Lower bounds and properties for the average number of colors in the non-equivalent colorings of a graph

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. We study the average number A(G) of colors in the non-equivalent colorings of a graph G. We conjecture several lower bounds on A(G), determine the value of this graph invariant for some classes of graphs and give general properties of A(G) which we will use for proving the validity of the conjectures for specific families of graphs, namely chordal graphs and graphs with maximum degree at most 2.