Graph Terminology

MAT115

graph: consists of a set of vertices (or nodes), and a set of edges (or arcs) that each connect a pair of vertices.
These are often depicted as circles, connected by segments, e.g.
- vertex: a point or node in a graph
- edge (or arc): a connection made between two vertices in a graph
"In the above Graph, the set of vertices V = {0,1,2,3,4} and the set of edges E = {01, 12, 23, 34, 04, 14, 13}." (Thanks to GeeksForGeeks for the image I'm using and this description.)
adjacent vertices: two distinct vertices are said be adjacent if there is an edge connecting them
So in the graph above, vertices 0 and 1 are adjacent (for example), connected by edge 01.
degree of a vertex: the number of edges incident to (arriving at) a given vertex
In the graph above, the degree of vertex 4 is three.
isolated vertex: a vertex of degree zero (no edges going in or out of it)
In the graph above, no vertex is isolated. But you can imagine that in a graph representing Facebook relationships between friends, for example, some friend may be left out of the group...:(
reachability: describes a relationship between two vertices in which one vertex is reachable from the other via a path
In the graph above, every vertex is reachable from every other.
parallel: where there is more than a single edge connecting two given vertices
If the black segment indicating the vertices in the graph above were actually an edge, then there would be parallel edges connecting vertices 2 and 3.
loop (or sling): where an edge joins a vertex to itself
There are no loops in the graph above.
path: a sequence of edges that begins at an initial vertex and ends at a terminal vertex
There is a path from vertex 0 to vertex 2, for example. In fact, there are several! (Can you find at least four different ones, that don't reuse any vertices? A mathematician then asks "Exactly how many such paths are there?")
length: the number of edges appearing in the sequence of a path
Are there two paths of length three from vertex 0 to vertex 2?
cycle (or circuit): a path that begins and terminates in the same vertex
You can see that there are lots of different cycles in the graph above.
digraph (or directed graph): a graph in which every edge is directed
The graph above is not directed. A road map, showing roads between intersections, might use directed edges -- because some roads are one-way: they leave one intersection to go to another, but not vice versa.
labeled graph: a graph in which each vertex is labeled
The graph above is labeled.
weight: a number assigned to an edge
The graph above does not have weighted edges.
weighted graph: a connected graph in which a number has been assigned to each edge
The graph above does is not weighted.
simple graph: a graph that contains no loops or parallel edges
The graph above is simple.
complete graph: a simple graph in which every pair of distinct vertices are adjacent
The graph above is not complete. For example, there is not edge joining vertex 0 to vertex 2 (so they are not adjacent).
connected graph: a graph in which for any given vertex in the graph, all the other vertices are reachable from it
The graph above is not connected.
Planar graph : a graph that can be drawn on paper (on the plane) such that no two edges intersect. Edges only come together at common vertices.
Euler path: a path on a graph that traces out each edge exactly once. No edge is traversed twice. If the path starts and ends at the same vertex, it becomes an Euler Cycle.
Face: A face is a region enclosed by edges. The edges make up its boundary. The outside of the graph counts as a face (like a circle has an inside and outside region).
A face is ultimately two-dimensional, or should be thought of that way.
Would you be surprised to know that a tetrahedron has four faces?

Footnotes:

The names of the edges tell which vertices they join -- in this case we could have just as well have called it "10", without confusion, because there is only one edge joining those two vertices. But, in general, there may be more than one edge joining two vertices.
incident: an edge that joins two specified vertices is said to be incident to them

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