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.

Walk-set induced connectedness in digital spaces

2017 ◽  
Vol 33 (2) ◽  
pp. 247-256
Author(s):  
JOSEF SLAPAL ◽  
Keyword(s):  
Jordan Curve ◽  
Digital Images ◽  
Simple Graph ◽  
Jordan Curve Theorem ◽  
Strong Product ◽  
Digital Plane ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Digital Spaces

In an undirected simple graph, we define connectedness induced by a set of walks of the same lengths. We show that the connectedness is preserved by the strong product of graphs with walk sets. This result is used to introduce a graph on the vertex set Z2 with sets of walks that is obtained as the strong product of a pair of copies of a graph on the vertex set Z with certain walk sets. It is proved that each of the walk sets in the graph introduced induces connectedness on Z2 that satisfies a digital analogue of the Jordan curve theorem. It follows that the graph with any of the walk sets provides a convenient structure on the digital plane Z2 for the study of digital images.

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 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 .

scholarly journals Path-set induced closure operators on graphs

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 863-871 ◽  
Author(s):  
Josef Slapal
Keyword(s):  
Image Analysis ◽  
Digital Image ◽  
Digital Image Analysis ◽  
Closure Operator ◽  
Simple Graph ◽  
Digital Topology ◽  
Closure Operators ◽  
Vertex Set ◽  
Positive Length ◽  
Path Connectedness

Given a simple graph, we associate with every set of paths of the same positive length a closure operator on the (vertex set of the) graph. These closure operators are then studied. In particular, it is shown that the connectedness with respect to them is a certain kind of path connectedness. Closure operators associated with sets of paths in some graphs with the vertex set Z2 are discussed which include the well known Marcus-Wyse and Khalimsky topologies used in digital topology. This demonstrates possible applications of the closure operators investigated in digital image analysis.

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>

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.

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.

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.

scholarly journals Vertex decomposable graph

2016 ◽  
Vol 99 (113) ◽  
pp. 203-209 ◽  
Author(s):  
N. Hajisharifi ◽  
S. Yassemi
Keyword(s):  
Simple Graph ◽  
Simple Graphs ◽  
Vertex Set ◽  
Vertex Decomposable ◽  
Vertex Sets

Let G be a simple graph on the vertex set V (G) and S = {x11,...,xn1} a subset of V (G). Let m1,...,mn ? 2 be integers and G1,...,Gn connected simple graphs on the vertex sets V (Gi) = {xi1,..., ximi} for i = 1,..., n. The graph G(G1,...,Gn) is obtained from G by attaching Gi to G at the vertex xi1 for i = 1,...,n. We give a characterization of G(G1,...,Gn) for being vertex decomposable. This generalizes a result due to Mousivand, Seyed Fakhari, and Yassemi.

scholarly journals The Polytope of Degree Partitions

10.37236/1072 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Amitava Bhattacharya ◽  
S. Sivasubramanian ◽  
Murali K. Srinivasan
Keyword(s):  
Convex Hull ◽  
Linear Inequality ◽  
Degree Sequence ◽  
Extreme Points ◽  
Main Tool ◽  
Simple Graph ◽  
Degree Sequences ◽  
Simple Graphs ◽  
Vertex Set ◽  
Partition Polytope

The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. The polytope of degree partitions (respectively, degree sequences) is the convex hull of degree partitions (respectively, degree sequences) of all simple graphs on the vertex set $[n]$. The polytope of degree sequences has been very well studied. In this paper we study the polytope of degree partitions. We show that adding the inequalities $x_1\geq x_2 \geq \cdots \geq x_n$ to a linear inequality description of the degree sequence polytope yields a linear inequality description of the degree partition polytope and we show that the extreme points of the degree partition polytope are the $2^{n-1}$ threshold partitions (these are precisely those extreme points of the degree sequence polytope that have weakly decreasing coordinates). We also show that the degree partition polytope has $2^{n-2}(2n-3)$ edges and $(n^2 -3n + 12)/2$ facets, for $n\geq 4$. Our main tool is an averaging transformation on real sequences defined by repeatedly averaging over the ascending runs.

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