🚧 Fin chapter 1

chapter_1/examples.md

@@ -715,3 +715,130 @@ Does $f = g$?
 
 Yes. Because the absolute value of any real number equals the square root of its
 square, $|x| = \sqrt{x^2}$ for all $x \in \mathbb{R}$. Hence $f = g$.
+
+---
+
+**Example 1.4.1 Terminology**
+
+Consider the following graph:
+
+![image 1_3_2_6](./1_3_2_6.png)
+
+a. Write the vertex set and edge set, and give a table showing the edge-endpoint
+function.
+
+b. Find all edges that are incident on $v_1$, all vertices that are adjacent to
+$v_1$, all edges that are adjacent to $e_1$, all loops, all parallel edges, all
+vertices that are adjacent to themselves, and all isolated vertices.
+
+**Solution**
+
+a.
+
+$$ \text{vertex set } = \{v_1, v_2, v_3, v_4, v_5, v_6\} $$
+
+$$ \text{edge set } = \{e_1, e_2, e_3, e_4, e_5, e_6, e_7\} $$
+
+| Edge  | Endpoints      |
+| ----- | -------------- |
+| $e_1$ | $\{v_1, v_2\}$ |
+| $e_2$ | $\{v_1, v_3\}$ |
+| $e_3$ | $\{v_1, v_3\}$ |
+| $e_4$ | $\{v_2, v_3\}$ |
+| $e_5$ | $\{v_5, v_6\}$ |
+| $e_6$ | $\{v_5\}$      |
+| $e_7$ | $\{v_6\}$      |
+
+b.
+
+$e_1$, $e_2$, and $e_3$ are incident on $v_1$.
+
+$v_1$ and $v_3$ are adjacent to $v_1$.
+
+$e_2$, $e_3$, and $e_4$ are adjacent to $e_1$.
+
+$e_6$ and $e_7$ are loops.
+
+$e_2$ and $e_3$ are parallel.
+
+$v_5$ and $v_6$ are adjacent to themselves.
+
+$v_4$ is an isolated vertex.
+
+---
+
+**Example 1.4.2 Drawing More Than One Picture for a Graph**$adjacent
+
+Page 49
+
+Consider the graph specified as follows:
+
+$$ \text{vertex set } = \{v_1, v_2, v_3, v_4\} $$
+
+$$ \text{edge set } = \{e_1, e_2, e_3, e_4\} $$
+
+edge-endpoint function:
+
+| Edge  | Endpoints      |
+| ----- | -------------- |
+| $e_1$ | $\{v_1, v_3\}$ |
+| $e_2$ | $\{v_2, v_4\}$ |
+| $e_3$ | $\{v_2, v_4\}$ |
+| $e_4$ | $\{v_3\}$      |
+
+Both drawings (a) and (b) shown below are pictorial representations of this
+graph (see Page 50).
+
+---
+
+**Example 1.4.3 Labeling Drawings to Show They Represent the Same Graph**
+
+Page 50
+
+Consider the two drawings shown in Figure 1.4.1. Label vertices and edges in
+such a way that both drawings represent the same graph.
+
+(see page 50)
+
+**Solution**
+
+Imagine putting one end of a piece of string at the top vertex of Figure
+1.4.1(a) (call this vertex $v_1$), then laying the string to the next adjacent
+vertex on the lower right (call this vertex $v_2$), then laying it to the next
+adjacent vertex on the upper left ($v_3$), and so forth, returning finally to
+the top vertex $v_1$. Call the first edge $e_1$, the second edge $e_2$, and so
+forth, as shown below.
+
+(see page 50)
+
+Now imagine picking up the piece of string, together with its labels, and
+repositioning it as follows:
+
+(see page 50)
+
+This is the same as Figure 1.4.1(b), so both drawings represent the graph with
+vertex set $\{v_1, v_2, v_3, v_4, v_5\}$, edge set
+$\{e_1, e_2, e_3, e_4, e_5\}$, and edge-endpoint function as follows:
+
+| Edge  | Endpoints      |
+| ----- | -------------- |
+| $e_1$ | $\{v_1, v_2\}$ |
+| $e_2$ | $\{v_2, v_3\}$ |
+| $e_3$ | $\{v_3, v_4\}$ |
+| $e_4$ | $\{v_4, v_5\}$ |
+| $e_5$ | $\{v_5, v_1\}$ |
+
+---
+
+**Example 1.4.4 Using a Graph to Represent a Network**
+
+Page 51
+
+Telephone, electric power, gas pipeline, and air transport systems can all be
+represented by graphs, as can computer networks - from small local area networks
+to the global Internet system that connects millions of computers worldwide.
+Questions that arise in the design of such systems involve choosing connecting
+edges to minimize cost, optimize a certain type of service, and so forth. A
+typical network, called a _hub-ad-spoke model_, is shown below.
+
+(see page 51)

chapter_1/notes.md

@@ -195,3 +195,46 @@ Page 42
 If $A$ and $B$ are sets and $F$ is a function from $A$ to $B$, then given any
 element $x$ in $A$, the unique element in $B$ that is related to $x$ by $F$ is
 denoted $F(x)$, which is read **"$F$ of $x$."**
+
+---
+
+Page 48
+
+**Definition**
+
+A **graph** $G$ consists of two finite sets: a nonempty set $V(G)$ of
+**vertices** and a set $E(G)$ of **edges**, where each edge is associated with a
+set consisting of either one or two vertices called its **endpoints**. The
+correspondence from edges to endpoints is called the **edge-endpoint function**.
+
+An edge with just one endpoint is called a **loop**, and two or more distinct
+edges with the same set of endpoints are said to be **parallel**. An edge is
+said to **connect** its endpoints; two vertices that are connected by an edge
+are called **adjacent**; and a vertex that is an endpoint of a loop is said to
+be **adjacent to itself**.
+
+An edge is said to be **incident on** each of its endpoints, and two edges
+incident on the same endpoint are called **adjacent**. A vertex on which no
+edges are incident is called **isolated**.
+
+---
+
+Page 52
+
+**Definition**
+
+A **directed graph**, or **digraph**, consists of two finite sets: a nonempty
+set $V(G)$ of vertices and a set $D(G)$ of directed edges, where each is
+associated with an ordered pair of vertices called its **endpoints**. If edge
+$e$ is associated with the pair $(v, w)$ of vertices, then $e$ is said to be the
+**(directed) edge** from $v$ to $w$.
+
+---
+
+Page 54
+
+**Definition**
+
+Let $G$ be a graph and $v$ a vertex of $G$. The **degree of $v$**, denoted
+**$deg(v)$**, equals the number of edges that are incident on $v$, with an edge
+that is a loop counted twice.

chapter_1/test_yourself.md

@@ -117,3 +117,65 @@ $(x, y) \in F$ and $(x, z) \in F$; $y = z$
 **Solution**
 
 a unique element of $B$ that is related to $x$ by $F$.
+
+---
+
+**Test Yourself**
+
+Page 57
+
+1. A graph consists of two finite sets: _______ and _______, where each edge is
+   associated with a set consisting of _______.
+
+**Solution**
+
+a finite; nonempty set of vertices; a finite set of edges.
+
+2. A loop in a graph is _______.
+
+**Solution**
+
+An edge that has a single endpoint.
+
+3. Two distinct edges in a graph are parallel if, and only if, _______.
+
+**Solution**
+
+They share the same set of endpoints.
+
+4. Two vertices are called adjacent if, and only if, _______.
+
+**Solution**
+
+They are connected by an edge.
+
+5. An edge is incident on _______.
+
+**Solution**
+
+Each of its endpoints.
+
+6. Two edges incident on the same endpoint are _______.
+
+**Solution**
+
+adjacent
+
+7. Two edges incident on the same endpoint are _______.
+
+**Solution**
+
+isolated
+
+8. In a directed graph, each edge is associated with _______.
+
+**Solution**
+
+an ordered pair of vertices called its endpoints
+
+9. The degree of a vertex in a graph is _______.
+
+**Solution**
+
+the number of edges that are incident on the vertex, with an edge that is a loop
+counted twice.