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 .

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.

scholarly journals PELABELAN TOTAL TAK REGULER PADA BEBERAPA GRAF

2018 ◽  
Vol 10 (2) ◽  
pp. 9
Author(s):  
Nugroho Arif Sudibyo ◽  
Siti Komsatun
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
The Given ◽  
Irregularity Strength

For a simple graph G with vertex set V (G) and edge set E(G), a labeling $\Phi:V(G)\cup U(G)\rightarrow\{1,2,...k\}$ is  called  a  vertex  irregular  total  k- labeling of G if for any two diferent vertices x and y, their weights wt(x) and wt(y) are distinct.  The weight wt(x) of a vertex x in G is the sum of its label and the labels of all edges incident with the given vertex x.  The total vertex irregularity strength of G, tvs(G), is the smallest positive integer k for which G has a vertex irregular total k-labeling.  In this paper, we study the total vertex irregularity strength of some class of graph.

On the edge irregularity strength of corona product of graphs with cycle

2020 ◽  
Vol 12 (06) ◽  
pp. 2050083
Author(s):  
I. Tarawneh ◽  
R. Hasni ◽  
A. Ahmad ◽  
G. C. Lau ◽  
S. M. Lee
Keyword(s):  
Simple Graph ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Corona Product ◽  
Irregularity Strength

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text], respectively. An edge irregular [Formula: see text]-labeling of [Formula: see text] is a labeling of [Formula: see text] with labels from the set [Formula: see text] in such a way that for any two different edges [Formula: see text] and [Formula: see text], their weights [Formula: see text] and [Formula: see text] are distinct. The weight of an edge [Formula: see text] in [Formula: see text] is the sum of the labels of the end vertices [Formula: see text] and [Formula: see text]. The minimum [Formula: see text] for which the graph [Formula: see text] has an edge irregular [Formula: see text]-labeling is called the edge irregularity strength of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the exact value of edge irregularity strength of corona product of graphs with cycle.

scholarly journals On the Total Vertex Irregular Labeling of Proper Interval Graphs

2020 ◽  
Vol 12 (4) ◽  
pp. 537-543
Author(s):  
A. Rana
Keyword(s):  
Efficient Algorithm ◽  
Simple Graph ◽  
Interval Graphs ◽  
Vertex Weight ◽  
Vertex Set ◽  
Positive Integers ◽  
Irregularity Strength

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers).  For a simple graph G = (V, E) with vertex set V and edge set E, a labeling  Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling  Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which  a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

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.

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.

scholarly journals Irregularity Strength of Regular Graphs

10.37236/806 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
Regular Graph ◽  
Simple Graph ◽  
Regular Graphs ◽  
Isolated Vertex ◽  
Irregularity Strength

Let $G$ be a simple graph with no isolated edges and at most one isolated vertex. For a positive integer $w$, a $w$-weighting of $G$ is a map $f:E(G)\rightarrow \{1,2,\ldots,w\}$. An irregularity strength of $G$, $s(G)$, is the smallest $w$ such that there is a $w$-weighting of $G$ for which $\sum_{e:u\in e}f(e)\neq\sum_{e:v\in e}f(e)$ for all pairs of different vertices $u,v\in V(G)$. A conjecture by Faudree and Lehel says that there is a constant $c$ such that $s(G)\le{n\over d}+c$ for each $d$-regular graph $G$, $d\ge 2$. We show that $s(G) < 16{n\over d}+6$. Consequently, we improve the results by Frieze, Gould, Karoński and Pfender (in some cases by a $\log n$ factor) in this area, as well as the recent result by Cuckler and Lazebnik.

scholarly journals A Note on Edge Irregularity Strength of Some Graphs

2020 ◽  
Vol 4 (1) ◽  
pp. 10
Author(s):  
I Nengah Suparta ◽  
I Gusti Putu Suharta
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Title Page ◽  
Edge Weight ◽  
Minimum Value ◽  
Join Graph ◽  
Chain Graphs ◽  
Positive Integers ◽  
Irregularity Strength

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Let </span><em>G</em><span>(</span><span><em>V</em>, <em>E</em></span><span>) </span><span>be a finite simple graph and </span><span>k </span><span>be some positive integer. A vertex </span><em>k</em><span>-labeling of graph </span><em>G</em>(<em>V,E</em>), Φ : <em>V</em> → {1,2,..., <em>k</em>}, is called edge irregular <em>k</em>-labeling if the edge weights of any two different edges in <em>G</em> are distinct, where the edge weight of <em>e</em> = <em>xy</em> ∈ <em>E</em>(<em>G</em>), w<sub>Φ</sub>(e), is defined as <em>w</em><sub>Φ</sub>(<em>e</em>) = Φ(<em>x</em>) + Φ(<em>y</em>). The edge irregularity strength for graph G is the minimum value of k such that Φ is irregular edge <em>k</em>-labeling for <em>G</em>. In this note we derive the edge irregularity strength of chain graphs <em>mK</em><sub>3</sub>−path for m ≢ 3 (mod4) and <em>C</em>[<em>C<sub>n</sub></em><sup>(<em>m</em>)</sup>] for all positive integers <em>n</em> ≡ 0 (mod 4) 3<em>n</em> and <em>m</em>. We also propose bounds for the edge irregularity strength of join graph <em>P<sub>m</sub></em> + <em>Ǩ<sub>n</sub></em> for all integers <em>m, n</em> ≥ 3.</p></div></div></div>

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