The question is asking for each of the following four trees, how many different ways are there of colouring the vertices with $k$ colours so that no two adjacent vertices are coloured the same colour?
I am really new to this kind of content in combinatorics. I know how to approach this question when actual colours are given, but when it comes to n colours i get confused when they say no two adjacent vertices.
How many different ways are there of coloring the vertices with k colors such that adjacent vertices are colored with different colors and so that two colorings of the graph are considered different if there is no rearrangement of the vertices so that they look the same
here i don't know what hes asking for when he says without rearrangement