Total Edge Irregularity Strength Research Articles

On the total edge irregularity strength of certain classes of cycle related graphs

For a graph G=(V,E), an edge irregular total k-labeling is a labeling of the vertices and edges of G with labels from the set {1, 2, ..., k } such that any two different edges have distinct weights. The sum of the label of edge uv and the labels of vertices u and v determines the weight of the edge uv. The smallest possible k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G. We determine the exact value of the total edge irregularity strength for some cycle related graphs.

On Tades of Disjoint Union of Some Graphs

Consider a total labeling ξ of a graph G . For every two different edges e and f of G , let w t ( e ) ≠ w t ( f ) where weight of e = x y is defined as w t ( e ) = | ξ ( e ) – ξ ( x ) – ξ ( y ) | . Then ξ is called edge irregular total absolute difference k -labeling of G . Let k be the minimum integer for which there is a graph G with edge irregular total absolute difference labeling. This k is called the total absolute difference edge irregularity strength of the graph G , denoted t a d e s ( G ) . We compute t a d e s of S C n , disjoint union of grid and zigzag graph.

Total edge irregularity strength for special types of square snake graphs

One of the extremely useful branches in graph theory is the labeling of a graph. Graph labeling plays a vital role in many fields such as database management, astronomy, coding theory, X-ray crystallography, communication network addressing and radar. A labeling of a connected simple graph GV,E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\left( {V,E} \\right)$$\\end{document} is a map that assign each element in G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G$$\\end{document} with a positive integer number. An edge irregular total λ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda^{\\!\\!\\!\\!\\!-}$$\\end{document}-labeling is a map β:VG∪EG→1,2,3,…,λ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta :V\\left( G \\right) \\cup E\\left( G \\right) \ o \\left\\{ 1,2,3, \\ldots ,\\lambda^{\\!\\!\\!\\!\\!-} \\right\\}$$\\end{document} such that Wβh≠Wβz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_\\beta \\left( h \\right) \ e W_\\beta \\left( z \\right)$$\\end{document} where Wβh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_\\beta \\left( h \\right)$$\\end{document} and Wβz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_\\beta \\left( z \\right)$$\\end{document} are weights for any two distinct edges. In this case, G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G$$\\end{document} has total edge irregularity strength (TEIS) if λ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda^{\\!\\!\\!\\!\\!-}$$\\end{document} is minimum. In this paper, a new family of graphs called square snake graphs is defined and denoted by C4,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{4,n}$$\\end{document}. Moreover, we define some related graphs of square snake graphs named double square snake graph DC4,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D\\left( {C_{4,n} } \\right)$$\\end{document}, triple square snake graph TC4,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\left( {C_{4,n} } \\right)$$\\end{document} and m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m$$\\end{document}-multiple square snake graph MmC4,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_m \\left( {C_{4,n} } \\right)$$\\end{document}. Finally, we determine TEIS for square snake graphs, double square snake graph, triple square snake graph and m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m$$\\end{document}-multiple square snake graph, which have many applications in coding theory and physics.

Total absolute difference edge irregularity strength of Tp-tree graphs

A total labeling ξ is defined to be an edge irregular total absolute difference k-labeling of the graph G if for every two different edges e and f of G there is wt(e) 6= wt(f) where weight of an edge e = xy is defined as wt(e) = |ξ(e) − ξ(x) − ξ(y)|. The minimum k for which the graph G has an edge irregular total absolute difference labeling is called the total absolute difference edge irregularity strength of the graph G, tades(G). In this paper, we determine the total absolute difference edge irregularity strength of the precise values for Tp-tree related graphs.

Total Edge Irregularity Strength of the Cartesian Product of Bipartite Graphs and Paths

For a simple graph G = (V (G), E(G)), a total labeling ∂ is called an edge irregular total k-labeling of G if ∂ : V (G) ∪ E(G) → {1, 2, . . . , k} such that for any two different edges uv and u'v' in E(G), we have wt∂(uv) not equal to wt∂(u'v') where wt∂(uv) = ∂(u) + ∂(v) + ∂(uv). The minimum k for which G has an edge irregulartotal k-labeling is called the total edge irregularity strength, denoted by tes(G). It is known that ceil((|E(G)|+2)/3) is a lower bound for the total edge irregularity strength of a graph G. In this paper we prove that if G is a bipartite graph for which this bound is tight then this is also true for Cartesian product of G with any path.

On (a,d)-total edge irregular labelings

A total edge irregular [Formula: see text]-labeling of a graph [Formula: see text], [Formula: see text] is a labeling of vertices and edges of [Formula: see text] in such a way that the weights of all edges are distinct. A total edge irregularity strength of graph [Formula: see text], denoted by [Formula: see text] is defined as the minimum [Formula: see text] for which a graph [Formula: see text] has a totally irregular total [Formula: see text]-labeling. In this paper, we define [Formula: see text]-total edge irregularity labeling and provide the irregularity strength for some well-known families of graphs.

Further Results on (a, d) -total Edge Irregularity Strength of Graphs

Further Results on (a, d) -total Edge Irregularity Strength of Graphs

Total Absolute Difference Edge Irregularity Strength of Graphs

We introduce a new graph characteristic, the total absolute difference edge irregularity strength. We obtain the estimation on the total absolute difference edge irregularity strength and determine the precise values for some families of graphs.

Generalized Arithmetic Staircase Graphs and Their Total Edge Irregularity Strengths

Let Γ=(VΓ,EΓ) be a simple undirected graph with finite vertex set VΓ and edge set EΓ. A total n-labeling α:VΓ∪EΓ→{1,2,…,n} is called a total edge irregular labeling on Γ if for any two different edges xy and x′y′ in EΓ the numbers α(x)+α(xy)+α(y) and α(x′)+α(x′y′)+α(y′) are distinct. The smallest positive integer n such that Γ can be labeled by a total edge irregular labeling is called the total edge irregularity strength of the graph Γ. In this paper, we provide the total edge irregularity strength of some asymmetric graphs and some symmetric graphs, namely generalized arithmetic staircase graphs and generalized double-staircase graphs, as the generalized forms of some existing staircase graphs. Moreover, we give the construction of the corresponding total edge irregular labelings.

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.

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.

Computing total edge irregularity strength for heptagonal snake graph and related graphs

A labeling of edges and vertices of a simple graph \(G(V,E)\) by a mapping \(\Lambda :V\left( G \right) \cup E\left( G \right) \to \left\{ { 1,2,3, \ldots ,\Psi } \right\}\) provided that any two pair of edges have distinct weights is called an edge irregular total \(\Psi\)-labeling. If \(\Psi\) is minimum and \(G\) admits an edge irregular total \(\Psi\) -labelling, then \(\Psi\) is called the total edge irregularity strength (TEIS) and denoted by \(\mathrm{tes}\left(G\right).\) In this paper, we start by defining new families of graphs called heptagonal snake graph \( {\mathrm{HPS}}_{\mathrm{n}} \),the double heptagonal snake graph \( D({\mathrm{HPS}}_{\mathrm{n}}), \) and an \(l-\) multiple heptagonal snake graph\( L({\mathrm{HPS}}_{\mathrm{n}}) \). We follow some steps to deduce the exact value of TEISs for the new families. We first labeled the vertices, and then the edges were labeled such that the weights of edges are different. After that, we calculated that exact value of TEISs for the new families.

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.

(a,d)-total edge irregularity strength of graphs

A new graph characteristic, (a,d)-total edge irregularity strength of graphs is introduced. (a,d)-edge irregular evaluations of some families of graphs has been made, upper and lower bounds of the above parameter are determined.

Total edge irregularity strength of some cycle related graphs

<p>An edge irregular total <em>k</em>-labeling <em>f</em> : <em>V</em> ∪ <em>E</em> → 1,2, ..., <em>k</em> of a graph <em>G</em> = (<em>V,E</em>) is a labeling of vertices and edges of <em>G</em> in such a way that for any two different edges <em>uv</em> and <em>u'v'</em>, their weights <em>f</em>(<em>u</em>)+<em>f</em>(<em>uv</em>)+<em>f</em>(<em>v</em>) and <em>f</em>(<em>u'</em>)+<em>f</em>(<em>u'v'</em>)+<em>f</em>(<em>v'</em>) are distinct. The total edge irregularity strength tes(<em>G</em>) is defined as the minimum <em>k</em> for which the graph <em>G</em> has an edge irregular total <em>k</em>-labeling. In this paper, we determine the total edge irregularity strength of new classes of graphs <em>C<sub>m</sub></em> @ <em>C<sub>n</sub></em>, <em>P<sub>m,n</sub></em>* and <em>C<sub>m,n</sub></em>* and hence we extend the validity of the conjecture tes(<em>G</em>) = max {⌈|<em>E</em>(<em>G</em>)|+2)/3⌉, ⌈(Δ(<em>G</em>)+1)/2⌉}<em> </em> for some more graphs.</p>

Computing the total edge irregularity strength for quintet snake graph and related graphs

In graph theory many useful mathematical disciplines, but the extremely useful one is graph labeling. A graph labeling is used in many applications in various fields like X-ray crystallography, computer science, data base management, coding theory, astronomy and physics. A labeling of edges and vertices of a simple graph G(V, E) by a mapping ¥ : V(G) ∪ E(G) → {1, 2, 3, … , ℜ} provided that any two different edges have distinct weights is called an edge irregular total ℜ -labeling. If ℜ is minimum and G admits an edge irregular total ℜ-labeling, then it is called the total edge irregularity strength (TEIS) denoted by tes(G). In this paper, we have introduced three definitions: the quintet snake graph QS n , the double quintet snake graph D(QS n) and the m–multiple quintet snake graph M(QS n). Moreover, the exact value of TEIS for the new graphs are investigated.

Total vertex irregularity strength of generalized prism graphs

A total k-labeling φ : V ∪ E → {1, 2, 3,…, k} is called a vertex irregular total k-labeling, if wt(a) ≠ wt(b), for x ≠ y ∈V, The weight of a vertex is defined as: , where N(a) is the set of neighbors of a. The minimum value of k for a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G) [7]. Recently, many authors are studying the two well-known modifications of irregularity strength of graphs, namely, the total edge irregularity strength and the total vertex irregularity strength of graphs. In this paper we study the total vertex irregularity strength of the generalized prism .

Total Edge Irregularity Strength of q Tuple Book Graphs

Let G(V, E) be a simple, undirected, and finite graph with a vertex set V and an edge set E. An edge irregular total k-labelling is a function f from the set V \cup E to the set of non-negative integer set (1, 2, ... , k) such that any two different edges in E have distinct weights. The weight of edge xy is defined as the sum of the label of vertex x, the label of vertex y and the label of edge xy. The minimum k for which the graph G can be labelled by an edge irregular total k-labelling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of triple book graphs, quadruplet book graphs and quintuplet book graphs. In this paper, we construct an edge irregular total of k-labelling and show the exact value of the total edge irregularity strength of q tuple book graphs.

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>