On local irregularity of the vertex coloring of the corona product of a tree graph

Let $G=(V,E)$ be a graph with a vertex set $V$ and an edge set $E$. The graph $G$ is said to be with a local irregular vertex coloring if there is a function $f$ called a local irregularity vertex coloring with the properties: (i) $l:(V(G)) \to \{ 1,2,...,k \} $ as a vertex irregular $k$-labeling and $w:V(G)\to N,$ for every $uv \in E(G),$ ${w(u)\neq w(v)}$ where $w(u)=\sum_{v\in N(u)}l(i)$ and (ii) $\mathrm{opt}(l)=\min\{ \max \{ l_{i}: l_{i} \text{ is a vertex irregular labeling}\}\}$. The chromatic number of the local irregularity vertex coloring of $G$ denoted by $\chi_{lis}(G)$, is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this paper, we study a local irregular vertex coloring of $P_m\bigodot G$ when $G$ is a family of tree graphs, centipede $C_n$, double star graph $(S_{2,n})$, Weed graph $(S_{3,n})$, and $E$ graph $(E_{3,n})$.