Math 38: Graph Theory
Dartmouth College
Spring 2004
Homework #4: Solutions
2.3.2. If T is a minimum-weight spanning tree of a weighted graph G, then the u, v -path in T is not necessarily a minimum-weight u, v -path in G. Consider the following graph: w v
Math 485, Graph Theory: Homework #4
Stephen G. Simpson
Due Friday, December 4, 2009
The assignment consists of Exercises 3.1.8, 3.1.19, 3.1.24, 3.1.25, 3.2.6, 6.1.12,
6.1.25, 6.1.35, 6.2.6 in the West textbook. Each exercise counts for 10 points.
Here are
21-484, Spring 2004, HW 5 Solutions
1. (3.1.21) Let G be an X ,Y -bigraph such that |N (S )| > |S | for every
S X with S = . Prove that for every edge e of G there is a
matching M that contains e and saturates X . In other words, prove
that every edge ext
Graph Theory, Spring 2012, Homework 1
1. (West 1.1.4)
=
From the denition of isomorphism, prove that G H if and only if G H .
=
Proof:
(=)
Suppose G H . Then, there is an isomorphism (which is also a bijection) f : V (G) V (H ) such
=
that
uv E (G) if an
Graph Theory, Spring 2012, Homework 2
1. (West 1.2.22)
Prove that a graph is connected if and only if for every partition of its vertices into two nonempty
sets, there is an edge with endpoints in both sets.
Proof:
(=)
Suppose G is a connected graph. Cons
Graph Theory, Spring 2012, Homework 3
1. (West 1.2.10)
Prove or disprove:
(a) Every Eulerian bipartite graph has an even number of edges.
This statement is TRUE.
Proof:
Suppose G is an Eulerian bipartite graph. We know, then, by a Theorem proved in class,
Graph Theory, Spring 2012, Homework 5
1. (West 3.3.1) Determine whether the graph (on page 145) has a 1-factor.
Solution: The graph does not have a 1-factor.
Proof: Let G be the graph pictured. Let S be the set of vertices in G which have degree 3. Then,
Graph Theory Midterm Examination
Applied Mathematics and Statistics 550.472/672
Spring 2012
Instructions.
You have nearly 48 hours to work on this exam. It is due on Friday, March 9 at 3:00 pm.
You should submit your papers to Shaer 103 (Engineering Acad
Graph Theory, Spring 2012, Homework 5
1. (West 5.1.1)
Solution:
2. (West 5.1.4)
Solution:
3. (West 5.1.7)
Solution:
4. (West 5.1.20)
Solution:
5. (West 5.1.22)
Solution:
Graph Theory, Spring 2012, Homework 7 and Homework 8
Homework 7:
1. (West 5.3.1)
Solution:
2. (West 5.3.3)
Solution:
3. (West 5.3.4)
Solution:
Homework 8:
1. (West 8.3.13)
Solution:
2. (West 8.3.17)
Solution:
3. (West 8.3.18)
Solution:
Graph Theory, Spring 2012, Homework 9
1. (West 7.1.1)
Solution:
2. (West 7.1.2)
Solution:
3. (West 7.1.4)
Solution:
4. (West 7.1.9)
Solution:
5. (West 7.1.11)
Solution:
6. (West 7.1.27)
Solution:
Graph Theory, Spring 2012, Homework 10
1. (West 7.2.4)
Solution:
2. (West 7.2.6)
Solution:
3. (West 7.2.7)
Solution:
4. (West 7.2.8)
Solution:
5. (West 7.2.27)
Solution:
6. (West 7.2.33) Prove that a simple graph is Hamiltonian if e(G)
connected if e(G)
Graph Theory, Spring 2012, Homework 11
1. (West 6.1.9)
Solution:
2. (West 6.1.20)
Solution:
3. (West 6.1.29)
Solution:
4. (West 6.1.30)
Solution:
5. (West 6.1.34)
Solution:
6. Prove that if a triangulation has vertices of degree 5 and 6 only, then it has
GRAPH THEORY FUNDAMENTAL DEFINITIONS
JOHNS HOPKINS UNIVERSITY
APPLIED MATHEMATICS & STATISTICS 550.472/672
Standard combinatorial notation:
Z: Set of integers.
N: Set of natural numbers1; i.e., the nonnegative integers cfw_0, 1, 2, 3, . . ..
For n N,
Welcome to Graph Theory
First Lectures
Applied Mathematics & Statistics 550.472/672
Spring 2012
About The Course
550.472/672 (Graph Theory)
First Lectures
Spring 2012
2 / 53
Who, Where, When, How
Instructor: Ed Scheinerman
Email: ers@jhu.edu
Ofce hours: B
Proof that Kruskals Greedy Algorithm
Produces a Minimum Weight Spanning Tree
Let G be a connected graph with n vertices and let w : E (G) R.
Suppose that Kruskals algorithm produces the spanning tree T by adding
the following edges in this order: e1 , e2
Tutte Matrix and Perfect Matchings
Let G be a graph with V (G) = [n]. Dene the Tutte matrix of G to be the n n-matrix
T (G) with
xi j
if i j and i < j,
[T (G)]i j = x ji if i j and i > j, and
0
otherwise
For example,
x12
0
x14
0
x
12
0
x23
0
T (C4 ) =
2 /22/12
Lives Forever Linked Through Kidney Transplant Chain 124 - NYTimes.com
Reprints
This copy is for your personal, noncommercial use only. You can order presentation-ready copies for
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L A B
B
Rn stands for all (real) n-vectors (columns) and Rmn stands for all (real) m n-matrices.
Usual typographic conventions apply: s R, x Rn , and A Rmn .
We write either x y or x, y for xt y = x1 y1 + + xn yn . The magnitude of a vector is
x = x x. V
Graph Laplacians
Professor Scheinerman
Johns Hopkins University
Applied Mathematics & Statistics 550.472/672
Graph Matrices
Graph Theory (JHU)
Graph Laplacians
550.472/672
2 / 51
Adjacency and degree matrices
Let G be a (simple) graph with V (G ) = [n].
A
Graph Theory Midterm Examination
Applied Mathematics and Statistics 550.472/672
Spring 2006
Instructions.
You have 24 hours for this exam. You may pick up this exam from Ms. Bechtel in the
Applied Mathematics & Statistics Department Ofce (Whitehead Hall,
Graph Theory Midterm Examination
Applied Mathematics and Statistics 550.472/672
Spring 2007
Instructions.
You have one day to work on this exam. You may choose one of the following two plans:
Take the exam from Wednesday, March 7 to Thursday, March 8: I
Graph Theory Midterm Examination
Applied Mathematics and Statistics 550.472/672
Spring 2010
Instructions.
You have one day to work on this exam. It is due in class on Wednesday, March 24 at
9:00 a.m.
Please be sure to write Graduate (those enrolled in 6
Graph Theory Midterm Examination
Applied Mathematics and Statistics 550.472/672
Spring 2011
Instructions.
You have 48 hours to work on this exam. It is due on Tuesday, March 15 at 12:00 noon. You
should submit your papers to Shaer 103 (Engineering Academ
Graph Theory Final Examination
Applied Mathematics and Statistics 550.472/672
Spring 2010
Instructions (read carefully):
You have two days to work on this exam based on the date you selected to begin working on
the exam. The exam is due at 9:00 a.m.
If
GRAPH THEORY FINAL EXAMINATION
APPLIED MATHEMATICS AND STATISTICS 550.472/672
SPRING 2011
Instructions (read carefully):
This exam is due on Thursday, June 19 at 12:00 noon. Please bring your paper to Shaer
103 (Engineering Advising Oce).
There are SIX