Graph theory: adjacency vs incident

Question

Okay, so I think if 2 vertices are adjacent to each other, they are incident to each other....or do I have it wrong? Is this just different terminology. I thought I was totally clear on this for my class, but now I am doubting myself reading the book and looking at my notes. I just want to know if I have it correct, and if I don't could someone explain to me what the difference is between the two. I found several wiki's and different university definitions, but none ever said that the two are alike and I'm confused and would like some reassurance. Thanks in advance.

Answer 1

Usually one speaks of adjacent vertices, but of incident edges.

Two vertices are called adjacent if they are connected by an edge.

Two edges are called incident, if they share a vertex.

Also, a vertex and an edge are called incident, if the vertex is one of the two vertices the edge connects.

Answer 2

If for two vertices $A$ and $B$ there is an edge $e$ joining them, we say that $A$ and $B$ are adjacent.

If two edges $e$ and $f$ have a common vertex $A$, the edges are called incident.

If the vertex $A$ is on edge $e$, the vertex $A$ is often said to be incident on $e$.

There is unfortunately some variation in usage. So you need to check the particular book or notes for the definition being used.

Answer 3

Excerpted from wikipedia:

• Two edges of a graph are called adjacent (sometimes coincident) if they share a common vertex.

• Similarly, two vertices are called adjacent if they share a common edge.

• An edge and a vertex on that edge are called incident.

This terminology seems very sensible to my ear.