The use of Matlab to compute the total vertex irregularity strength of generalized uniform cactus chain graphs with pendant vertices

<div class="page" title="Page 1"><div class="layoutArea"><div class="column">Given graph G(V,E). We use the notion of total k-labeling which is edge irregular. The notion of total edge irregularity strength (tes) of graph G means the minimum integer k used in the edge irregular total k-labeling of G. A cactus graph G is a connected graph where no edge lies in more than one cycle. A cactus graph consisting of some blocks where each block is cycle Cn with same size n is named an n-uniform cactus graph. If each cycle of the cactus graph has no more than two cut-vertices and each cut-vertex is shared by exactly two cycles, then G is called n-uniform cactus chain graph. In this paper, we determine tes of n-uniform cactus chain graphs C(Cnr) of length r for some n ≡ 0 mod 3. We also investigate tes of related chain graphs, i.e. tadpole chain graphs Tr(4,n) and Tr(5,n) of length r. Our results are as follows: tes(C(Cnr)) = ⌈(nr + 2)/3⌉ ; tes(Tr(4,n)) = ⌈((5+n)r+2)/3⌉ ; tes(Tr(5,n)) = ⌈((5+n)r+2)/3⌉.</div></div></div>

Download Full-text

On total vertex irregularity strength of generalized uniform cactus chain graphs with pendant vertices

Journal of Discrete Mathematical Sciences and Cryptography ◽

10.1080/09720529.2020.1830562 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1369-1380

Author(s):

Isnaini Rosyida ◽

Mulyono ◽

Diari Indriati

Keyword(s):

Chain Graphs ◽

Irregularity Strength

Download Full-text

A Note on Edge Irregularity Strength of Some Graphs

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2020.4.1.2 ◽

2020 ◽

Vol 4 (1) ◽

pp. 10

Author(s):

I Nengah Suparta ◽

I Gusti Putu Suharta

Keyword(s):

Positive Integer ◽

Simple Graph ◽

Title Page ◽

Edge Weight ◽

Minimum Value ◽

Join Graph ◽

Chain Graphs ◽

Positive Integers ◽

Irregularity Strength

<div class="page" title="Page 1"><div class="layoutArea"><div class="column">Let G(V, E) be a finite simple graph and k be some positive integer. A vertex k-labeling of graph G(V,E), Φ : V → {1,2,..., k}, is called edge irregular k-labeling if the edge weights of any two different edges in G are distinct, where the edge weight of e = xy ∈ E(G), wΦ(e), is defined as wΦ(e) = Φ(x) + Φ(y). The edge irregularity strength for graph G is the minimum value of k such that Φ is irregular edge k-labeling for G. In this note we derive the edge irregularity strength of chain graphs mK3−path for m ≢ 3 (mod4) and C[Cn(m)] for all positive integers n ≡ 0 (mod 4) 3n and m. We also propose bounds for the edge irregularity strength of join graph Pm + Ǩn for all integers m, n ≥ 3.</div></div></div>

Download Full-text

The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽

10.3390/sym13040605 ◽

2021 ◽

Vol 13 (4) ◽

pp. 605

Author(s):

Martin Bača ◽

Zuzana Kimáková ◽

Marcela Lascsáková ◽

Andrea Semaničová-Feňovčíková

Keyword(s):

Simple Graph ◽

Isolated Vertex ◽

Positive Integers ◽

Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

Download Full-text