Modular Irregularity Strength on Some Corona Products of Graphs

We consider a finite graph with vertex set $V(G)$ and edge set $E(G)$. Let $G$ be a graph of order $n$ and $f$ be an edge $k$-labeling that is a mapping from the set of edged of $G$ to the set of numbers from $1,2,...,k$. The labeling $f$ is called modular irregular labeling of the graph $G$ if there exist a bijection $w_f: V(G) \to \mathbb{Z}_n$ defined by $w_f(u) =\sum_{uv\in E(u)}{f(uv)}\pmod{n}$ where $\mathbb{Z}_n$ is a group of integer modulo $n$. The modular weight of a vertex $u \in V(G)$ is the value of $w_f(u)$. The modular irregularity strength of $G$, denoted by $\mathrm{ms}(G)$, is defined as the smallest integer $k$ such that $G$ admits a modular irregular labeling with $k$ as its maximum label. In this research, we focus on the corona product of a circulant graph with a null graph of order $p$, we have results for the corona product of $G$ and $H$ where $G$ is a $d$-regular graph containing a perfect matching and $H$ is a graph of order 3 by determining the modular irregularity strength $\mathrm{ms}(G \odot C_3)=n+1$ and $\mathrm{ms}(G \odot (P_2 \cup P_1) )=\frac{3n}{2} $. Lastly, we find the modular irregularity strength $\mathrm{ms}(G \odot P_5)=\left\lceil \frac{5n+1}{3}\right\rceil$.