On The Total Edge and Vertex Irregularity Strength of Some Graphs Obtained from Star

Let $G=(V(G),E(G))$ be a graph and $k$ be a positive integer. A total $k$-labeling of $G$ is a map $f: V(G)\cup E(G)\rightarrow \{1,2,\ldots,k \}$. The edge weight $uv$ under the labeling $f$ is denoted by $w_f(uv)$ and defined by $w_f(uv)=f(u)+f(uv)+f(v)$. The vertex weight $v$ under the labeling $f$ is denoted by $w_f(v)$ and defined by $w_f(v) = f(v) + \sum_{uv \in{E(G)}} {f(uv)}$. A total $k$-labeling of $G$ is called an edge irregular total $k$-labeling of $G$ if $w_f(e_1)\neq w_f(e_2)$ for every two distinct edges $e_1$ and $e_2$ in $E(G)$. The total edge irregularity strength of $G$, denoted by $tes(G)$, is the minimum $k$ for which $G$ has an edge irregular total $k$-labeling. A total $k$-labeling of $G$ is called a vertex irregular total $k$-labeling of $G$ if $w_f(v_1)\neq w_f(v_2)$ for every two distinct vertices $v_1$ and $v_2$ in $V(G)$. The total vertex irregularity strength of $G$, denoted by $tvs(G)$, is the minimum $k$ for which $G$ has a vertex irregular total $k$-labeling. In this paper, we determine the total edge irregularity strength and the total vertex irregularity strength of some graphs obtained from star, which are gear, fungus, and some copies of stars.