph with vertex set $G$ and two vertices are adjacent whenever one of them is a positive power of the other. In this paper, for a finite noncyclic nilpotent group $G$, we study the minimum degree $\delta(\mathcal{P}(G))$ of $\mathcal{P}(G)$. Under some conditions involving the prime divisors of $|G|$ and the Sylow subgroups of $G$, we identify certain vertices associated with the generators of maximal cyclic subgroups of $G$ such that $\delta(\mathcal{P}(G))$ is equal to the degree of one of these vertices. As an application, we obtain $\delta(\mathcal{P}(G))$ for some classes of finite noncyclic abelian groups $G$.

Characterizing finite nilpotent groups associated with a graph theoretic equality

2021-01-01

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

Laplacian eigenvalues of the zero divisor graph of the ring $\mathbb{Z}_{n}$

2019-01-01

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

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Vertex connectivity of the power graph of a finite cyclic group II

2018-01-01

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph $\mathcal{P}(G)$ of a given finite group $G$ is the simple undirected graph whose vertices are the elements of $G$, in which two distinct vertices are adjacent if … The power graph $\mathcal{P}(G)$ of a given finite group $G$ is the simple undirected graph whose vertices are the elements of $G$, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity $κ(\mathcal{P}(G))$ of $\mathcal{P}(G)$ is the minimum number of vertices which need to be removed from $G$ so that the induced subgraph of $\mathcal{P}(G)$ on the remaining vertices is disconnected or has only one vertex. For a positive integer $n$, let $C_n$ be the cyclic group of order $n$. Suppose that the prime power decomposition of $n$ is given by $n =p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $r\geq 1$, $n_1,n_2,\ldots, n_r$ are positive integers and $p_1,p_2,\ldots,p_r$ are prime numbers with $p_1

On the minimum degree of the power graph of a finite cyclic group

2019-01-01

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

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On the minimum cut-sets of the power graph of a finite cyclic group

2022-01-01

Sanjay Mukherjee , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a … The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum cut-sets of $\mathcal{P}(C_n)$ are characterized in \cite{cps} for $r\leq 3$. In this paper, for $r\geq 4$, we identify certain cut-sets of $\mathcal{P}(C_n)$ such that any minimum cut-set of $\mathcal{P}(C_n)$ must be one of them.

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Algebraic connectivity of connected graphs with fixed number of pendant vertices

2010-01-01

A. K. Lal , Kamal Lochan Patra , Binod Kumar Sahoo

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Minimal cut-sets in the power graphs of certain finite non-cyclic groups

2018-01-01

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

2010-01-01

Kamal Lochan Patra , Binod Kumar Sahoo

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On vertex connectivity of zero-divisor graphs of finite commutative rings

2024-04-17

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

On the minimum cut-sets of the power graph of a finite cyclic group, II

2025-05-21

Sanjay Mukherjee , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph [Formula: see text] of a finite group [Formula: see text] is the simple graph with vertex set [Formula: see text] and two distinct vertices are adjacent if … The power graph [Formula: see text] of a finite group [Formula: see text] is the simple graph with vertex set [Formula: see text] and two distinct vertices are adjacent if one of them is a power of the other. Let [Formula: see text] where [Formula: see text] are primes with [Formula: see text] and [Formula: see text] are positive integers. For the cyclic group [Formula: see text] of order [Formula: see text], the minimum cut-sets of [Formula: see text] are characterized in [S. Chattopadhyay, K. L. Patra and B. K. Sahoo, Vertex connectivity of the power graph of a finite cyclic group, Discrete Appl. Math. 266 (2019) 259–271] for [Formula: see text]. Recently, in [S. Mukherjee, K. L. Patra and B. K. Sahoo, On the minimum cut-sets of the power graph of a finite cyclic group, J. Algebra Appl. 23(11) (2024) 2450176], certain cut-sets of [Formula: see text] are identified such that any minimum cut-set of [Formula: see text] must be one of them. In this paper, for [Formula: see text], we explicitly determine the minimum cut-sets, in particular, the vertex connectivity of [Formula: see text] when: (i) [Formula: see text], (ii) [Formula: see text] and [Formula: see text] and (iii) [Formula: see text], [Formula: see text], [Formula: see text].

On the domination number of proper power graphs of finite groups

2025-05-09

Sudip Bera , Hiranya Kishore Dey , Kamal Lochan Patra +1 more

On the minimum cut-sets of the power graph of a finite cyclic group, II

2025-01-30

Sanjay Mukherjee , Kamal Lochan Patra , Binod Kumar Sahoo

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a power … The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a power of the other. Let $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r},$ where $p_1,p_2,\ldots,p_r$ are primes with $p_1<p_2<\cdots <p_r$ and $n_1,n_2,\ldots, n_r$ are positive integers. For the cyclic group $C_n$ of order $n$, the minimum cut-sets of $\mathcal{P}(C_n)$ are characterized in \cite{cps} for $r\leq 3$. Recently, in \cite{MPS}, certain cut-sets of $\mathcal{P}(C_n)$ are identified such that any minimum cut-set of $\mathcal{P}(C_n)$ must be one of them. In this paper, for $r\geq 4$, we explicitly determine the minimum cut-sets, in particular, the vertex connectivity of $\mathcal{P}(C_n)$ when: (i) $n_r\geq 2$, (ii) $r=4$ and $n_r=1$, and (iii) $r=5$, $n_r=1$, $p_1\geq 3$.

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Blocking sets of secant and tangent lines with respect to a quadric of $$\text{ PG }(n,q)$$

2025-01-17

Bart De Bruyn , Puspendu Pradhan , Binod Kumar Sahoo

On Blocking Sets of the Tangent Lines to a Nonsingular Quadric in $\mathrm{PG}(3,q)$, $q$ Prime

2024-12-26

Bart De Bruyn , Francesco Pavese , Puspendu Pradhan +1 more

On vertex connectivity of zero-divisor graphs of finite commutative rings

2024-04-17

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

On the minimum cut-sets of the power graph of a finite cyclic group

2023-04-11

Sanjay Mukherjee , Kamal Lochan Patra , Binod Kumar Sahoo

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A characterization of the family of secant lines to a hyperbolic quadric in PG(3,q), q odd, Part II

2022-11-16

Bart De Bruyn , Puspendu Pradhan , Binod Kumar Sahoo +1 more

On the minimum degree of power graphs of finite nilpotent groups

2022-07-18

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

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On the minimum cut-sets of the power graph of a finite cyclic group

2022-01-01

Sanjay Mukherjee , Kamal Lochan Patra , Binod Kumar Sahoo

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Blocking sets of tangent and external lines to an elliptic quadric in PG(3, q)

2021-10-01

Bart De Bruyn , Puspendu Pradhan , Binod Kumar Sahoo

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Characterizing finite nilpotent groups associated with a graph theoretic equality

2021-09-28

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

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Characterizing finite nilpotent groups associated with a graph theoretic equality

2021-05-27

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

On the minimum degree of power graphs of finite nilpotent groups

2021-01-01

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

Characterizing finite nilpotent groups associated with a graph theoretic equality

2021-01-01

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

Minimal cut-sets in the power graphs of certain finite non-cyclic groups

2020-10-16

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

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Minimum size blocking sets of certain line sets with respect to an elliptic quadric in $PG(3,q)$

2020-07-30

Bart De Bruyn , Puspendu Pradhan , Binod Kumar Sahoo

Proper divisor graph of a positive integer

2020-01-01

Hitesh Kumar , Kamal Lochan Patra , Binod Kumar Sahoo

On the minimum degree of the power graph of a finite cyclic group

2019-12-18

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

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Erratum: Vertex connectivity of the power graph of a finite cyclic group II

2019-11-01

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

Laplacian eigenvalues of the zero divisor graph of the ring <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>

2019-08-20

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

On the minimum degree of the power graph of a finite cyclic group

2019-05-26

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

Laplacian eigenvalues of the zero divisor graph of the ring $\mathbb{Z}_{n}$

2019-03-19

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

Vertex connectivity of the power graph of a finite cyclic group II

2019-02-08

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

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Blocking sets of tangent lines to a hyperbolic quadric in PG(3, 3)

2019-01-21

Bart De Bruyn , Binod Kumar Sahoo , Bikramaditya Sahu

Laplacian eigenvalues of the zero divisor graph of the ring $\mathbb{Z}_{n}$

2019-01-01

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

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On the minimum degree of the power graph of a finite cyclic group

2019-01-01

Ramesh Prasad Panda , Kamal Lochan Patra , Binod Kumar Sahoo

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Blocking sets of tangent and external lines to a hyperbolic quadric in $$\varvec{PG(3,q)}$$ P G ( 3 , q ) , $$\varvec{q}$$ q even

2018-12-17

Binod Kumar Sahoo , Bikramaditya Sahu

On minimum size blocking sets of the outer tangents to a hyperbolic quadric in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="normal">PG</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

2018-11-19

Bart De Bruyn , Binod Kumar Sahoo

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Blocking sets of certain line sets related to a hyperbolic quadric in PG(3, q)

2018-07-20

Binod Kumar Sahoo , Bikramaditya Sahu

Blocking sets of tangent and external lines to a hyperbolic quadric in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>P</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

2018-07-14

Bart De Bruyn , Binod Kumar Sahoo , Bikramaditya Sahu

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Vertex connectivity of the power graph of a finite cyclic group

2018-06-25

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

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Minimal cut-sets in the power graph of certain finite non-cyclic groups

2018-02-21

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

Vertex connectivity of the power graph of a finite cyclic group II

2018-01-01

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

Minimal cut-sets in the power graphs of certain finite non-cyclic groups

2018-01-01

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

Bounds for the Laplacian spectral radius of graphs

2017-10-30

Kamal Lochan Patra , Binod Kumar Sahoo

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Existence of non-abelian representations of the near hexagon Q(5,2)⊗Q(5,2)

2016-03-17

Binod Kumar Sahoo

Minimum size blocking sets of certain line sets related to a conic in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>P</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

2016-02-23

Kamal Lochan Patra , Binod Kumar Sahoo , Bikramaditya Sahu

Revisiting Eisenstein-type criterion over integers

2016-01-01

Akash Jena , Binod Kumar Sahoo

Polarized non-abelian representations of slim near-polar spaces

2015-12-10

Bart De Bruyn , Binod Kumar Sahoo

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Binary codes of the symplectic generalized quadrangle of even order

2015-01-28

Binod Kumar Sahoo , N. S. Narasimha Sastry

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

2013-12-01

Kamal Lochan Patra , Binod Kumar Sahoo

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Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

2010-12-04

Kamal Lochan Patra , Binod Kumar Sahoo

Algebraic Connectivity of Connected Graphs with Fixed Number of Pendant Vertices

2010-08-20

A. K. Lal , Kamal Lochan Patra , Binod Kumar Sahoo

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Non-abelian representations of the slim dense near hexagons on 81 and 243 points

2010-06-02

B. De Bruyn , Binod Kumar Sahoo , N. S. Narasimha Sastry

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Non-abelian representations of the slim dense near hexagons on 81 and 243 points

2010-03-24

Bart De Bruyn , Binod Kumar Sahoo , N. S. Narasimha Sastry

Algebraic connectivity of connected graphs with fixed number of pendant vertices

2010-03-24

A. K. Lal , Kamal Lochan Patra , Binod Kumar Sahoo

Algebraic connectivity of connected graphs with fixed number of pendant vertices

2010-01-01

A. K. Lal , Kamal Lochan Patra , Binod Kumar Sahoo

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Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

2010-01-01

Kamal Lochan Patra , Binod Kumar Sahoo

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A non-abelian representation of the dual polar space DQ(2n,2)

2009-01-01

Kamal Lochan Patra , Binod Kumar Sahoo

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Coauthor	Papers Together
Kamal Lochan Patra	34
Sriparna Chattopadhyay	11
Bart De Bruyn	10
Ramesh Prasad Panda	8
Bikramaditya Sahu	6
N. S. Narasimha Sastry	6
Puspendu Pradhan	5
Sanjay Mukherjee	4
A. K. Lal	3
Francesco Pavese	1
B. De Bruyn	1
Sudip Bera	1
Akash Jena	1
Hitesh Kumar	1
Hiranya Kishore Dey	1

A characterization of flat spaces in a finite geometry and the uniqueness of the hamming and the MacDonald codes

1966-06-01

R. C. Bose , Robert C. Burton

Finite Generalized Quadrangles

2009-04-08

Stanley E. Payne , J. A. Thas

Undirected power graphs of semigroups

2009-01-26

Ivy Chakrabarty , Susmita Ghosh , M. K. Sen

Power graphs: A survey

2013-11-13

Jemal Abawajy , Andrei Kelarev , Morshed Chowdhury

This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems … This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.

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On Blocking Sets of External Lines to a Hyperbolic Quadric in PG(3, q), q Even

2009-01-29

Paola Biondi , Pia Maria Lo Re

An Euler totient sum inequality

2016-01-11

Brian Curtin , Gholam Reza Pourgholi

Certain properties of the power graph associated with a finite group

2014-02-10

A. R. Moghaddamfar , S. Rahbariyan , W. J. Shi

The power graph [Formula: see text] of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and … The power graph [Formula: see text] of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other. The subgraph [Formula: see text] of [Formula: see text] is obtained by deleting the vertex 1 (the identity element of G). In this paper, we first investigate some properties of the power graph [Formula: see text] and its subgraph [Formula: see text]. We next provide necessary and sufficient conditions for a power graph [Formula: see text] to be a strongly regular graph, a bipartite graph or a planar graph. Finally, we obtain some infinite families of finite groups G for which the power graph [Formula: see text] contains some cut-edges.

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A characterization of finite symplectic polar spaces of odd prime order

2006-04-19

Binod Kumar Sahoo , N. S. Narasimha Sastry

Blocking Sets of Certain Line Sets Related to a Conic

2006-04-17

Angela Aguglia , Massimo Giulietti

Blocking sets of certain line sets related to a hyperbolic quadric in PG(3, q)

2018-07-20

Binod Kumar Sahoo , Bikramaditya Sahu

Non-Abelian Representations of Some Sporadic Geometries

1996-04-01

А. А. Иванов , Dmitrii V. Ṗasechnik , Sergey Shpectorov

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Edge-maximality of power graphs of finite cyclic groups

2013-12-09

Brian Curtin , Gholam Reza Pourgholi

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Vertex connectivity of the power graph of a finite cyclic group

2018-06-25

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

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On connectedness of power graphs of finite groups

2017-09-18

Ramesh Prasad Panda , K. V. Krishna

The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power … The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.

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Blocking Sets Of External Lines To A Conic In PG(2,q), q ODD

2006-08-01

Angela Aguglia , Gábor Korchmáros

Blocking sets of tangent and external lines to a hyperbolic quadric in $$\varvec{PG(3,q)}$$ P G ( 3 , q ) , $$\varvec{q}$$ q even

2018-12-17

Binod Kumar Sahoo , Bikramaditya Sahu

Algebraic connectivity of graphs

1973-01-01

Miroslav Fiedler

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Near polygons and Fischer spaces

1994-03-01

A.E. Brouwer , Arjeh M. Cohen , H.A. Wilbrink +1 more

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On the power graph of a finite group

2012-01-01

Mahsa Mirzargar , Али Реза Ашрафи , M.J. Nadjafi-Arani

The power graph P(G) of a group G is the graph whose vertex set is the group elements and two elements are adjacent if one is a power of the … The power graph P(G) of a group G is the graph whose vertex set is the group elements and two elements are adjacent if one is a power of the other. In this paper, we consider some graph theoretical properties of a power graph P(G) that can be related to its group theoretical properties. As consequences of our results, simple proofs for some earlier results are presented.

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Binary codes of the symplectic generalized quadrangle of even order

2015-01-28

Binod Kumar Sahoo , N. S. Narasimha Sastry

Some results on the power graphs of finite groups

2015-01-01

Doostabadi Alireza , Emy Zairah Ahmad , Abbas Jafarzadeh

We classify planar graphs and complete power graphs of groups and show that the only infinite group with a complete power graph is the Prüfer group p ∞ .Clique and … We classify planar graphs and complete power graphs of groups and show that the only infinite group with a complete power graph is the Prüfer group p ∞ .Clique and chromatic numbers and the automorphism group of power graphs are investigated.We also prove that the reduced power graph of a group G is regular if and only if G is a cyclic p-group or exp(G) = p for some prime number p.

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Blocking sets of external lines to a conic in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>P</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math> even

2005-11-12

Massimo Giulietti

Minimum size blocking sets of certain line sets related to a conic in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>P</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

2016-02-23

Kamal Lochan Patra , Binod Kumar Sahoo , Bikramaditya Sahu

Minimal cut-sets in the power graphs of certain finite non-cyclic groups

2020-10-16

Sriparna Chattopadhyay , Kamal Lochan Patra , Binod Kumar Sahoo

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On the order of a non-abelian representation group of a slim dense near hexagon

2008-03-03

Binod Kumar Sahoo , N. S. Narasimha Sastry

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Blocking sets of nonsecant lines to a conic in PG(2,q), q odd

2004-12-14

Angela Aguglia , Gábor Korchmáros

Abstract In a previous paper 1 , all point sets of minimum size in PG (2, q ), blocking all external lines to a given irreducible conic ${\cal C}$ , … Abstract In a previous paper 1 , all point sets of minimum size in PG (2, q ), blocking all external lines to a given irreducible conic ${\cal C}$ , have been determined for every odd q . Here we obtain a similar classification for those point sets of minimum size, which meet every external and tangent line to ${\cal C}$ . © 2004 Wiley Periodicals, Inc.

On the chromatic number of the power graph of a finite group

2015-04-27

Xuanlong Ma , Min Feng

On the Connectivity of Proper Power Graphs of Finite Groups

2015-07-06

Alireza Doostabadi , Mohammad Farrokhi Derakhshandeh Ghouchan

We study the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a nontrivial partition, and symmetric and alternating groups. Also, for such … We study the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a nontrivial partition, and symmetric and alternating groups. Also, for such a group, the corresponding proper power graph has diameter at most 26 whenever it is connected.

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The power graph of a finite group, II

2010-01-01

Peter J‎. Cameron

The directed power graph of a group G is the digraph with vertex set G, having an arc from y to x whenever x is a power of y; the … The directed power graph of a group G is the digraph with vertex set G, having an arc from y to x whenever x is a power of y; the undirected power graph has an edge joining x and y whenever one is a power of the other. We show that, for a finite group, the undirected power graph determines the directed power graph up to isomorphism. As a consequence, two finite groups which have isomorphic undirected power graphs have the same number of elements of each order.

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The full automorphism group of the power (di)graph of a finite group

2015-11-03

Min Feng , Xuanlong Ma , Kaishun Wang

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Non-Abelian Representations of Geometries

2018-12-29

А. А. Иванов

Let $\mathcal{G}$ be a geometry in which the elements of one type are called points and the elements of some other type are called …  Let $\mathcal{G}$ be a geometry in which the elements of one type are called points and the elements of some other type are called lines. Suppose that every line is incident to exactly $p + 1$ points where $p$ is a prime number. A (non-abelian) representation of $\mathcal{G}$ is a pair $(R, \psi)$, where $R$ is a group and $\psi$ is a mapping of the set of points of $\mathcal{G}$ into the set of subgroups of order $p$ in $R$ such that $R$ is generated by the image of $\psi$ and whenever $\{x_{\infty}, x_0, \dots, x_{p-1}\}$ is the set of points incident to a line, the subgroups $\psi (x_{\infty}), \psi (x_0), \dots, \psi (x_{p-1})$ are pairwise different and generate in $R$ a subgroup of order $p^2$. In this article we discuss representations of some classical and sporadic geometries and their applications to certain problems in algebraic combinatorics and group theory.

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The structure of near polygons with quads

1983-06-01

A.E. Brouwer , H.A. Wilbrink

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The Zero-Divisor Graph of a Commutative Ring

1999-07-01

David F. Anderson , Philip S. Livingston

Blocking sets of tangent and external lines to a hyperbolic quadric in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>P</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>

2018-07-14

Bart De Bruyn , Binod Kumar Sahoo , Bikramaditya Sahu

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Finite Soluble Groups

1992-12-31

Klaus Doerk , Trevor Hawkes

The Laplacian Spectrum of a Graph

1990-04-01

Robert Grone , Russell Merris , V. S. Sunder

Let G be a graph. The Laplacian matrix $L(G) = D(G) - A(G)$ is the difference of the diagonal matrix of vertex degrees and the 0-1 adjacency matrix. Various aspects … Let G be a graph. The Laplacian matrix $L(G) = D(G) - A(G)$ is the difference of the diagonal matrix of vertex degrees and the 0-1 adjacency matrix. Various aspects of the spectrum of $L(G)$ are investigated. Particular attention is given to multiplicities of integer eigenvalues and to the effect on the spectrum of various modifications of G.

Coloring of commutative rings

1988-07-01

István Beck

Near n-gons and line systems

1980-03-01

Ernest E. Shult , Arthur Yanushka

Maximal intersecting families and affine regular polygons in PG(2, q)

1989-09-01

Endre Boros , Zoltán Füredi , Jeff Kahn

Laplacian matrices of graphs: a survey

1994-01-01

Russell Merris

Blocking sets of tangent lines to a hyperbolic quadric in PG(3, 3)

2019-01-21

Bart De Bruyn , Binod Kumar Sahoo , Bikramaditya Sahu

On iteration digraph and zero-divisor graph of the ring ℤ n

2014-09-01

JU Teng-xia , WU Mei-yun

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Maximizing Algebraic Connectivity Over Unicyclic Graphs

2003-08-28

Shaun Fallat , Steve Kirkland , Sukanta Pati

We consider the class of unicyclic graphs on n vertices with girth g, and over that class, we attempt to determine which graph maximizes the algebraic connectivity. When g is … We consider the class of unicyclic graphs on n vertices with girth g, and over that class, we attempt to determine which graph maximizes the algebraic connectivity. When g is fixed, we show that there is an N such that for each n>N, the maximizing graph consists of a g cycle with n−g pendant vertices adjacent to a common vertex on the cycle. We also provide a bound on N. On the other hand, when g is large relative to n, we show that this graph does not maximize the algebraic connectivity, and we give a partial discussion of the nature of the maximizing graph in that situation.

The power graph of a finite group

2010-03-15

Peter J‎. Cameron , Susmita Ghosh

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Computing the Characteristic polynomial of a graph

1974-01-01

Allen J. Schwenk

The strong metric dimension of the power graph of a finite group

2018-01-09

Xuanlong Ma , Min Feng , Kaishun Wang

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On the foundations of polar geometry, II

1990-01-01

Francis Buekenhout

On the zero-divisor graph of a commutative ring

2003-10-14

Saieed Akbari , Abdolmajid Mohammadian

Generalized Quadrangles with a Spread of Symmetry

1999-11-01

Bart De Bruyn

Minimal covering of all chords of a conic in $PG(2,q)$, $q$ even

2006-01-01

Angela Aguglia , Gábor Korchmáros , Alessandro Siciliano

In this paper we determine the minimal blocking sets of chords of an irreducible conic $\mathcal C$ in the desarguesian projective plane $PG(2,q)$, $q$ even. Similar results on blocking sets … In this paper we determine the minimal blocking sets of chords of an irreducible conic $\mathcal C$ in the desarguesian projective plane $PG(2,q)$, $q$ even. Similar results on blocking sets of external lines, as well as of nonsecant lines, are given in [1], [3], and [2].

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