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Assignment6.tm
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<TeXmacs|1.99.5>
<style|generic>
<\body>
<doc-data|<doc-title|Discrete Math Assignment
6>|<doc-author|<author-data|<author-name|Yuejiam Mo>|<\author-affiliation>
<english|>11510511
</author-affiliation>>>>
<with|font-series|bold|P650 Ex.11>: Let <with|font-shape|italic|G> be a
simple graph. Show that the relation <with|font-shape|italic|R> on the set
of vertices of <with|font-shape|italic|G> such that
<with|font-shape|italic|uRv> if and only if there is an edge associated to
{<with|font-shape|italic|u, v><with|font-shape|italic|>} is a symmetric,
irreflexive relation on <with|font-shape|italic|G.>
<with|font-series|bold|Proof:>
If <math|uRv,>there is an edge associated with <math|<around*|{|u,v|}>.>
Since this is undirected, this edge can be presented as <math|vRu>. Thus
<with|font-shape|italic|R> is symmetric. No loop in simple graph, which
mean <with|font-shape|italic|R> is irreflexive.\
<with|font-series|bold|QED.>
\;
<\with|font-series|bold>
P668 Ex. 64: <with|font-series|medium|Show that if
<with|font-shape|italic|G> is a simple graph with
<with|font-shape|italic|n> vertices and <with|font-shape|italic|e> edges,
then <math|e\<leqslant\>v<rsup|2>/4><with|font-shape|italic|.>>\
</with>
<\with|font-series|bold>
Proof:<with|font-series|medium|>
</with>
Let the partition be <with|font-series|bold|>
\;
\;
<\with|font-series|bold>
P668 Ex. 66: <with|font-series|medium|Describe an algorithm to decide
whether a graph is bipartite based on the fact that a graph is bipartite
if and only if it is possible to color its vertices two different colors
so that no two vertices of the same color are adjacent.>
<with|font-series|medium|<with|font-series|bold|Solve:> >
<with|font-series|medium|We start by coloring any vertex red. Then we
color all the vertices adjacent to this vertex blue. Then we color all
the vertices red, then color all the vertices adjacent to red vertices
blue. If we ever are in the position of trying to color a vertex with the
color opposite to the color it already has, then we stop and know that
the graph is not bipartite. If the process terminates before all the
vertices have been colored, then we color some uncolor vertex red and
begin the process again. Eventually we will have either colored all the
vertices or stopped and decided that the graph is not biparitite.>\
\;
</with>
<with|font-series|bold|P676 Ex. 28: >What is the sum of the entries in a
row of the adjacency matrix for an undicrected graph? For a directed graph?
\;
<with|font-series|bold|P676 Ex. 29 >What is the sum of the entries in a
column of the adjacency
matrix for an undirected graph? For a directed graph?
<with|font-series|bold|P676 Ex. 32> Find an adjacency matrix for each of
these graphs.
a) <with|font-shape|italic|<math|K<rsub|n>>><space|2em>b)
<math|C<rsub|n>><space|1em>c) <math|W<rsub|n>><space|1em>d)
<math|K<rsub|m,n>><space|1em>e) <math|Q<rsub|n>>
\;
<with|font-series|bold|P677 Ex. 45 >Show that isomorphism of simple graphs
is an equiva-
lence relation.
<\with|font-series|bold>
P677 Ex. 52\
</with>
Show that if G is a self-complementary simple graph with v vertices, then v
\<equiv\> 0 or 1 (mod 4).
\;
<\with|font-series|bold>
P690 Ex. 20 <with|font-series|medium|Use paths either to show that these
graphs are not isomorphic or to find an isomorphism between these
graphs.<image|<tuple|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\;
</with>
<\with|font-series|bold>
P691 Ex. 28 <with|font-series|medium|Show that every connected graph with
<with|font-shape|italic|n> vertices has at least n-1 edges.>
\;
</with>
<\with|font-series|bold>
P704 Ex. 10 <with|font-series|medium|Can someone cross an the bridges
shown in this map exactly once and returen to the starting
point?<image|<tuple|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\;
</with>
<\with|font-series|bold>
P716 Ex. 2 <with|font-series|medium|Find the length of a shortest path
between <with|font-shape|italic|a> and <with|font-shape|italic|z> in the
given weighted graph> 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</with>
<\with|font-series|bold>
P716 Ex. 14 <with|font-series|medium|Explain how to find a path with the
least nubmer of edges between two vertices in an undirected graph by
considering it as a shortest path problesm in a weighted graph.>
\;
</with>
<\with|font-series|bold>
P725 Ex. 6 <with|font-series|medium|Determine whether the given graph is
planar. If so, draw it so that no edges cross.>
<image|<tuple|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|png>|216px|123px||>
</with>
<\with|font-series|bold>
P726 Ex. 17 <with|font-series|medium|Suppose that a connected planar
simple graph with <with|font-shape|italic|e> edges and
<with|font-shape|italic|v> vertices contains no simple circuits of length
4 or less. Show that <with|font-shape|italic|e><math|\<leqslant\><around*|(|5/3|)>v-<around*|(|10/3|)>>
if <with|font-shape|italic|v <math|\<geqslant\>>>4.>
\;
</with>
<\with|font-series|bold>
P756 Ex. 16 <with|font-series|medium|||Which complete bipartite graphs
<math|K<rsub|m,n>>, where are <with|font-shape|italic|m >and
<with|font-shape|italic|n >are positive integers, are trees?>
</with>
<\with|font-series|bold>
P757 Ex. 46 <with|font-series|medium|How many vertices, leaves and
internal verices does the rooted Fibonacci tree
<with|font-shape|italic|<math|T<rsub|n>>> are defined recursively in the
following way. <math|T<rsub|1> and> <math|T<rsub|2>> are both the rooted
tree consisting of a single vertex, and for
<math|n=3,4\<ldots\>\<ldots\>,> the rooted tree <math|T<rsub|n>> is
constructed from a root with <math|T<rsub|n-1> >as its left subtree and
<math|T<rsub|n-2>> as its right subtree.>
\;
</with>
<\with|font-series|bold>
P784 Ex. 24 <with|font-series|medium|What is the value of each of these
postfix expressions?>
a) \<minus\> \<ast\> 2 / 8 4 3
b) \<uparrow\> \<minus\> \<ast\> 3 3 \<ast\> 4 2 5
c) + \<minus\> \<uparrow\> 3 2 \<uparrow\> 2 3 / 6 \<minus\> 4 2
d) \<ast\> + 3 + 3 \<uparrow\> 3 + 3 3 3
</with>
<\with|font-series|bold>
P795 Ex. 11 <with|font-series|medium|How many different spanning trees
does each of these simple graphs have?>
\;
\;
\;
</with>
a) <math|K<rsub|3>><space|1em>b) <math|K<rsub|4>> c)<math|K<rsub|5>>
<with|font-series|bold|P795 Ex. 12 >How many noinsomorphic spanning trees
does each of these simple graphs have?
a) <math|K<rsub|3>><space|1em>b) <math|K<rsub|4>> c)<math|K<rsub|5>>
\;
\;
<with|font-series|bold|P802 Ex. 2 >Use Prim's algorithm to find a minimum
spanning tree for the given weighted 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<with|font-series|bold|P802 Ex. 6 >Use Kruskal's algorithm to find minimum
spanning tree for the weighted graph Ex.2
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