scholarly journals Modular Irregular Labeling on Double-Star and Friendship Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
K. A. Sugeng ◽  
Z. Z. Barack ◽  
N. Hinding ◽  
R. Simanjuntak
Keyword(s):  
Double Star ◽  
Star Graph ◽  
Edge Label ◽  
Vertex Weight ◽  
Graph Classes ◽  
Incident Edge ◽  
Irregularity Strength

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

scholarly journals Graceful and Magic Labeling in Special Fuzzy Graphs

2019 ◽  
Vol 8 (3) ◽  
pp. 5320-5328
Keyword(s):  
Double Star ◽  
Star Graph ◽  
Fuzzy Graphs

In this paper, we present an algorithm to find fuzzy labeling of a star graph K1,n , bi star graph Bn,n and double star graph K1,n,n . We prove that star graph K1,n at most 89 edges are fuzzy graceful iff it admits fuzzy magic graph. Also we prove that bi star graph Bn,n having at most 59 edges are fuzzy graceful iff it is a fuzzy bi magic graph .We prove that fuzzy labeled double star graph K1,n,n at most 30 edges is fuzzy graceful.

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 Sequence variations of the 1-2-3 conjecture and irregularity strength

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
Ben Seamone ◽  
Brett Stevens
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Minimum Degree ◽  
List Size ◽  
Edge Weighting ◽  
Incident Edge ◽  
Sequence Variations ◽  
Distinct Sequence ◽  
International Audience ◽  
Irregularity Strength

Graph Theory International audience Karonski, Luczak, and Thomason (2004) conjecture that, for any connected graph G on at least three vertices, there exists an edge weighting from 1, 2, 3 such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) make a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than 1, 2, 3. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of E(G), and so two variations arise - one where we may choose any ordering of E(G) and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights.

scholarly journals The diachromatic number of double star graph

2021 ◽  
Vol 2106 (1) ◽  
pp. 012024
Author(s):  
Nilamsari Kusumastuti ◽  
Raventino ◽  
Fransiskus Fran
Keyword(s):  
Oriented Graph ◽  
Internal Vertex ◽  
Double Star ◽  
Star Graph

Abstract We are interested in the extension for the concept of complete colouring for oriented graph G → that has been proposed in many different notions by several authors (Edwards, Sopena, and Araujo-Pardo in 2013, 2014, and 2018, respectively). An oriented colouring is complete if for every ordered pair of colours, at least one arc in G → whose endpoints are coloured with these colours. The diachromatic number, dac ( G → ) , is the greatest number of colours in a complete oriented colouring. In this paper, we establish the formula of diachromatic numbers for double star graph, k 1 , n , n → , over all possible orientations on the graph. In particular, if din (u) = 0 (resp. dout(u) = 0)and din (wi ) = 1 (resp. dout (w 1) = 1) for all i, then dac ( k 1 , n , n → ) = ⌊ n ⌋ + 1 , where u is the internal vertex and w i , i ∈ {1,…, n}, is the pendant vertices of the digraph.

scholarly journals Total vertex irregularity strength of comb product of two cycles

2018 ◽  
Vol 197 ◽  
pp. 01007
Author(s):  
Rismawati Ramdani ◽  
Muhammad Ali Ramdhani
Keyword(s):  
Positive Integer ◽  
Connected Graphs ◽  
Vertex Weight ◽  
Irregularity Strength

Let G = (V (G),E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f : V (G) ∪ E(G) → {1,2...,k}. The vertex weight v under the labeling f is denoted by Wf(v) and defined by Wf(v) = f(v) + Σuv∈E(G)f(uv). A total k-labeling of G is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of G, denoted by tvs(G), is the minimum k such that G has a vertex irregular total k-labeling. This labeling was introduced by Bača, Jendrol', Miller, and Ryan in 2007. Let G and H be two connected graphs. Let o be a vertex of H . The comb product between G and H, in the vertex o, denoted by G⊳o H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. In this paper, we determine the total vertex irregularity strength of comb product of Cn and Cm where m ∈ {1,2}.

b-Chromatic sum of a graph

2015 ◽  
Vol 07 (04) ◽  
pp. 1550040 ◽  
Author(s):  
P. C. Lisna ◽  
M. S. Sunitha
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Double Star ◽  
Star Graph ◽  
Proper Coloring ◽  
Color Class ◽  
Wheel Graph ◽  
Chromatic Sum

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.

scholarly journals A Research of Achromatic and B-Chromatic Number of Multistar-Related Graphs

2019 ◽  
Vol 8 (6S) ◽  
pp. 141-144
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Double Star ◽  
Star Graph ◽  
Triple Star

The structural properties, achromatic number and b-chromatic number of central graph of double star graph and triple star graph have been studied in [6]. In this paper, these studies have been carried out, for the central graph of any multi star graph and central graph of shadow graph of star graph and double star graph.

scholarly journals The total disjoint irregularity strength of some certain graphs

2020 ◽  
Vol 4 (2) ◽  
pp. 91
Author(s):  
Meilin I Tilukay ◽  
A. N. M. Salman
Keyword(s):  
Lower Bound ◽  
Title Page ◽  
Edge Weight ◽  
Complete Graphs ◽  
Vertex Weight ◽  
Minimum Value ◽  
Irregularity Strength

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Under a totally irregular total </span><em>k</em><span>-labeling of a graph </span><span><em>G</em> </span><span>= (</span><span><em>V</em>,<em>E</em></span><span>), we found that for some certain graphs, the edge-weight set </span><em>W</em><span>(</span><em>E</em><span>) and the vertex-weight set </span><em>W</em><span>(</span><em>V</em><span>) of </span><span><em>G</em> </span><span>which are induced by </span><span><em>k</em> </span><span>= </span><span>ts</span><span>(</span><em>G</em><span>), </span><em>W</em><span>(</span><em>E</em><span>) </span><span>∩ </span><em>W</em><span>(</span><em>V</em><span>) is a non empty set. For which </span><span>k</span><span>, a graph </span><span>G </span><span>has a totally irregular total labeling if </span><em>W</em><span>(</span><em>E</em><span>) </span><span>∩ </span><em>W</em><span>(</span><em>V</em><span>) = </span><span>∅</span><span>? We introduce the total disjoint irregularity strength, denoted by </span><span>ds</span><span>(</span><em>G</em><span>), as the minimum value </span><span><em>k</em> </span><span>where this condition satisfied. We provide the lower bound of </span><span>ds</span><span>(</span><em>G</em><span>) and determine the total disjoint irregularity strength of cycles, paths, stars, and complete graphs.</span></p></div></div></div>

scholarly journals Harmonious coloring on double star graph families

2012 ◽  
Vol 43 (2) ◽  
pp. 153-158 ◽  
Author(s):  
Vernold Vivin.J ◽  
Venkatachalam M. ◽  
Kaliraj K.
Keyword(s):  
Chromatic Number ◽  
Line Graph ◽  
Double Star ◽  
Star Graph ◽  
Graph Families

In this present paper, we have proved for the line graph of double star graph, the harmonious chromatic number and the achromatic number are equal. As a motivation this work can be extended by classifying the different families of graphs for which these two numbers are equal.

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