15.12: Chapter 12
- Page ID
- 129941
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Your Turn
![A graph represents common boundaries between regions of Oahu. The five vertices are L, N, W, C, and S. Edges connect L with N and C. Edges connect N with W and C. Edges connect W with C and S. An edge connects S with C.](https://math.libretexts.org/@api/deki/files/110092/CS_Figure_12_01_015.png?revision=1)
Graph 1:
![One graph labeled graph 1. The graph has four vertices labeled 1, 2, 2, and 1. The edges connect 1 2, 2 2, and 2 1.](https://math.libretexts.org/@api/deki/files/110090/CS_Figure_12_02_026a.png?revision=1)
Graph 2:
![One graph labeled graph 2. The graph has four vertices labeled 1, 3, 1, and 1. The edges connect 1 3, 3 1, and 3 1.](https://math.libretexts.org/@api/deki/files/110088/CS_Figure_12_02_026b.png?revision=1)
![Graph F has four vertices: V, W, X, and Z. The edges connect X W, X V, V W, X Z, and W Z.](https://math.libretexts.org/@api/deki/files/110089/CS_Figure_12_02_03b.png?revision=1)
Quadrilateral: (b, c, d, e or a, b, e, d)
Pentagon: (a, b, c, d, e)
Graph C1 has 4 vertices and Graph C2 has 5.
Graph C1 has no vertex of degree 4, but Graph C2 has one vertex of degree 4.
They do not have the same cycles. For example, Graph C2 has a pentagon cycle, but Graph C1 does not.
- The total number of vertices in each graph is different. The Diamonds graph has 17 vertices while Dots graph has only 16.
- The degrees of vertices differ. The Diamonds graph has vertices of degrees 4 while the Dots graph does not.
- The graphs have different sizes of cyclic subgraphs. The Diamonds graph has 4 squares (4-cycles), while the Dots graph has 3 squares. Also, the Dots graph has 8-cycles while the Diamonds graph does not.
- a − q, d − s, c − p, and b − r.
- a − p, d − s, c − q, and b − r.
- a − q, d − r, c − p, and b − s.
- a − p, d − r, c − q, and b − s.
![The complement of graph H is displayed. The vertices are A, B, C, D, and E. The edges connect B E, B D, and E C.](https://math.libretexts.org/@api/deki/files/110091/CS_Figure_12_03_079.png?revision=1)
![A graph with seven vertices. It has 2 vertices in red, 3 vertices in blue, and 2 vertices in purple. Edges from the first red vertex lead to two blue vertices and two purple vertices. The first purple and the first blue vertices are connected by an edge. The second purple and the second blue vertices are connected by an edge. The second purple connects with the second red and third blue via edges. The second red and third blue are connected via an edge.](https://math.libretexts.org/@api/deki/files/110093/CS_Figure_12_04_121.png?revision=1)
![A graph represents common boundaries between regions of Oahu. The five vertices are L, N, W, C, and S. L and W are in purple. N and S are in blue. C is in red. Edges from L connect with N and C. Edges from N connect with W and C. Edges from W connect with C and S. An edge from S connects with S.](https://math.libretexts.org/@api/deki/files/110095/CS_Figure_12_04_124.png?revision=1)
![A graph of a neighborhood has 16 vertices arranged in 4 rows and 4 columns. The vertices are connected to form 9 squares. On each side, a curved edge connects the center two vertices.](https://math.libretexts.org/@api/deki/files/110094/CS_Figure_12_05_151.png?revision=1)
![Graph Z has four vertices: a, b, c, and d. The edges connect a b, b d, d c, and c a. A double edge connects a to d.](https://math.libretexts.org/@api/deki/files/110096/CS_Figure_12_05_161b.png?revision=1)
v → w → x → u → z → y → w → u
- V → W → X → Y → Z → V
- V → W → X → Z → Y → V
- V → W → Y → X → Z → V
- V → W → Y → Z → X → V
- V → W → Z → X → Y → V
- V → W → Z → Y → X → V
- V → X → W → Y → Z → V
- V → X → W → Z → Y → V
- V → X → Y → W → Z → V
- V → X → Y → Z → W → V (reverse of 6)
- V → X → Z → W → Y → V
- V → X → Z → Y → W → V (reverse of 4)
- V → Y → X → W → Z → V
- V → Y → X → Z → W → V (reverse of 5)
- V → Y → W → X → Z → V
- V → Y → W → Z → X → V (reverse of 11)
- V → Y → Z → X → W → V (reverse of 2)
- V → Y → Z → W → X → V (reverse of 8)
- V → Z → X → Y → W → V (reverse of 3)
- V → Z → X → W → Y → V (reverse of 15)
- V → Z → Y → X → W → V (reverse of 1)
- V → Z → Y → W → X → V (reverse of 7)
- V → Z → W → X → Y → V (reverse of 13)
- V → Z → W → Y → X → V (reverse of 9)
![Three graphs. Each graph has 6 vertices and 5 edges.](https://math.libretexts.org/@api/deki/files/110097/CS_Figure_12_10_265a.png?revision=1)
![Three graphs. Each graph has 6 vertices and 5 edges.](https://math.libretexts.org/@api/deki/files/110098/CS_Figure_12_10_265b.png?revision=1)
![Three graphs. Each graph has 6 vertices and 5 edges.](https://math.libretexts.org/@api/deki/files/110099/CS_Figure_12_10_265c.png?revision=1)
List A: be, eh, hi, gi, bg
List B: ac and ad, ac and af, ac and cd, ac and cf, ad and af, ad and cd, ad and cf, af and cd, af and cf, or cd and cf.
![Two weighted graphs. The vertices in each graph are as follows: U, V, W, X, and Y. The edges in the first graph are as follows. U W, 37. W X, 45. W V, 68. V Y, 24. The edges in the second graph are as follows. U W, 37. W X, 45. X Y, 68. Y V, 24.](https://math.libretexts.org/@api/deki/files/110100/CS_Figure_12_10_280.png?revision=1)
Check Your Understanding
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