69 lines
2.6 KiB
Typst
69 lines
2.6 KiB
Typst
#import "@local/dtu-template:0.5.1":*
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= 10.1
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== 11
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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.
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#solution[
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- *Symmetric:* If $u$ is connected to $v$, then $v$ is connected to $u$
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- *Irreflexive:* Because you are looking at a simple graph, then there will not be a loop.
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]
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= 10.2
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== 5
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_Can a simple graph exist with 15 vertices each of degree five?_
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#solution[
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No because the total degree would be uneven
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]
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== 43
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_How many edges does a graph have if its degree sequence is $5, 2, 2, 2, 2, 1$? Draw such a graph._
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#solution[
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$(5+2+2+2+2+1)/2=7$
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]
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== (53)
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= 10.3
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== (7)
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_Represent the graph in Exercise 3 with an adjacency matrix._
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#image("Exercise-pictures/10.3-7.png",width: 40%)
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#set math.mat(delim: "[")
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#solution[
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$mat("",a,b,c,d;a,1,1,1,1;b,0,0,0,1;c,1,1,0,0;d,0,1,1,1)$
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]
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== 57
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_For which integers $n$ is $C_n$ self-complementary?_
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#solution[
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n=5 only
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]
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= 10.4
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== 9
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_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._
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#solution[
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Because if a mathematician has a finite Erdos number, then there must be a path of mathematicians who have collaborated connecting them to Erdo.
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]
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== 11
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_Determine whether each of these graphs is strongly connected and if not, whether it is weakly connected._
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#image("Exercise-pictures/10.4-11.png", width: 50%)
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#solution[
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/ a): No. Because of a. But it is weekly connected
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/ b): No. Because of a-e-b-a. But it is weekly connected.
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/ c): Neither, because of the two subgraphs are not connected
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]
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= 10.5
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_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._
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== (1)
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#image("Exercise-pictures/10.5-1.png", width: 40%)
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#solution[
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Neither
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]
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== 3
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#image("Exercise-pictures/10.5-3.png", width: 40%)
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#solution[
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No Euler circuit because a has degree 3. It has a path $a,b,e,b,d,e,a,c,e,c,d$
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]
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== 13
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_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._
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#image("Exercise-pictures/10.5-13.png",width: 40%)
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#solution[
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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.
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]
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