Exam 10: A: Graphs

Suppose you have a graph G with vertices v₁, v₂, . . . , v₁₇. Explain how you would use the adjacency matrix A to ﬁnd (a) The number of paths from $v _ { 5 } \text { to } v _ { 3 }$ of length 12. (b) The length of a shortest path from $v _ { 5 } \text { to } v _ { 3 }$ .

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Question 21

either give an example or prove that there are none. -A simple digraph with indegrees 0, 1, 2 and outdegrees 0, 1, 2.

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Question 22

either give an example or prove that there are none. -A planar graph with 8 vertices, 12 edges, and 6 regions.

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Question 23

ﬁnd $\kappa ( G ) , \lambda ( G ) \text {, and } \min _ { \nu \in V } \operatorname { deg } ( \nu ) \text {, and determine which of the two inequalities }$ $\text { in } \kappa ( G ) \leq \lambda ( G ) \leq \min _ { \nu \in V } \operatorname { deg } ( \nu ) \text { are strict }$ - $ﬁnd \kappa ( G ) , \lambda ( G ) \text {, and } \min _ { \nu \in V } \operatorname { deg } ( \nu ) \text {, and determine which of the two inequalities } \text { in } \kappa ( G ) \leq \lambda ( G ) \leq \min _ { \nu \in V } \operatorname { deg } ( \nu ) \text { are strict } -$

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Question 24

the grid graph G_m,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: $the grid graph Gm,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -For which positive integers m and n does G _ { m , n } have an Euler path but no Euler circuit?$ -For which positive integers m and n does $G _ { m , n }$ have an Euler path but no Euler circuit?

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Question 25

ﬁll in the blanks. -If G is a planar connected graph with 20 vertices, each of degree 3, then G has ____ regions.

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Question 26

ﬁll in the blanks. -List all positive integers n such that $Q _ { n }$ has a Hamilton circuit ____.

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Question 27

either give an example or prove that there are none. -A connected simple planar graph with 5 regions and 8 vertices, each of degree 3.

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Question 28

Find the strongly connected components of the graph.

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Question 29

the grid graph G_m,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: $the grid graph Gm,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find a formula for the number of regions (including the inﬁnite region) of G _ { m , n }$ -Find a formula for the number of regions (including the inﬁnite region) of $G _ { m , n }$

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Question 30

refer to a cubic graph, i.e., a graph that is simple and has every vertex of degree 3. -Draw a cubic graph with 7 vertices, or else prove that there are none.

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Question 31

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for K_3,2.

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Question 32

ﬁll in the blanks. -The adjacency matrix for $\mathrm { K } _ { m , n }$ has ____ columns.

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Question 33

ﬁll in the blanks. -If a regular graph G has 10 vertices and 45 edges, then each vertex of G has ____ degree .

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Question 34

the grid graph G_m,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: $the grid graph Gm,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find a formula for the number of vertices of G _ { m , n }$ -Find a formula for the number of vertices of $G _ { m , n }$

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Question 35

either give an example or prove that there are none. -A simple digraph with indegrees 1, 1, 1 and outdegrees 1, 1, 1.

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Question 36

Determine whether this graph is planar.

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Question 37

either give an example or prove that there are none. -A graph with a Hamilton circuit but no Hamilton path.

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Question 38

ﬁll in the blanks. -The adjacency matrix for $Q _ { 4 }$ has ____ entries.

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Question 39

either give an example or prove that there are none. -A simple graph with 8 vertices, whose degrees are 0, 1, 2, 3, 4, 5, 6, 7.

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Question 40

Showing 21 - 40 of 131

Exam 10: A: Graphs

either give an example or prove that there are none. -A simple digraph with indegrees 0, 1, 2 and outdegrees 0, 1, 2.

either give an example or prove that there are none. -A planar graph with 8 vertices, 12 edges, and 6 regions.

the grid graph Gm,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -For which positive integers m and n does Gm,nG _ { m , n }Gm,n​ have an Euler path but no Euler circuit?

ﬁll in the blanks. -If G is a planar connected graph with 20 vertices, each of degree 3, then G has ____ regions.

ﬁll in the blanks. -List all positive integers n such that QnQ _ { n }Qn​ has a Hamilton circuit ____.

either give an example or prove that there are none. -A connected simple planar graph with 5 regions and 8 vertices, each of degree 3.

Find the strongly connected components of the graph.

the grid graph Gm,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find a formula for the number of regions (including the inﬁnite region) of Gm,nG _ { m , n }Gm,n​

refer to a cubic graph, i.e., a graph that is simple and has every vertex of degree 3. -Draw a cubic graph with 7 vertices, or else prove that there are none.

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for K3,2.

ﬁll in the blanks. -The adjacency matrix for Km,n\mathrm { K } _ { m , n }Km,n​ has ____ columns.

ﬁll in the blanks. -If a regular graph G has 10 vertices and 45 edges, then each vertex of G has ____ degree .

the grid graph Gm,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find a formula for the number of vertices of Gm,nG _ { m , n }Gm,n​

either give an example or prove that there are none. -A simple digraph with indegrees 1, 1, 1 and outdegrees 1, 1, 1.

Determine whether this graph is planar.

either give an example or prove that there are none. -A graph with a Hamilton circuit but no Hamilton path.

ﬁll in the blanks. -The adjacency matrix for Q4Q _ { 4 }Q4​ has ____ entries.

either give an example or prove that there are none. -A simple graph with 8 vertices, whose degrees are 0, 1, 2, 3, 4, 5, 6, 7.

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

ﬁll in the blanks. -List all positive integers n such that $Q _ { n }$ has a Hamilton circuit ____.

Find the vertex-chromatic number, the edge-chromatic number, and the region-chromatic number for K_3,2.

ﬁll in the blanks. -The adjacency matrix for $\mathrm { K } _ { m , n }$ has ____ columns.

ﬁll in the blanks. -The adjacency matrix for $Q _ { 4 }$ has ____ entries.