Narrow Results

Coloring the vertices of a graph $G$ according to certain conditions can be considered as a random experiment and a discrete random variable (r.v.) $X$ can be defined as the number of vertices having a particular color in the proper coloring of $G$ and a probability mass function for this random variable can be defined accordingly. In this paper, we extend the concepts of mean and variance to a modified injective graph coloring and determine the values of these parameters for a number of standard graphs.

Coloring the vertices of a graph $G$ according to certain conditions is a random experiment and a discrete random variable $X$ is defined as the number of vertices having a particular color in the given type of coloring of $G$ and a probability mass function for this random variable can be defined accordingly. An equitable coloring of a graph $G$ is a proper coloring ${𝒞}$ of $G$ which an assignment of colors to the vertices of $G$ such that the numbers of vertices in any two color classes differ by at most one. In this paper, we extend the concepts of arithmetic mean and variance, the two major statistical parameters, to the theory of equitable graph coloring and hence determine the values of these parameters for a number of standard graphs.

Coloring the vertices of a graph $G$ subject to given conditions can be considered as a random experiment and corresponding to this experiment, a discrete random variable $X$ can be defined as the color of a vertex chosen at random, with respect to the given type of coloring of $G$ and a probability mass function for this random variable can be defined accordingly. A proper coloring $ℂ$ of a graph $G$, which assigns colors to the vertices of $G$ such that the numbers of vertices in any two color classes differ by at most one, is called an equitable coloring of $G$. In this paper, we study two statistical parameters of certain graphs, with respect to their equitable colorings.