A sequence of polynomial pairs assigned to a graph

In this note, we introduce a new degree-based descriptive parameter, namely, the degree polynomial-pair (DPP), for the edges of a simple graph. This notion leads to a concept, namely, the degree polynomial-pair sequence (DPPS) in graphs. We show that the DPPS of a graph gives more information about the graph than its degree polynomial sequence does, but it still does not identify the graph uniquely. We obtain the DPPS for some well-known graphs. Also we prove a theorem in which a necessary condition for the graphic realizability of a sequence of polynomial pairs is given. Several open problems concerning these subjects are given as well.