scholarly journals Local Inclusive Distance Vertex Irregular Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1673
Author(s):  
Kiki Ariyanti Sugeng ◽  
Denny Riama Silaban ◽  
Martin Bača ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Graph Invariant ◽  
Irregular Graphs ◽  
Irregularity Strength

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs.

scholarly journals The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 605
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Marcela Lascsáková ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Isolated Vertex ◽  
Positive Integers ◽  
Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

On H-irregularity strength of cycles and diamond graphs

2021 ◽  
pp. 2150157
Author(s):  
Hayat Labane ◽  
Isma Bouchemakh ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Strength Formula ◽  
Irregularity Strength

A simple graph [Formula: see text] admits an [Formula: see text]-covering if every edge in [Formula: see text] belongs to at least one subgraph of [Formula: see text] isomorphic to a given graph [Formula: see text]. The graph [Formula: see text] admits an [Formula: see text]-irregular total[Formula: see text]-labeling [Formula: see text] if [Formula: see text] admits an [Formula: see text]-covering and for every two different subgraphs [Formula: see text] and [Formula: see text] isomorphic to [Formula: see text], there is [Formula: see text], where [Formula: see text] is the associated [Formula: see text]-weight. The total[Formula: see text]-irregularity strength of [Formula: see text] is [Formula: see text]. In this paper, we give the exact values of [Formula: see text], where [Formula: see text]. For the versions edge and vertex [Formula: see text]-irregularity strength [Formula: see text] and [Formula: see text], respectively, we determine the exact values of [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the diamond graph.

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 .

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 Computing Total Edge Irregularity Strength for Heptagonal Snake Graph and Related Graphs

2021 ◽  
Author(s):  
Fatma Salama
Keyword(s):  
Simple Graph ◽  
New Family ◽  
Irregularity Strength ◽  
Total Edge Irregularity Strength

Abstract A labeling of edges and vertices of a simple graph 𝐺(𝑉,𝐸) by a mapping Ŧ:𝑉(𝐺) ∪ 𝐸(𝐺)→{ 1,2,3,…,Ћ} provided that any two pair of edges have distinct weights is called an edge irregular total Ћ-labeling. If Ћ is minimum and 𝐺 admits an edge irregular total Ћ -labelling, then Ћ is called the total edge irregularity strength (TEIS) and denoted by 𝑡𝑒𝑠(𝐺). In this paper, the definitions of the heptagonal snake graph HPSn ,the double heptagonal snake graph 𝐷(HPSn) and an 𝑙−multiple heptagonal snake graph 𝐿(HPSn) have been introduced. The exact value of TEISs for the new family has also been investigated.

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>

scholarly journals On bounds for harmonic topological index

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 311-317 ◽  
Author(s):  
Marjan Matejic ◽  
Igor Milovanovic ◽  
Emina Milovanovic
Keyword(s):  
Topological Index ◽  
Upper Bounds ◽  
Simple Graph ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Harmonic Index

Let G=(V,E), V = {1,2,..., n}, E = {e1,e2,..., em}, be a simple graph with n vertices and m edges. Denote by d1 ? d2 ?... ? dn > 0 and d(e1) ? d(e2) ?... ? d(em), sequences of vertex and edge degrees, respectively. If i-th and j-th vertices of the graph G are adjacent, it is denoted as i ~ j. Graph invariant referred to as harmonic index is defined as H(G)= ? i~j 2/di+dj. Lower and upper bounds for invariant H(G) are obtained.

scholarly journals Irregularity and Modular Irregularity Strength of Wheels

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2710
Author(s):  
Martin Bača ◽  
Muhammad Imran ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Graph Labeling ◽  
Edge Label ◽  
Labeling Scheme ◽  
Natural Modification ◽  
Maximum Multiplicity ◽  
Irregularity Strength

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel edges required to ensure that all vertices have distinct degrees? This problem is known as the graph labeling problem and, for its solution, Chartrand et al. introduced irregular assignments. The irregularity strength of a graph G is known as the maximal edge label used in an irregular assignment, minimized over all irregular assignments. Thus, the irregularity strength of a simple graph G is equal to the smallest maximum multiplicity of an edge of G in order to create an irregular multigraph from G. In the present paper, we show the existence of a required irregular labeling scheme that proves the exact value of the irregularity strength of wheels. Then, we modify this irregular mapping in six cases and obtain labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength.

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