Statistics-lab™可以为您提供williams.edu MATH334 Graph Theory图论课程的代写代考和辅导服务！

MATH334 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.

MATH334 Graph Theory HELP（EXAM HELP， ONLINE TUTOR）

Proof: Assume $G$ is disconnected. To prove $\bar{G}$ is connected, we must show there is an $x-y$ path for any pair of vertices $x, y$. If $x$ and $y$ are not adjacent in $G$ then $x y \in E(\bar{G})$. Thus the edge $x y$ is itself a $x-y$ path. Otherwise, $x y \in E(G)$ and so $x$ and $y$ are in the same component of $G$ and not adjacent in $\bar{G}$. Since $G$ is not connected, there must exist some vertex $z$ in a different component from $x$ and $y$, and so $z$ cannot be adjacent to either of $x$ or $y$. This implies $x z, y z \in E(\bar{G})$, and so $x z y$ is a $x-y$ path in $\bar{G}$. Note that every pair of vertices in $\bar{G}$ fall into one of these two cases and so satisfy $d(x, y) \leq 2$. Thus $\bar{G}$ is connected and $\operatorname{diam}(\bar{G}) \leq 2$

Proof: Since $G$ is simple and $\operatorname{rad}(G)=r \geq 3$, we know that $\epsilon(v) \geq r$ for all vertices in $G$. In particular, there exists some path wxyz such that $w$ is not adjacent to $y$ and $z$, and $x$ is not adjacent to $z$. Since $r \geq 3$ we know $\epsilon(x) \geq 3$ and so there must exist another vertex $v$ (not one of $w, y, z$ ) such that $d(x, v) \geq 3$. Thus $v$ is not adjacent to either of $x$ or $y$. Moreover, $v$ cannot be adjacent to both $w$ and $z$ since otherwise $d(w, z)<3$. Thus at least one of the edges $v w$ or $v z$, but possibly both, cannot exist in $G$.

In either case, we have that the distance in $\bar{G}$ between any two of these vertices is at most 2 , and since this holds for any collection of vertices in $G$, we see that $\operatorname{rad}(\bar{G}) \leq 2$

Since the radius measures the shortest distance between two vertices and the diameter the longest, we should expect that these quantities cannot be too far away from one another. In geometry when we study circles, we define the diameter to be twice the radius. But is the same true in graph theory? As we have already seen above, the diameter need not be equal to twice the radius, but as the result below proves it cannot be larger than twice the radius.

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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