i need your help as i have the below exercise:
Two graphs are chromatically equivalent if they have the same chromatic polynomial. A unicyclic graph is a connected graph containing exactly one cycle. Which unicyclic graphs of the same order are chromatically equivalent?
1st step: what i need to do is to compute the chromatic polynomial of unicyclic graph with order n ( n-vertices) and cycle length k. 2nd step: consider two unicyclic graphs G and G' with n order and length k and k' respectively, and deduce something on the values k, k'.
i cannot find anything on internet which can help me.
what i think is that: the chromatic polynomial of unicyclic graph it should be P_k(x)=x(x-1)^n-1. right?
but then i do not know what else to do. Please help. thanks