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

Isnaini Rosyida, Diari Indriati
2020 Indonesian Journal of Combinatorics
<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Given graph </span>G<span>(</span><span>V,E</span><span>)</span><span>. We use the notion of total </span>k<span>-labeling which is edge irregular. The notion </span>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
more » ... onsisting of some blocks where each block is cycle C<sub>n</sub> 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(C<sub>n</sub><sup>r</sup>) of length r for some n ≡ 0 mod 3. We also investigate tes of related chain graphs, i.e. tadpole chain graphs T<sub>r</sub>(4,n) and T<sub>r</sub>(5,n) of length r. Our results are as follows: tes(C(C<sub>n</sub><sup>r</sup>)) = ⌈(nr + 2)/3⌉ ; tes(T<sub>r</sub>(4,n)) = ⌈((5+n)r+2)/3⌉ ; tes(T<sub>r</sub>(5,n)) = ⌈((5+n)r+2)/3⌉.</p></div></div></div>