Help me in proving an interesting result

Sir my supervisor give me a special type of graph. Corresponding to each vertex of graph there exist a polynomial(BY RULE). My graph has 15 verteices. So i find15 polynomial after very lenghty calculations. Its Very interesting for me all 15 polynomials were the same. So we find a very good result for given graph that "There exists a unique polynomial for each vertex of the Graph(say g)". Can anybody give me a slight idea that how i prove this result mathematically. I will be very thankful.

Perhaps if you explained how the polynomials are related to the graph or to anything at all, we would be able to help you.