What little i know is that a discrete graph and an empty graph has chromatic number $1$, while a complete graph with $n$ vertices has a chromatic number $n$. I am curious to know whether there is any known formula to calculate the chromatic number of a simple graph $G$ just by knowing its number of vertices ( order) and number of edges (size)? Any reference (if possible) in this regards would also be appreciated.
The problem of finding the chromatic number of a graph in general in an NP-complete problem.
https://www2.cs.duke.edu/courses/fall05/cps230/L-24.pdf (see the last page)
So there is no general formula to calculate the chromatic number based on the number of vertices and edges. There are a number of types of graphs for which we know the chromatic number (e.g., cycles), and we know a number of bounds on the chromatic number (both upper and lower). But there is no known formula based only on vertices and edges.