6.4.0: Graph Theory
- Page ID
- 76153
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)There are several definitions that are important to understand before delving into Graph Theory. They are:
- A graph is a picture of dots called vertices and lines called edges.
- An edge that starts and ends at the same vertex is called a loop.
- If there are two or more edges directly connecting the same two vertices, then these edges are called multiple edges.
- If there is a way to get from one vertex of a graph to all the other vertices of the graph, then the graph is connected.
- If there is even one vertex of a graph that cannot be reached from every other vertex, then the graph is disconnected.
In the above graph, the vertices are U, V, W, and Z and the edges are UV, VV, VW, UW, WZ1, and WZ2.
This is a connected graph. VV is a loop. WZ1, and WZ2 are multiple edges.
The graph in Figure \(\PageIndex{2}\) is connected while the graph in Figure \(\PageIndex{3}\) is disconnected.
- Order of a Network: the number of vertices in the entire network or graph
- Adjacent Vertices: two vertices that are connected by an edge
- Adjacent Edges: two edges that share a common vertex
- Degree of a Vertex: the number of edges at that vertex
- Path: a sequence of vertices with each vertex adjacent to the next one that starts and ends at different vertices and travels over any edge only once
- Circuit: a path that starts and ends at the same vertex
- Bridge: an edge such that if it were removed from a connected graph, the graph would become disconnected
In the above graph the following is true:
- Vertex A is adjacent to vertex B, vertex C, vertex D, and vertex E.
- Vertex F is adjacent to vertex C, and vertex D.
- Edge DF is adjacent to edge BD, edge AD, edge CF, and edge DE.
The degrees of the vertices:
| A | 4 |
| B | 4 |
| C | 4 |
| D | 4 |
| E | 4 |
| F | 2 |
Here are some paths in the above graph: (there are many more than listed)
A,B,D
A,B,C,E
F,D,E,B,C
Here are some circuits in the above graph: (there are many more than listed)
B,A,D,B
B,C,F,D,B
F,C, E, D, F
The above graph does not have any bridges.