  In this paper, we introduce a new concept namely degree polynomial for
vertices of a simple graph. This notion leads to a concept namely degree
polynomial sequence which is stronger than the concept of degree sequence.
After obtaining the degree polynomial sequence for some well-known graphs, we
prove a theorem which gives a necessary condition for realizability of a
sequence of polynomials with coefficients in positive integers. Also we
calculate the degree polynomial for vertises of join, Cartesian product, tensor
product, and lexicographic product of two simple graphs and for vertices of the
complement of a simple graph. Some examples, counterexamples, and open problems
concerning to this subjects, is given as well
