Proper Injective Coloring Parameters of Some Wheel-Related Graphs

Any vertex coloring protocol of a graph can be viewed as a random experiment of assigning colors to the vertices, such that the random variable of this experiment is the number of vertices given a specific color in that coloring. Based on this, the statistical parameters of mean and variance have been extended as the chromatic mean and chromatic variance for various vertex colorings of graphs, in the literature. In this paper, the chromatic mean and chromatic variance of graphs with respect to its proper injective coloring are investigated for certain wheel-related graphs, by determining the optimal injective coloring pattern for the graphs with optimal chromatic sum.