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

Abstract

<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>