Total Irregularity Strength Research Articles

Total Face Irregularity Strength of Certain Graphs

The edge k-labeling ψ of G is defined by a mapping from EG to a set of integers 1,2,…,k, where the integer weight assigned to the vertex x∈VG is given as wψx=∑ψxy, such that the sum is taken over every vertex of y∈VG that is adjacent to x and the integer weights of adjacent vertices must be distinct for all vertices with x≠y. An irregular assignment of G using atmost k labels which is considered to be a minimum k is defined as irregularity strength of a graph G and can be denoted as sG. There are also further works on familiar irregular assignments, such as edge irregular labelings, vertex irregular total labelings, edge irregular total labelings, and face irregular entire k-labelings of plane graphs. A plane graph can be defined as a graph that is embedded in the plane in which no two lines will be intersected. In a plane graph the number of regions present are called faces and we denote it as F. The concept of total face irregularity strength is defined by the motivation of irregular networks and entire irregular face k-labeling. In our paper, we have obtained a minimum bound for the total face irregularity strength of two-connected plane graphs like cycle-of-ladder, C-necklace graph, P-necklace graph, sibling tree, and triangular graph.

On the Total Irregularity Strength of the Corona Product of a Path with Path


 This paper deals with the totally irregular total labeling of the corona product of a path with path, cycle, and star. The results gave the exact values of the total irregularity strength of pm \dot Pn and for integer 2 \leq m \leq 3 and n\geq 3

The total vertex irregularity strength of symmetric cubic graphs of the Foster's Census

<p>Foster (1932) performed a mathematical census for all connected symmetric cubic (trivalent) graphs of order <em>n</em> with <em>n </em>≤ 512. This census then was continued by Conder et al. (2006) and they obtained the complete list of all connected symmetric cubic graphs with order <em>n</em> ≤ 768. In this paper, we determine the total vertex irregularity strength of such graphs obtained by Foster. As a result, all the values of the total vertex irregularity strengths of the symmetric cubic graphs of order <em>n</em> from Foster census strengthen the conjecture stated by Nurdin, Baskoro, Gaos & Salman (2010), namely ⌈(<em>n</em>+3)/4⌉.</p>

The total irregularity strength of m copies of the friendship graph

This paper deals with the totally irregular total labeling of the disjoin union of friendship graphs. The results shows that the disjoin union of copies of the friendship graph is a totally irregular total graph with the exact values of the total irregularity strength equals to its edge irregular total strength.

Total Irregularity Strengths of a Disjoint Union of (4,6)-Fullerene Graphs

A fullerene is a polyhedral molecule made entirely of carbon atoms other than graphite and diamond. Mathematical chemistry or Chemical graph theory as a combination of chemistry and graph theory studies the physical properties of mathematically modeled chemical compounds. Graph labeling is a vital area of graph theory that has applications in both physical and social sciences. For example, in chemistry vertex labeling is used in the constitution of valence isomers and transition labeling to study chemical reaction networks. A (4,6)-fullerene graph is a mathematical model of a (4,6)-fullerene molecule. It is a cubic-regular planar graph whose faces are squares and hexagons. In this article, we study the total vertex irregularity strength and the total edge irregularity strength of a disjoint union of (4,6)-fullerene graphs and provide their exact values.

TOTAL EDGE AND VERTEX IRREGULAR STRENGTH OF TWITTER NETWORK

Twitter data can be converted into a graph where users can represent the vertices. Then the edges can be represented as relationships between users. This research focused on determining the total edge irregularity strength (tes) and the total vertices irregularity strength (tvs) of the Twitter network. The value could be determined by finding the greatest lower bound and the smallest upper bound. The lower bound was determined by using the properties, characteristics of the Twitter network graph along with the supporting theorems from previous studies, while the upper bound is determined through the construction of the total irregular labeling function on the Twitter network. The results in this study are the tes(TW)=18 and tvs(TW)=16.

Total Irregularity Strengths of an Arbitrary Disjoint Union of (3,6)- Fullerenes.

A fullerene graph is a mathematical model of a fullerene molecule. A fullerene molecule, or simply a fullerene, is a polyhedral molecule made entirely of carbon atoms other than graphite and diamond. Chemical graph theory is a combination of chemistry and graph theory, where theoretical graph concepts are used to study the physical properties of mathematically modeled chemical compounds. Graph labeling is a vital area of graph theory that has application not only within mathematics but also in computer science, coding theory, medicine, communication networking, chemistry, among other fields. For example, in chemistry, vertex labeling is being used in the constitution of valence isomers and transition labeling to study chemical reaction networks Method: In terms of graphs, vertices represent atoms while edges stand for bonds between the atoms. By tvs (tes) we mean the least positive integer for which a graph has a vertex (edge) irregular total labeling such that no two vertices (edges) have the same weights. A (3,6)-fullerene graph is a non-classical fullerene whose faces are triangles and hexagons Results: Here, we study the total vertex (edge) irregularity strength of an arbitrary disjoint union of (3,6)-fullerene graphs and provide their exact values. The lower bound for tvs (tes) depends on the number of vertices. Minimum and maximum degree of a graph exist in literature, while to get different weights, one can use sufficiently large numbers, but it is of no interest. Here, by proving that the lower bound is the upper bound, we close the case for (3,6)-fullerene graphs.

Some Types of Irregular Labeling of Diamond Networks on Ten Vertices

There are three interesting parameters in irregular networks based on total labelling, i.e. the total vertex irregularity strength, the total edge irregularity strength, and the total irregularity strength of a graph. Besides that, there is a parameter based on edge labelling, i.e., the irregular labelling. In this paper, we determined the four parameters for diamond graph on eight vertices.

Complete bipartite graph is a totally irregular total graph

A graph G is called a totally irregular total k -graph if it has a totally irregular total k-labeling λ : V ∪ E → 1, 2, ... , k , that is a total labeling such that for any pair of different vertices x and y of G, their weights wt(x) and wt(y) are distinct, and for any pair of different edges e and f of G, their weights wt(e) and wt(f) are distinct. The minimum value k under labeling λ is called the total irregularity strength of G, denoted by ts(G). For special cases of a complete bipartite graph K m , n , the t s ( K 1, n ) and the t s ( K n , n ) are already determined for any positive integer n. Completing the results, this paper deals with the total irregularity strength of complete bipartite graph K m , n for any positive integer m and n.

On the total irregularity strength of convex polytope graphs

A vertex (edge) irregular total k-labeling ? of a graph G is a labeling of the vertices and edges of G with labels from the set {1,2,...,k} in such a way that any two different vertices (edges) have distinct weights. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x, whereas the weight of an edge is the sum of label of the edge and the vertices incident to that edge. The minimum k for which the graph G has a vertex (edge) irregular total k-labeling is called the total vertex (edge) irregularity strength of G. In this paper, we are dealing with infinite classes of convex polytopes generated by prism graph and antiprism graph. We have determined the exact value of their total vertex irregularity strength and total edge irregularity strength.

Total Face Irregularity Strength of Grid and Wheel Graph under K-Labeling of Type (1, 1, 0)

In this study, we used grids and wheel graphs G = V , E , F , which are simple, finite, plane, and undirected graphs with V as the vertex set, E as the edge set, and F as the face set. The article addresses the problem to find the face irregularity strength of some families of generalized plane graphs under k -labeling of type α , β , γ . In this labeling, a graph is assigning positive integers to graph vertices, graph edges, or graph faces. A minimum integer k for which a total label of all verteices and edges of a plane graph has distinct face weights is called k -labeling of a graph. The integer k is named as total face irregularity strength of the graph and denoted as tfs G . We also discussed a special case of total face irregularity strength of plane graphs under k -labeling of type (1, 1, 0). The results will be verified by using figures and examples.

The total irregularity strength of caterpillars with odd number of internal vertices of degree three

Given a graph G consisting of vertex set V and edget set E, repectively. Assume G is simple, connected, and the edges do not have direction. A function λ that maps V ∪ E into a set of κ-integers is named a totally irregular total k-labelling if no vertices have the same weight and also the edges of G get distinct weights. We call the minimum number k for which G has totally irregular total k-labelling as total irregularity strength of G, ts(G). In this article, we construct labels of vertices and edges of caterpillar graphs which have q internal vertices of degree 3 where q is 5,7, and 9. We obtain the exact values of ts in the following: n + 2 if the caterpillars have q=5 internal vertices, n + 3 for q=7, and n + 4 for q=9.

On total edge irregularity strength for some special types of uniform theta snake graphs

<abstract> A labeling of a connected, simple and undirected graph <italic>G</italic>(<italic>V, E</italic>) is a map that assigns the elements of a graph <italic>G</italic> with positive numbers. Many types of labeling for graph are found and one of them is a total edge irregularity strength (TEIS) of <italic>G</italic>, which denoted by <italic>tes</italic>(<italic>G</italic>). In the current paper, we defined a new type of family of graph called uniform theta snake graph, $\theta_{n}(t, m)$. Also, the exact values of total edge irregularity strengths for some special types of the new family have been determined. </abstract>

Total irregularity strength of some cubic graphs

Let G = (V;E) be a graph. A total labeling ψ : V ⋃ E → {1, 2, ....k} is called totally irregular total k-labeling of G if every two distinct vertices u and v in V (G) satisfy wt(u) ≠wt(v); and every two distinct edges u1u2 and v1v2 in E(G) satisfy wt(u1u2) ≠ wt(v1v2); where wt(u) = ψ (u) + ∑uv∊E(G) ψ(uv) and wt(u1u2) = ψ(u1) + ψ(u1u2) + ψ(u2): The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G): In this paper, we determine the exact value of the total irregularity strength of cubic graphs.

The total disjoint irregularity strength of some certain graphs

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

Totally irregular total labeling of some caterpillar graphs

Assume that G ( V,E ) is a graph with V and E as its vertex and edge sets, respectively. We have G is simple, connected, and undirected. Given a function λ from a union of V and E into a set of k -integers from 1 until k . We call the function λ as a totally irregular total k -labeling if the set of weights of vertices and edges consists of different numbers. For any u ∈ V , we have a weight wt ( u )=λ( u )+ ∑ {uy ∈ E} λ( uy ). Also, it is defined a weight wt ( e )= λ( u )+ λ( uv ) + λ( v ) for each e = uv ∈ E . A minimum k used in k -total labeling λ is named as a total irregularity strength of G , symbolized by ts( G ). We discuss results on ts of some caterpillar graphs in this paper. The results are ts (S {p,2,2,q} ) = ⌈ (p+q-1)/2 ⌉ for p , q greater than or equal to 3, while ts(S {p,2,2,2,p} ) = ⌈(2p-1)/2 ⌉, p ≥ 4.

On total H-irregularity strength of graphs

Let G be a graph with vertex set V and edge set E. Total labeling f : V(G) ∪ E(G) → {1, 2, 3, …, l} called a total l-labeling of a graph G. The total l-labeling is a total H-irregular l-labeling of graph G if for H ⊆ G, the total H-weights wtf (H) = ∑ v∈V(H) f(v) + ∑ e∈E(H) f(e) are distinct. The irregularity strength s(G) of a graph G is known as the minimum k for which G has an irregular assignment using labels at most k. The total H–irregular a–labeling from the minimum where the graph G is called the total H–irregularity strength of G, is denoted by tHs(G). In this paper, we have obtained tHs from linegrid, buttrefly, hexagonal and diamond graphs. To obtain the tHs, we begin to study the total irregularity strength of graph G with subgraph H.

On the Total Irregularity Strength of M-Copy Cycles and M-Copy Paths

On the Total Irregularity Strength of M-Copy Cycles and M-Copy Paths

The total face irregularity strength of some plane graphs

A face irregular total k-labeling λ:V∪E→{1,2,…,k} of a 2-connected plane graph G is a labeling of vertices and edges such that their face-weights are pairwise distinct. The weight of a face f under a labeling λ is the sum of the labels of all vertices and edges surrounding f. The minimum value k for which G has a face irregular total k-labeling is called the total face irregularity strength of G, denoted by tfs(G). The lower bound of tfs(G) is provided along with the exact value of two certain plane graphs. Improving the results, this paper deals with the total face irregularity strength of the disjoint union of multiple copies of a plane graph G. We estimate the bounds of tfs(G) and prove that the lower bound is sharp for G isomorphic to a cycle, a book with m polygonal pages, or a wheel.

Total irregularity strength of m-copies of rhombus graph

Let G = (V, E) is a graph. Total k-labeling on G is a mapping from V ∪ E to {1,2,3,…,k}. The weight of the vertex v ∈ V is the sum of the label of v and the labels of all the edges incident with v. While the weight of the edge uv ∈ E is the sum of the label of uv and the label of u and v. A total k-labeling of G is called a totally irregular total labeling, if the weight of every two distinct vertices are different and the weight of every two distinct edges are different. The minimum k such that a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G). In this paper, we obtain ts of m-copies of rhombus graph