scholarly journals The radius of unit graphs of rings

2021 ◽  
Vol 6 (10) ◽  
pp. 11508-11515
Author(s):  
Zhiqun Li ◽  
◽  
Huadong Su
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Unit Graph ◽  
Vertex Set ◽  
Ring Extensions ◽  
Nonzero Identity

<abstract><p>Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.</p></abstract>

scholarly journals On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nurdin Hinding ◽  
Hye Kyung Kim ◽  
Nurtiti Sunusi ◽  
Riskawati Mise
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
Cluster Graph ◽  
Cluster Graphs ◽  
Irregularity Strength

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ ​ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

Finite commutative rings with higher genus unit graphs

2014 ◽  
Vol 14 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Huadong Su ◽  
Kenta Noguchi ◽  
Yiqiang Zhou
Keyword(s):  
Nonnegative Integer ◽  
Commutative Rings ◽  
Orientable Surface ◽  
Simple Graph ◽  
Isomorphism Classes ◽  
Unit Graph ◽  
Higher Genus ◽  
Vertex Set ◽  
Finite Commutative Rings

Let R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three.

Domination number of unit graph of ℤn

2020 ◽  
Vol 12 (05) ◽  
pp. 2050059
Author(s):  
Huadong Su ◽  
Liying Yang
Keyword(s):  
Domination Number ◽  
Simple Graph ◽  
Prime Divisors ◽  
Unit Graph ◽  
Vertex Set

The unit graph of a ring [Formula: see text] is the simple graph, denoted by [Formula: see text], whose vertex set is [Formula: see text], and in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a unit of [Formula: see text]. In this paper, we completely determine the domination number of [Formula: see text] when [Formula: see text] has exactly three prime divisors.

scholarly journals Line graph of unit graphs associated with finite commutative rings

2021 ◽  
Author(s):  
Pranjali ◽  
Amit Kumar ◽  
Pooja Sharma
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Common Vertex ◽  
Unit Graph ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Nonzero Identity

For a given graph G, its line graph denoted by L(G) is a graph whose vertex set V (L(G)) = E(G) and {e1, e2} ∈ E(L(G)) if e1 and e2 are incident to a common vertex in G. Let R be a finite commutative ring with nonzero identity and G(R) denotes the unit graph associated with R. In this manuscript, we have studied the line graph L(G(R)) of unit graph G(R)  associated with R. In the course of the investigation, several basic properties, viz., diameter, girth, clique, and chromatic number of L(G(R)) have been determined. Further, we have derived sufficient conditions for L(G(R)) to be Planar and Hamiltonian

SIGNED TOTAL {K}-DOMINATION AND {K}-DOMATIC NUMBERS OF GRAPHS

2012 ◽  
Vol 04 (01) ◽  
pp. 1250006
Author(s):  
S. M. SHEIKHOLESLAMI ◽  
L. VOLKMANN
Keyword(s):  
Positive Integer ◽  
Upper Bounds ◽  
Simple Graph ◽  
Domatic Number ◽  
Vertex Set ◽  
Dominating Function

Let k be a positive integer, and let G be a simple graph with vertex set V(G). A function f : V(G) → {±1, ±2, …, ±k} is called a signed total {k}-dominating function if ∑u∈N(v) f(u) ≥ k for each vertex v ∈ V(G). A set {f1, f2, …, fd} of signed total {k}-dominating functions on G with the property that [Formula: see text] for each v∈V(G), is called a signed total {k}-dominating family (of functions) on G. The maximum number of functions in a signed total {k}-dominating family on G is the signed total {k}-domatic number of G, denoted by [Formula: see text]. Note that [Formula: see text] is the classical signed total domatic number dS(G). In this paper, we initiate the study of signed total k-domatic numbers in graphs, and we present some sharp upper bounds for [Formula: see text]. In addition, we determine [Formula: see text] for several classes of graphs. Some of our results are extensions of known properties of the signed total domatic number.

scholarly journals Path-induced closure operators on graphs for defining digital Jordan surfaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1374-1380
Author(s):  
Josef Šlapal
Keyword(s):  
Positive Integer ◽  
Closure Operator ◽  
Simple Graph ◽  
Closure Operators ◽  
Adjacency Graph ◽  
Digital Space ◽  
Vertex Set

Abstract Given a simple graph with the vertex set X, we discuss a closure operator on X induced by a set of paths with identical lengths in the graph. We introduce a certain set of paths of the same length in the 2-adjacency graph on the digital line ℤ and consider the closure operators on ℤm (m a positive integer) that are induced by a special product of m copies of the introduced set of paths. We focus on the case m = 3 and show that the closure operator considered provides the digital space ℤ3 with a connectedness that may be used for defining digital surfaces satisfying a Jordan surface theorem.

scholarly journals Galois connections between sets of paths and closure operators in simple graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 1573-1581 ◽  
Author(s):  
Josef Šlapal
Keyword(s):  
Positive Integer ◽  
Digital Images ◽  
Galois Connection ◽  
Simple Graph ◽  
Closure Operators ◽  
Simple Graphs ◽  
Galois Connections ◽  
Digital Plane ◽  
Vertex Set

AbstractFor every positive integer n,we introduce and discuss an isotone Galois connection between the sets of paths of lengths n in a simple graph and the closure operators on the (vertex set of the) graph. We consider certain sets of paths in a particular graph on the digital line Z and study the closure operators associated, in the Galois connection discussed, with these sets of paths. We also focus on the closure operators on the digital plane Z2 associated with a special product of the sets of paths considered and show that these closure operators may be used as background structures on the plane for the study of digital images.

Bounds on the k-tuple domatic number of a graph

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
Lutz Volkmann
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Dominating Set ◽  
Simple Graph ◽  
Dominating Sets ◽  
Domatic Number ◽  
Vertex Set

AbstractLet k be a positive integer, and let G be a simple graph with vertex set V (G). A vertex of a graph G dominates itself and all vertices adjacent to it. A subset S ⊆ V (G) is a k-tuple dominating set of G if each vertex of V (G) is dominated by at least k vertices in S. The k-tuple domatic number of G is the largest number of sets in a partition of V (G) into k-tuple dominating sets.In this paper, we present a lower bound on the k-tuple domatic number, and we establish Nordhaus-Gaddum inequalities. Some of our results extends those for the classical domatic number.

On the normalized Laplacian spectrum of the comaximal graphs

2021 ◽  
pp. 2250094
Author(s):  
Mojgan Afkhami
Keyword(s):  
Commutative Ring ◽  
Simple Graph ◽  
Laplacian Spectrum ◽  
Induced Subgraph ◽  
Randic Index ◽  
Ring Of Integers ◽  
Laplacian Energy ◽  
Comaximal Graph ◽  
Vertex Set ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. The comaximal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be an induced subgraph of [Formula: see text] with nonunit elements of [Formula: see text] as vertices. In this paper, we describe the normalized Laplacian spectrum of [Formula: see text], and we determine it for some values of [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text]. Moreover, we investigate the normalized Laplacian energy and general Randic index of [Formula: see text].

The diameter of power graphs of symmetric groups

2018 ◽  
Vol 17 (12) ◽  
pp. 1850234 ◽  
Author(s):  
Kobra Pourghobadi ◽  
Sayyed Heidar Jafari
Keyword(s):  
Positive Integer ◽  
Finite Groups ◽  
Symmetric Groups ◽  
Simple Graph ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Power

The power graph of a group [Formula: see text] is the simple graph [Formula: see text], with vertex-set [Formula: see text] and vertices [Formula: see text] and [Formula: see text] are adjacent, if and only if [Formula: see text] and either [Formula: see text] or [Formula: see text] for some positive integer [Formula: see text]. The proper power graph of [Formula: see text], denoted [Formula: see text], is the graph obtained from [Formula: see text] by deleting the vertex [Formula: see text]. In [On the connectivity of proper power graphs of finite groups, Comm. Algebra 43 (2015) 4305–4319], it is proved that if [Formula: see text] and neither [Formula: see text] nor [Formula: see text] is a prime, then [Formula: see text] is connected and [Formula: see text]. In this paper, we improve the diameter bound of [Formula: see text] for which [Formula: see text] is connected. We show that [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. We also describe a number of short paths in these power graphs.

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