Local Total Distance Irregularity Labeling of Graph

Let G be a simple, connected and undirected graph with vertex set V and edge set E. A total k-labeling is defined as totally irregular total k-labeling if the weights of any two different both vertices and edges are distinct. The weight of vertex x is defined as , while the weight of edge xy is . A minimum k for which G has totally irregular total k-labeling is mentioned as total irregularity strength of G and denoted by ts(G). This paper contains investigation of totally irregular total k-labeling and determination of their total irregularity strengths for caterpillar graphs with each internal vertex between two stars has degree three. The results are and for n > 4:

Total labellings, irregularity strengths and colourings

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

Let G(V, E) be a simple graph. For a labeling ϕ : V (G) ∪ E(G) → {1, 2, …, k} the weight of a vertex x is defined as wt(x) = ϕ(x) + ∑y∈N(x) ϕ (xy), where N(x) is the set of neighbors of x and y. The labeling ϕ is called a vertex irregular total k-labeling if for every pair of distinct vertices x and y we have wt(x) ≠ wt(y). The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G and is denoted by tvs(G). In this paper, we determine total vertex irregularity strengths of trees with maximum degree four and a subdivision of a double-star.

A total labeling Φ : V (G) ∪ E(G) → {1, 2, ... , k} is called a vertex irregular total k-labeling of a graph G if different vertices in G have different weights. The weight of a vertex is defined as the sum of the labels of its incident edges and the label of that vertex. The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, denoted by tvs(G). In this paper we deal with the total vertex irregularity strength of uniform theta graphs and centralized uniform theta graphs. Theta graph is a closer representation of bipolar electric or magnetic fields so labeling of various theta graphs can help the law of physics in future.

Let &lt;em&gt;G &lt;/em&gt;= (&lt;em&gt;V&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;),&lt;em&gt;E&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;)) be a graph and &lt;em&gt;k&lt;/em&gt; be a positive integer. A total &lt;em&gt;k&lt;/em&gt;-labeling of &lt;em&gt;G&lt;/em&gt; is a map &lt;em&gt;f&lt;/em&gt; : &lt;em&gt;V&lt;/em&gt; ∪ &lt;em&gt;E&lt;/em&gt; → {1,2,3,...,&lt;em&gt;k&lt;/em&gt;}. The vertex weight &lt;em&gt;v&lt;/em&gt; under the labeling &lt;em&gt;f&lt;/em&gt; is denoted by w_&lt;em&gt;f&lt;/em&gt;(&lt;em&gt;v&lt;/em&gt;) and defined by &lt;em&gt;w&lt;/em&gt;_&lt;em&gt;f&lt;/em&gt;(&lt;em&gt;v&lt;/em&gt;) = &lt;em&gt;f&lt;/em&gt;(&lt;em&gt;v&lt;/em&gt;) + \sum_{uv \in{E(G)}} {&lt;em&gt;f&lt;/em&gt;(&lt;em&gt;uv&lt;/em&gt;)}. A total &lt;em&gt;k&lt;/em&gt;-labeling of &lt;em&gt;G&lt;/em&gt; is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of &lt;em&gt;G&lt;/em&gt;, denoted by &lt;em&gt;tvs&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;), is the minimum &lt;em&gt;k&lt;/em&gt; such that &lt;em&gt;G&lt;/em&gt; has a vertex irregular total &lt;em&gt;k&lt;/em&gt;-labeling. This labelings were introduced by Baca, Jendrol, Miller, and Ryan in 2007. Let &lt;em&gt;G&lt;/em&gt; and &lt;em&gt;H&lt;/em&gt; be two connected graphs. Let &lt;em&gt;o&lt;/em&gt; be a vertex of &lt;em&gt;H&lt;/em&gt;. The comb product between &lt;em&gt;G&lt;/em&gt; and &lt;em&gt;H&lt;/em&gt;, denoted by &lt;em&gt;G &lt;/em&gt;\rhd_o &lt;em&gt;H&lt;/em&gt;, is a graph obtained by taking one copy of &lt;em&gt;G&lt;/em&gt; and |&lt;em&gt;V&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;)| copies of &lt;em&gt;H&lt;/em&gt; and grafting the i-th copy of &lt;em&gt;H&lt;/em&gt; at the vertex &lt;em&gt;o&lt;/em&gt; to the i-th vertex of &lt;em&gt;G&lt;/em&gt;. In this paper, we determine the total vertex irregularity strength of comb product of two cycles and two stars.

Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling 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 book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.

All graph in this paper is simple and connected graph where $V(G)$ is vertex set and $E(G)$ is edge set. Let function $f : V(G)\longrightarrow \{0, 2,..., 2k_v\}$ as vertex labeling and a function $f: E(G)\longrightarrow \{1, 2,..., k_e\}$ as edge labeling where $k=max\{2k_v,k_e\}$ for $k_v,k_e$ are natural number. The weight of vertex $ u,v\in V(G) $ under $f$ is $w(u)=f(u)+ \Sigma_{uv \in E(G)} f(uv)$. In other words, the function $f$ is called local vertex irregular reflexive labeling if every two adjacent vertices has distinct weight and weight of a vertex is defined as the sum of the labels of vertex and the labels of all edges incident this vertex When we assign each vertex of $G$ with a color of the vertex weight $w(uv)$, thus we say the graph G admits a local vertex irregular reflexive coloring. The minimum number of colors produced from local vertex irregular reflexive coloring of graph $G$ is reflexive local irregular chromatic number denoted by $\chi_{lrvs}(G).$ Furthermore, the minimum $k$ required such that $\chi_{lrvs}(G)=\chi(G)$ is called a local reflexive vertex color strength, denoted by \emph{lrvcs}$(G)$. In this paper, we learn about the local vertex irregular reflexive coloring and obtain \emph{lrvcs}$(G)$ of wheel related graphs.

Given a graph G(V,E) a labeling @ : V ∪E → {1,2,...,k} is called an edge irregular total k-labeling if for every pair of distinct edges uv and xy, @(u)+@(uv)+@(v) 6 @(x)+@(xy)+@(y). The minimum k for which G has an edge irregular total k-labeling is called the total edge irregularity strength. In this paper we consider series composition of uniform theta graphs and obtain its total edge irregularity strength. We have determined the exact value of the total edge irregularity strength of this graph. We have further given an algorithm to prove the result.

Total edge irregularity strength of complete graphs and complete bipartite graphs

An edge irregular total k-labeling f : V ∪ E → {1, 2, 3,...,k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way thatfor any two different edges uv and u'v' their weights f (u) + f (uv) + f (v) and f (u') + f (u'v') + f (v') are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of disjoint union of p isomorphic double wheel graphs and disjoint union of p consecutive non-isomorphic double wheel graphs.

LetG= (V, E)be a finite, simple and undirected graph with a vertex setVand an edge setE. An edge irregular totalk-labelling is a functionf : V ᴗE → {1,2,…,k}such that for any two different edgesxyandx’y’inE, their weights are distinct. The weight of edgexyis the sum of label of edgexy, labels of vertexxand of vertexy. The minimumkfor which the graphGadmits an edge irregular totalk-labelling is called the total edge irregularity strength ofG, denoted bytes(G). We have determined the total edge irregularity strength of book graphs, double book graphs and triple book graphs. In this paper, we show the exact value of the total edge irregularity strength of quadruplet book graphs and quintuplet book graphs.

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

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

Let H= (T,S), be a finite simple graph, T(H)= T and S(H)= S, respectively, are the sets of vertices and edges on H. Let σ:T∪S→1,2,· · · ,k, be a total k-labeling on H and wσ(x), be a weight of x∈T while using σ labeling, which is evaluated based on the total number of all vertices labels in the neighborhood x and its incident edges. If every x∈T has a different weight, then σ is a distance vertex irregular total k-labeling (DVITL). Total distance vertex irregularity strength of H (tdis(H) is defined as the least k for which H has a DVITL. Our research investigates the DVITL of the path (Pr) and cycle (Cr) graphs. We establish a lower bound and then calculate the precise value of tdis(Pr) and tdis(Cr).