# Graph theory: adjacency vs incident

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.

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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.

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I would go so far as to say that vertex-edge incidence is the more common usage. – Erick Wong Sep 4 '12 at 0:13
Okay, thank you so much. I am now reviewing what I have and I had thought they were both referring to the vertices for both cases. This makes more sense now. – pqsk Sep 4 '12 at 0:19
@ErickWong: that seems right, considering objects like the incidence matrix. Thank you for the insight, I will modify my sentence. – Gregor Bruns Sep 4 '12 at 0:27

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.

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Thank you for your answer. It makes it more clear. I'm going to mark Bruns' as the answer, since I feel that it was more clear to me, but thank you so much for your input. Very useful as well. – pqsk Sep 4 '12 at 0:20

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.

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It does to me too now. Lol....my eyes were seeing vertex for everything. I did not realize that for an incident it was referring to the edges. The problem when you are working non-stop day in and day out and then going to school on your off time. :-S – pqsk Sep 4 '12 at 0:24

An edge "e" in a graph (Undirected or directed ) that is associated with the pair of vertices n and q is said to be incident on n and q, and n and q are said to be incident on e and to be adjacent vertices.

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