Proper Magic Sigma Coloring of Specific Graphs

A coloring $\varphi: V(G)\rightarrow \{1,2,\ldots,k \}$ is called a magic sigma coloring of $G$ if the sum of colors of all the vertices in the neighborhood of each vertex of $G$ is the same. A graph that admits such a coloring is said to be magic sigma colorable. The minimum number $k$ required in a magic sigma coloring of a graph $G$ is called the \emph{magic sigma chromatic number}, denoted by $\sigma_{m}(G)$. These concepts have been extensively studied and motivate us to define a new type of coloring as follows. A coloring $\varphi: V(G)\rightarrow \{1,2,\ldots,k \}$ is called a \emph{proper magic sigma coloring} of $G$ if it is both a magic sigma coloring and a proper vertex coloring. The minimum number $k$ required for a proper magic sigma coloring of $G$ is called the proper magic sigma chromatic number, denoted by $\sigma_{p,m}(G)$. In this work, we introduce the concept of the proper magic sigma chromatic number and study its properties. Additionally, we determine $\sigma_{p,m}(G)$ for several specific graphs.