#import "@local/dtu-template:0.5.1":*

= 10.1
== 11
Let $G$ be a simple graph. Show that the relation $R$ on the set of vertices of $G$ such that $u R v$ if and only if there is an edge associated to ${u, v}$ is a symmetric, irreflexive relation on G.
#solution[
- *Symmetric:* If $u$ is connected to $v$, then $v$ is connected to $u$
- *Irreflexive:* Because you are looking at a simple graph, then there will not be a loop.
]
= 10.2
== 5
_Can a simple graph exist with 15 vertices each of degree five?_
#solution[
No because the total degree would be uneven
]
== 43
_How many edges does a graph have if its degree sequence is $5, 2, 2, 2, 2, 1$? Draw such a graph._
#solution[
$(5+2+2+2+2+1)/2=7$
]
== (53)
= 10.3
== (7)
_Represent the graph in Exercise 3 with an adjacency matrix._
#image("Exercise-pictures/10.3-7.png",width: 40%)
#set math.mat(delim: "[")
#solution[
$mat("",a,b,c,d;a,1,1,1,1;b,0,0,0,1;c,1,1,0,0;d,0,1,1,1)$
]
== 57
_For which integers $n$ is $C_n$ self-complementary?_
#solution[
n=5 only
]
= 10.4
== 9
_Explain why in the collaboration graph of mathematicians (see Example 3 in Section 10.1) a vertex representing a mathematician is in the same connected component as the vertex representing Paul Erdős if and only if that mathematician has a finite Erdős number._

#solution[
Because if a mathematician has a finite Erdos number, then there must be a path of mathematicians who have collaborated connecting them to Erdo.
]
== 11
_Determine whether each of these graphs is strongly connected and if not, whether it is weakly connected._
#image("Exercise-pictures/10.4-11.png", width: 50%)
#solution[
/ a): No. Because of a. But it is weekly connected
/ b): No. Because of a-e-b-a. But it is weekly connected.
/ c): Neither, because of the two subgraphs are not connected
]
= 10.5
_In Exercises 1–8 determine whether the given graph has an Euler circuit. Construct such a circuit when one exists. If no Euler circuit exists, determine whether the graph has an Euler path and construct such a path if one exists._
== (1)
#image("Exercise-pictures/10.5-1.png", width: 40%)
#solution[
Neither
]
== 3
#image("Exercise-pictures/10.5-3.png", width: 40%)
#solution[
No Euler circuit because a has degree 3. It has a path $a,b,e,b,d,e,a,c,e,c,d$
]
== 13
_In Exercises 13–15 determine whether the picture shown can be drawn with a pencil in a continuous motion without lifting the pencil or retracing part of the picture._
#image("Exercise-pictures/10.5-13.png",width: 40%)
#solution[
Yes, draw the triangles first, and then after draw the pentagon in the middle. Also, imagine if every time the lines cross, its a vertex, all the vertices have even edges.
]