Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree

Let $G=(V,E)$ be a graph (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets).

Distinct vertices $u,v$ are adjacent, or neighbours, when $\{u,v\}\in E$. A vertex $v$ and an edge $e$ are incident when $v\in e$. The open neighbourhood and closed neighbourhood of $v$ are

$N_G(v):=\{\,u\in V:\{u,v\}\in E\,\},\qquad N_G[v]:=N_G(v)\cup\{v\}.$

The degree of $v$ is $\deg_G(v):=|N_G(v)|$, equivalently the number of edges incident with $v$. A graph is $r$-regular when every vertex has degree $r$; it is cubic when it is $3$-regular. The multiset of the vertex degrees, usually written in nonincreasing order, is the degree sequence.

When $V\ne\varnothing$, the minimum degree and maximum degree are

$\delta(G):=\min_{v\in V}\deg_G(v),\qquad \Delta(G):=\max_{v\in V}\deg_G(v).$

They are defined because $V$ is a nonempty finite set (The cardinality $\lvert A\rvert$ of a finite set). Neither $\delta(G)$ nor $\Delta(G)$ is defined for the null graph.