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