GRAPH THEORY
REMEMBER
graphs are supposed to indicate connections between things
graph theory
study of the mathematical properties of graphs
basic graph definitions/notation 1 (continued)
a graph is CONNECTED if it is all one piece; stated in another way, from any starting vertex we can trace along the edges of the graph to any other vertex WITHOUT ANY JUMPS!
basic graph definitions/notation 2
a graph is disconnected/unconnected if it has more than one piece
planar graphs
a graph is planar if it can be drawn in such a way that its edges do not cross; to determine if a graph is planar we have to consider isomorphic versions of the graph
n-colorable
a graph is this if it can be colored with n colors so that adjacent vertices do not have the same color
vertex set
a collection given by the labels we put on the vertices; EX: {A, B, C, D}
euler circuit theorem 2
a connected graph contains a euler path when all but two vertices have even degree; the path starts and ends at the vertices with odd degree
euler circuit theorem 1
a connected graph has an euler circuit exact when every vertex has an even degree
chromatic number 2 theorem
a graph has chromatic number 2 exactly when there are NO circuits with an ODD number of vertices
euler path
a route through a graph that covers EVERY edge exactly once with an ending vertex that is different than the starting vertex; a graph is semi eulerian if there is an euler path
eulerian graphs
a route through a graph that covers every edge EXACTLY ONCE with an ending vertex that is the same as the starting vertex which is called an euler circuit; a graph is Eulerian if there is an euler circuit
circuit
a route through adjacent vertices that starts at a certain vertex and ends at the same vertex as the route started
graph
a set of points called vertices and lines called edges that connect some of the vertices
edge set
collection of the edges, described using the endpoint vertices of each edge; EX: {AB, AC, AD, BC, CD}
step 3
define the isomorphism
general tips (continued..)
edges always represent the connections, but these depend on the "things" you choose to represent your vertices
the four color theorem
every planar graph is 4-colorable; if a graph is planar then it has chromatic number 1, 2, 3, 4; you can color ANY map with 4 or fewer colors
coloring the vertices of a graph
every vertex must be colored; any two vertices that are connected by an edge must have a different color
degree of a face
for a planar graph drawn without edges crossing, the number of edges bordering a particular face
sum of degrees of vertices theorem
for any graph the sum of the degrees of the vertices equals twice the number of edges; stated in a slightly different way, Dv=2e says that Dv is ALWAYS an even number
modeling with graphs (general tips)
graphs always represent how "things" are connected or related; using graphs means recognizing the things and what kinds of connections are being described
isomorphic graphs
graphs that have the same number of vertices and identical connections (but may look different; some graphs are the "same" even though they aren't drawn in the same way
graph isomorphism
if 2 graphs are isomorphic then there is a graph isomorphism that describes how they are the same; in practice this is: a relabeling of the vertices of graph 1 so that each corresponds to the "same" vertex of graph 2; this relabeling is done so that any edge of graph 1 has a corresponding edge of graph 2 under the new labels; a good place to start is to find the degrees of the vertices of BOTH graphs
faces of a planar graphs
in any planar graph, drawn with no intersections, the edges divide the planes into different regions
path or trail
is a route through adjacent vertices that starts at a certain vertex and ends at a different vertex
isomorphic graphs theorem 2
isomorphic graphs have the same degree lists; if the degree lists are DIFFERENT, the 2 graphs are not isomorphic
isomorphic graphs theorem 1
isomorphic graphs must have the same number of vertices and edges; v1=v2; e1=e2; it is important to note that having v1=v2 and e1=e2 is NOT a guarantee that 2 graphs will be isomorphic
graph isomorphic procedure: step 1
list the degrees, in ascending order, of both graphs
modeling with graphs (maps, places, and movement)
maps or places: this is a scenario where the things are states, regions, rooms, or some other locations; the connection between locations is some kind of proximity measure, either a border or passage of some kind; vertices are physical or geographical locations; edges whenever there is a way to pass from one location to an adjacent location
faces of the graph
means BOTH the interior AND the exterior faces; usually denote the number of faces of a planar graph by f; BEFORE YOU COUNT FACES, IT IS IMPORTANT TO FIRST DRAW A PLANAR GRAPH SO THAT NO EDGES CROSS
modeling with graphs (networks and scheduling)
networks--this is a scenario where people (things) are connected to other people in some way; vertices are people; edges between people whenever there is an association, like friendship
degree of a vertex
number of edges attached to that vertex
basic graphs definitions/notation 2 (continued)
the connected pieces are called the COMPONENTS; we will usually use the letter c to describe the number of components a graph has; a connected graph has c=1; any graph with c>1 must be disconnected
exterior (or infinite) face
the region surrounding the planar graph
interior faces
the regions enclosed by the planar graph
chromatic number
the smallest possible number of colors needed to color the vertices
sum of the degrees for faces
the sum of the degrees of all faces is equal to twice the number of edges; Df=2e
scheduling
this is a scenario where people (things) are connected to other people, but through conflict; we want the graph to represent the conflict in some way; vertices usually represent people; edges between people whenever there is a conflict
traffic or movement
this is a scenario where the things are traffic or paths of objects; connections between traffic/paths represents possible accidents; vertices are flows of traffic or paths; edges whenever flow along one path can affect flow among another
step 2
understand the connections of the vertices
Euler's general formula (components)
v-e+f-c=1
Euler's formula
v-e+f=2; only applies to connected graphs
general tips (continued)
vertices always represent the "things" being connected
basic graph definitions/notation 1 (continued)
we usually use the letter e to indicate the number of edges of a graph; this is just a count of things that appear in the edge set
basic graph definitions/notation 1
we usually use the letter v to indicate the number of vertices of a graph; this is just a count of things that appear in the vertex set; sometimes we call this number the ORDER of the graph