On Total Edge Irregularity Strength of Some Graphs Related to Double Fan Graphs

2020 ◽  
Vol 87 (1-2) ◽  
pp. 83
Author(s):  
Husnul Khotimah ◽  
Yeni Susanti
Keyword(s):  
Positive Integer ◽  
Undirected Graph ◽  
Ladder Graph ◽  
Vertex Set ◽  
Fan Graph ◽  
Irregularity Strength ◽  
Total Edge Irregularity Strength

Let <em>G = (V(G),E(G))</em> be a simple, connected, undirected graph with non empty vertex set <em>V(G)</em> and edge set<em> E(G)</em>. The function <em>f : V(G) ∪ E(G) ↦ </em>{1,2, ...,k} (for some positive integer k) is called an edge irregular total <em>k</em>−labeling where each two edges <em>ab</em> and<em> cd</em>, having distinct weights, that are<em> f (a)+ f (ab)+ f (b) ≠ f (c)+ f (cd)+ f (d).</em> The minimum <em>k</em> for which <em>G</em> has an edge irregular total <em>k</em>−labeling is denoted by tes<em>(G)</em> and called total edge irregularity strength of graph <em>G</em>. In this paper, we determine the exact value of the total edge irregularity strength of double fan ladder graph, centralized double fan graph, and generalized parachute graph with upper path.

scholarly journals Total edge irregularity strength of quadruplet and quintuplet book graphs

2021 ◽  
Vol 36 ◽  
pp. 03004
Author(s):  
Lucia Ratnasari ◽  
Sri Wahyuni ◽  
Yeni Susanti ◽  
Diah Junia Eksi Palupi
Keyword(s):  
Undirected Graph ◽  
Vertex Set ◽  
Irregularity Strength ◽  
Total Edge Irregularity Strength

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-labelling is a function f : V ᴗE → {1,2,…,k} such that for any two different edges xy and x’y’ in E, their weights are distinct. The weight of edge xy is the sum of label of edge xy, labels of vertex x and of vertex y. The minimum k for which the graph G admits an edge irregular total k-labelling is called the total edge irregularity strength of G, denoted by tes(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.

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 .

Total edge irregularity strength of ladder-related graphs

2018 ◽  
Vol 13 (04) ◽  
pp. 2050072 ◽  
Author(s):  
Lucia Ratnasari ◽  
Yeni Susanti
Keyword(s):  
Undirected Graph ◽  
Ladder Graphs ◽  
Irregularity Strength ◽  
Total Edge Irregularity Strength

An edge irregular total [Formula: see text]-labeling on simple and undirected graph [Formula: see text] is a map [Formula: see text] such that for any different edge [Formula: see text] and [Formula: see text] their weights [Formula: see text] and [Formula: see text] are distinct. The minimum [Formula: see text] for which the graph [Formula: see text] has an edge irregular total [Formula: see text]-labeling is called the total edge irregularity strength of [Formula: see text] and is denoted by tes[Formula: see text]. In this paper, we determine the exact value of the total edge irregularity strength of families of ladder-related graphs, namely triangular ladder graphs, diagonal ladder graphs and circular triangular ladder graphs.

scholarly journals Total Vertex Irregularity Strength of the Disjoint Union of Sun Graphs

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Slamin ◽  
Dafik ◽  
Wyse Winnona
Keyword(s):  
Positive Integer ◽  
Disjoint Union ◽  
Minimum Value ◽  
Vertex Set ◽  
Irregularity Strength

A vertex irregular total -labeling of a graph with vertex set and edge set is an assignment of positive integer labels to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of , denoted by is the minimum value of the largest label over all such irregular assignment. In this paper, we consider the total vertex irregularity strengths of disjoint union of isomorphic sun graphs, , disjoint union of consecutive nonisomorphic sun graphs, , and disjoint union of any two nonisomorphic sun graphs .

Hamiltonian trace graph of matrices

2021 ◽  
pp. 2250001
Author(s):  
M. Sivagami ◽  
T. Tamizh Chelvam
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Undirected Graph ◽  
Matrix Ring ◽  
Ideal Formula ◽  
The Matrix ◽  
Vertex Set ◽  
Matrix Formula ◽  
The Ideal

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a positive integer and [Formula: see text] be the set of all [Formula: see text] matrices over [Formula: see text] For a matrix [Formula: see text] Tr[Formula: see text] is the trace of [Formula: see text] The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text][Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] The ideal-based trace graph of the matrix ring [Formula: see text] with respect to an ideal [Formula: see text] of [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] In this paper, we investigate some properties and structure of [Formula: see text] Further, it is proved that both [Formula: see text] and [Formula: see text] are Hamiltonian.

scholarly journals On The Total Edge and Vertex Irregularity Strength of Some Graphs Obtained from Star

2019 ◽  
Vol 25 (3) ◽  
pp. 314-324
Author(s):  
Rismawati Ramdani ◽  
A.N.M Salman ◽  
Hilda Assiyatun
Keyword(s):  
Positive Integer ◽  
Edge Weight ◽  
Vertex Weight ◽  
Irregularity Strength ◽  
Total Edge 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 $f: V(G)\cup E(G)\rightarrow \{1,2,\ldots,k \}$. The edge weight $uv$ under the labeling $f$ is denoted by $w_f(uv)$ and defined by $w_f(uv)=f(u)+f(uv)+f(v)$. The vertex weight $v$ under the labeling $f$ is denoted by $w_f(v)$ and defined by $w_f(v) = f(v) + \sum_{uv \in{E(G)}} {f(uv)}$. A total $k$-labeling of $G$ is called an edge irregular total $k$-labeling of $G$ if  $w_f(e_1)\neq w_f(e_2)$ for every two distinct edges $e_1$ and $e_2$  in $E(G)$.  The total edge irregularity strength of $G$, denoted by $tes(G)$, is the minimum $k$ for which $G$ has an edge irregular total $k$-labeling.  A total $k$-labeling of $G$ is called a vertex irregular total $k$-labeling of $G$ if  $w_f(v_1)\neq w_f(v_2)$ for every two distinct vertices $v_1$ and $v_2$ in $V(G)$.  The total vertex irregularity strength of $G$, denoted by $tvs(G)$, is the minimum $k$ for which $G$ has a vertex irregular total $k$-labeling.  In this paper, we determine the total edge irregularity strength and the total vertex irregularity strength of some graphs obtained from star, which are gear, fungus, and some copies of stars.

scholarly journals Associate Ring Graphs

2010 ◽  
Vol 9 (1) ◽  
pp. 31-40
Author(s):  
M. James Subhakar
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Equivalence Relation ◽  
Undirected Graph ◽  
Ring Of Integers ◽  
Vertex Set ◽  
Ring Graph

R is a commutative ring with unity. The associate ring graph AG(R) is the graph with the vertex set V = R • {0} and edge set E = {(a, b) / a, b are associates and a ≠ b}. Since the relation of being associate is on equivalence relation, this graph is an undirected graph and also each component is complete. In this paper, I present some of the interesting results majority of which are for the ring of integers modulo n, n is a positive integer.

scholarly journals PELABELAN TOTAL TAK REGULER PADA BEBERAPA GRAF

2018 ◽  
Vol 10 (2) ◽  
pp. 9
Author(s):  
Nugroho Arif Sudibyo ◽  
Siti Komsatun
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
The Given ◽  
Irregularity Strength

For a simple graph G with vertex set V (G) and edge set E(G), a labeling $\Phi:V(G)\cup U(G)\rightarrow\{1,2,...k\}$ is  called  a  vertex  irregular  total  k- labeling of G if for any two diferent vertices x and y, their weights wt(x) and wt(y) are distinct.  The weight wt(x) of a vertex x in G is the sum of its label and the labels of all edges incident with the given vertex x.  The total vertex irregularity strength of G, tvs(G), is the smallest positive integer k for which G has a vertex irregular total k-labeling.  In this paper, we study the total vertex irregularity strength of some class of graph.

scholarly journals Total Edge Irregularity Strength of q Tuple Book Graphs

2021 ◽  
Vol 13 (1) ◽  
pp. 16
Author(s):  
Lucia Ratnasari ◽  
Sri Wahyuni ◽  
Yeni Susanti ◽  
Diah Junia Eksi Palupi
Keyword(s):  
Finite Graph ◽  
Negative Integer ◽  
Vertex Set ◽  
Irregularity Strength ◽  
Total Edge Irregularity Strength

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.

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