Reflexive Edge Strength in Certain Graphs with Dominant Vertex

Consider a basic, connected graph G with an edge set of $E(G)$ and a vertex set of $V(G)$. The functions $f_e$ and $f_v$, which take $k=max\{k_e, 2k_v\}$, from the edge set to the first $k_e$ natural number and the non-negative even number up to $2k_v$, respectively, are the components of total $k$-labeling. An \textit{edge irregular reflexive $k$ labeling} of the graph $G$ is the total $k$-labeling, if for every two different edges $x_1x_2$ and $x_1'x_2'$ of $G$, $wt(x_1x_2) \neq wt(x_1'x_2')$, where $wt(x_1x_2)=f_v(x_1)+f_e(x_1x_2)+f_v(x_2)$. The reflexive edge strength of graph $G$ is defined as the minimal $k$ for graph $G$ with an edge irregular reflexive $k$-labeling; it is denoted by $res(G)$. The $res(G)$, where $G$ are the book, triangular book, Jahangir, and helm graphs, was found in this work.