Degree polynomial of vertices in a graph and its behavior under graph operations

TL;DR

This paper introduces the degree polynomial of vertices in a graph, explores its properties under various graph operations, and establishes conditions for polynomial sequence realizability, advancing graph polynomial theory.

Contribution

It defines the degree polynomial concept, analyzes its behavior under graph operations, and provides realizability conditions for polynomial sequences.

Findings

Degree polynomial sequence is stronger than degree sequence.

Explicit formulas for degree polynomials under graph operations.

Necessary conditions for polynomial sequence realizability.

Abstract

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