scholarly journals Computing total edge irregularity strength of some n-uniform cactus chain graphs and related chain graphs

2020 ◽  
Vol 4 (1) ◽  
pp. 53 ◽  
Author(s):  
Isnaini Rosyida ◽  
Diari Indriati
Keyword(s):  
Connected Graph ◽  
Title Page ◽  
Chain Graph ◽  
Cactus Graph ◽  
Cut Vertex ◽  
Minimum Integer ◽  
Chain Graphs ◽  
Irregularity Strength ◽  
Total Edge Irregularity Strength

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Given graph </span><em>G</em><span>(</span><span><em>V</em>,<em>E</em></span><span>)</span><span>. We use the notion of total </span><em>k</em><span>-labeling which is edge irregular. The notion </span>of total edge irregularity strength (tes) of graph <em>G</em> means the minimum integer <em>k</em> used in the edge irregular total k-labeling of <em>G</em>. A cactus graph <em>G</em> 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 <em>C<sub>n</sub></em> with same size <em>n</em> is named an <em>n</em>-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 <em>G</em> is called <em>n</em>-uniform cactus chain graph. In this paper, we determine tes of n-uniform cactus chain graphs <em>C</em>(<em>C<sub>n</sub><sup>r</sup></em>) of length <em>r</em> for some <em>n</em> ≡ 0 mod 3. We also investigate tes of related chain graphs, i.e. tadpole chain graphs <em>T<sub>r</sub></em>(4,<em>n</em>) and <em>T<sub>r</sub></em>(5,<em>n</em>) of length <em>r</em>. Our results are as follows: tes(<em>C</em>(<em>C<sub>n</sub><sup>r</sup></em>)) = ⌈(<em>nr</em> + 2)/3⌉ ; tes(<em>T<sub>r</sub></em>(4,<em>n</em>)) = ⌈((5+<em>n</em>)<em>r</em>+2)/3⌉ ; tes(<em>T<sub>r</sub></em>(5,<em>n</em>)) = ⌈((5+<em>n</em>)<em>r</em>+2)/3⌉.</p></div></div></div>

scholarly journals A Note on Edge Irregularity Strength of Some Graphs

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

scholarly journals Multiplicative Zagreb indices of cacti

2016 ◽  
Vol 08 (03) ◽  
pp. 1650040 ◽  
Author(s):  
Shaohui Wang ◽  
Bing Wei
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
Connected Graph ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Material Chemistry ◽  
Cactus Graph ◽  
Cactus Graphs

Let [Formula: see text] be multiplicative Zagreb index of a graph [Formula: see text]. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which is a generalization of trees and has been the interest of researchers in the field of material chemistry and graph theory. In this paper, we use a new tool to obtain the upper and lower bounds of [Formula: see text] for all cactus graphs and characterize the corresponding extremal graphs.

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

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