数学代写|Math475 Graph Theory
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Math475 Graph Theory课程简介
The content of this course includes general enumeration methods, which involves counting techniques used to solve combinatorial problems. The course also covers difference equations, which are used to model and solve recurrence relations. Another topic covered is generating functions, which are used to convert combinatorial problems into algebraic problems that can be solved using calculus and algebra.
In addition, the course includes elements of graph theory, which is the study of graphs and their properties. This includes matrix representations of graphs, which are used to analyze the structure of graphs. The course also covers applications of graph theory to transport networks, where graphs are used to model and analyze transportation systems.
Matching theory and graphical algorithms are also covered in this course. Matching theory is the study of matching problems, where we try to match elements from one set to another according to certain criteria. Graphical algorithms are algorithms that operate on graphs, and are used to solve problems such as shortest path and maximum flow problems.
PREREQUISITES
Sample Textbooks
First Course in Graph Theory, by Gary Chartrand
Introduction to Enumerative Combinatorics, by Miklos Bona
Applications
Computer science, physics, economics, biology, chemistry
If you like this course, you might also consider the following courses
MATH 401, MATH 405, MATH416, Study abroad program Budapest Semesters of Mathematics
Additional Notes
Students interested in grad school in STAT or computer science should consider this course. A large element of the course involves puzzles that are very easy to understand, but requiring thinking outside the box.
Math475 Graph Theory HELP（EXAM HELP， ONLINE TUTOR）
2.6.7 Draw the graph with the adjacency matrix $A$. The vertices are in the order $a, b, c, d$
$$
A=\left(\begin{array}{llll}
0 & 1 & 1 & 1 \
1 & 0 & 1 & 0 \
1 & 1 & 0 & 1 \
1 & 0 & 1 & 0
\end{array}\right)
$$
2.6.15 Draw the digraph with the adjacency matrix $A$. The vertices are in the order $a, b, c, d$
$$
A=\left(\begin{array}{llll}
0 & 1 & 1 & 1 \
0 & 0 & 1 & 0 \
1 & 1 & 0 & 1 \
1 & 0 & 1 & 0
\end{array}\right)
$$
2.6.17 For the graph with the adjacency matrix $A$. (The vertices are in the order $a, b, c, d$ )
$$
A=\left(\begin{array}{llll}
2 & 1 & 1 & 1 \
1 & 0 & 1 & 0 \
1 & 1 & 0 & 1 \
1 & 0 & 1 & 0
\end{array}\right)
$$
Draw the graph $G$ and compute $A^2$ and show $A^2[a, a]$ is the number of paths of length two in $G$ from $a$ to itself. Show $A^2[d, a]$ does the same for paths of length two from $d$ to $a$.
2.6.7:
The graph with the given adjacency matrix is:
cssCopy codea -- b
| \/ |
| /\ |
c -- d
2.6.15:
The directed graph with the given adjacency matrix is:
markdownCopy codea -> b -> c <- d
^ |
|______________|
2.6.17:
The undirected graph with the given adjacency matrix is:
cssCopy code a -- b -- c
| |
d---------
Computing $A^2$, we have:
A^2=\left(\begin{array}{cccc} 3 & 1 & 3 & 1 \\ 1 & 1 & 1 & 1 \\ 3 & 1 & 3 & 1 \\ 1 & 1 & 1 & 1 \end{array}\right)A2=⎝⎛3131111131311111⎠⎞
$A^2[a,a]$ counts the number of paths of length 2 from $a$ to itself in the graph $G$. There are 3 such paths: $a \to b \to a$, $a \to c \to a$, and $a \to d \to a$. Therefore, $A^2[a,a] = 3$.
$A^2[d,a]$ counts the number of paths of length 2 from $d$ to $a$ in the graph $G$. There is only 1 such path: $d \to b \to a$. Therefore, $A^2[d,a] = 1$.
1.6.1 Formulate the personnel-assignment problem [Application 1.3.1] as a maximum flow problem (Hint: add an artificial source and sink to the bipartite graph) (Hint page 262)
1.7.1 A 20-vertex graph has 62-edges. Every vertex has degree 3 or 7 . How many vertices have degree 3 ?
1.7.2 Either draw a 3-regular 7-vertix graph or prove that none exits.
1.7.3 Prove that no 5-vertex 7 -edge simple graph has diameter greater than 2.
1.6.1: The personnel-assignment problem can be formulated as a maximum flow problem as follows:
Consider the bipartite graph $G=(V,E)$ where $V= P \cup J \cup {s,t}$, where $P$ is the set of people, $J$ is the set of jobs, $s$ is an artificial source and $t$ is an artificial sink. For each person $i\in P$ and each job $j \in J$ such that $i$ is qualified for $j$, create an edge $(i,j)$ with capacity 1.
Connect $s$ to each person $i \in P$ with an edge $(s,i)$ and capacity $q_i$, the maximum number of jobs that person $i$ can be assigned to.
Connect each job $j \in J$ to $t$ with an edge $(j,t)$ and capacity 1.
The maximum flow from $s$ to $t$ in this graph corresponds to the maximum number of job assignments that can be made subject to the constraints that each person is assigned at most $q_i$ jobs and each job is assigned to at most one person. The assignment is found by taking the set of edges with non-zero flow, which corresponds to the set of job assignments.
1.7.1: Let $x$ be the number of vertices with degree 3 and $y$ be the number of vertices with degree 7. Then we have:
$$3x+7y = 2|E|=124$$
Since there are 20 vertices, we also have $x+y=20$. Solving these two equations gives $x=8$ and $y=12$. Therefore, there are 8 vertices with degree 3.
1.7.2: There are many 3-regular 7-vertex graphs, one example is the cycle graph $C_7$. Another example is the complete bipartite graph $K_{3,4}$.
To prove that no such graph exists, we can use the fact that the sum of the degrees of the vertices in any graph is twice the number of edges. In a 3-regular graph with 7 vertices, the sum of the degrees is $3 \times 7 = 21$, so the number of edges must be $21/2 = 10.5$, which is not an integer. Therefore, such a graph cannot exist.
1.7.3: Let $G$ be a simple graph with 5 vertices and 7 edges, and let $diam(G)$ be the diameter of $G$. Suppose for contradiction that $diam(G) > 2$. Then there exist two vertices $u$ and $v$ such that $d(u,v) > 2$.
Since $G$ has only 5 vertices, there must be at least one vertex $w$ that is adjacent to both $u$ and $v$. Without loss of generality, assume that $w$ is closer to $u$ than $v$ in terms of shortest path length. Then we have $d(u,w) \leq 1$ and $d(w,v) \leq 1$, so $d(u,v) \leq 2$, which contradicts the assumption that $d(u,v) > 2$. Therefore, we must have $diam(G) \leq 2$.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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