--- ## Definition > [!tldr] Definition > Two [[Graph|vertices]] $v$ and $w$ in a [[graph]] are said to be **adjacent** if they are the endpoints of an [[Graph|edge]]. We also say that $v$ and $w$ are **neighbors**. > > An [[Graph|edge]] in a [[graph]] is **incident** to a vertex if that vertex is one of its endpoints. Two edges are said to be **incident** to each other if they have an endpoint in common. Notes: - The word "incident" has two different uses: One to describe the relationship of an edge to a vertex, and another to describe the relationship between two edges. Be sure to be clear on the context when using that term. ## Examples and Non-Examples Consider this graph: ![[degree-graph-example.png]] In this graph: - Vertices $1$ and $2$ are adjacent. So are vertices $2$ and $6$. But vertex $1$ is *not* adjacent to vertex $6$. - The complete list of neighbors of vertex $3$ is: $1,2,4,5$. Vertex $3$ is adjacent to all of these. But it is not adjacent to vertex $6$. - The edge $\{2,5\}$ is incident to vertices $2$ and $5$. In fact every edge is incident to its two endpoints -- and to no other vertices. - The edges $\{2,5\}$ and $\{5,6\}$ are incident with each other. But $\{2,5\}$ is not incident with $\{1,3\}$.