Definition:Degree of Polynomial

Graph Theory
Let $$G = (V,E)$$ be a graph. The degree of a vertex $$v \in V$$ is the number of vertices adjacent to $$v$$ in $$V$$.

That is, $$deg_G(v) = | \{u \in V : \{ u, v \} \in E \} |$$.

Polynomial
Let $$\left({R, +, \circ}\right)$$ be a commutative ring whose zero is $$0_R$$.

Let $$\left({D, +, \circ}\right)$$ be an integral domain such that $$D$$ is a subring of $$R$$.

Let $$X \in R$$ be transcendental over $$D$$.

Let $$D \left[{X}\right]$$ be the ring of polynomial forms in $$X$$ over $$D$$.

Let $$f$$ be a non-zero element of $$D \left[{X}\right]$$.

By Unique Representation in Polynomial Forms‎, there is one way of expressing $$f$$ as a polynomial:

$$f \in D \left[{X}\right]: f = \sum_{k=0}^n {a_k \circ X^k}$$

In particular, the coefficients $$a_0, a_1, \ldots, a_n$$ are uniquely determined by $$f$$.

The number $$n$$ is called the degree of $$f$$.

A polynomial of degree zero is a non-zero element of $$D$$.

Null Polynomial
The null polynomial $$0_R \in D \left[{X}\right]$$ does not have a degree.