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0;
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src=3D"gutierrez_files/image002.jpg" align=3Dleft alt=3D"Small Mission Coll=
ege Logo"
v:shapes=3D"Picture_x0020_2"><![endif]><!--[if gte vml 1]><v:shape id=3D"Pi=
cture_x0020_3"
 o:spid=3D"_x0000_s1045" type=3D"#_x0000_t75" alt=3D"Small Math Dept Logo" =
style=3D'position:absolute;
 margin-left:0;margin-top:0;width:24pt;height:24pt;z-index:7;visibility:vis=
ible;
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src=3D"gutierrez_files/image003.jpg" align=3Dleft alt=3D"Small Math Dept Lo=
go"
v:shapes=3D"Picture_x0020_3"><![endif]><b><span style=3D'mso-font-kerning:1=
.0pt;
mso-fareast-language:KO'>Math Department, </span></b><st1:place><st1:PlaceN=
ame><b><span
  style=3D'mso-font-kerning:1.0pt;mso-fareast-language:KO'>Mission</span></=
b></st1:PlaceName><b><span
 style=3D'mso-font-kerning:1.0pt;mso-fareast-language:KO'> </span></b><st1:=
PlaceType><b><span
  style=3D'mso-font-kerning:1.0pt;mso-fareast-language:KO'>College</span></=
b></st1:PlaceType></st1:place><b><span
style=3D'mso-font-kerning:1.0pt;mso-fareast-language:KO'>, </span></b><st1:=
place><st1:City><b><span
  style=3D'mso-font-kerning:1.0pt;mso-fareast-language:KO'>Santa Clara</spa=
n></b></st1:City><b><span
 style=3D'mso-font-kerning:1.0pt;mso-fareast-language:KO'>, </span></b><st1=
:State><b><span
  style=3D'mso-font-kerning:1.0pt;mso-fareast-language:KO'>California</span=
></b></st1:State></st1:place><span
style=3D'mso-font-kerning:1.0pt;mso-fareast-language:KO'> <o:p></o:p></span=
></p>

<p class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-alt=
:auto'><b><span
style=3D'mso-fareast-language:KO'>Go to <a
href=3D"http://www.missioncollege.org/depts/math/default.html">Math Dept Ma=
in
Page</a> | Go to <a href=3D"http://www.missioncollege.org/default.html">Mis=
sion
College Main Page</a></span></b><span style=3D'mso-fareast-language:KO'> </=
span><span
style=3D'font-family:"Arial Unicode MS";mso-fareast-language:KO'><o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-alt=
:auto'><b><span
style=3D'color:black;mso-fareast-language:KO'>This paper was written as an
assignment for Ian Walton's Math G - Math for liberal Arts Students - at </=
span></b><st1:place><st1:PlaceName><b><span
  style=3D'color:black;mso-fareast-language:KO'>Mission</span></b></st1:Pla=
ceName><b><span
 style=3D'color:black;mso-fareast-language:KO'> </span></b><st1:PlaceType><=
b><span
  style=3D'color:black;mso-fareast-language:KO'>College</span></b></st1:Pla=
ceType></st1:place><b><span
style=3D'color:black;mso-fareast-language:KO'>. If you use material from th=
is
paper, please acknowledge it. </span></b><span style=3D'mso-fareast-languag=
e:
KO'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-alt=
:auto'><b><span
style=3D'color:black;mso-fareast-language:KO'>To explore other such papers =
go to </span></b><b><span
style=3D'mso-fareast-language:KO'><a
href=3D"http://www.missioncollege.org/depts/math/gproject.htm">the Math G
Projects Page</a></span></b><span style=3D'font-size:11.0pt;font-family:Cal=
ibri'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span lang=3DES style=3D'fo=
nt-family:
Arial;mso-ansi-language:ES'>Math G &#8211; Final<span style=3D'mso-tab-coun=
t:
7'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;=
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nb=
sp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Selena
Gutierrez<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'>Dr.
Walton<span style=3D'mso-tab-count:8'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span><st1:date
Year=3D"2010" Day=3D"10" Month=3D"5"><span style=3D'font-family:Arial'>May =
10, 2010</span></st1:date><span
style=3D'font-family:Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'><span
style=3D'mso-tab-count:5'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp; </span>Graph
Theory<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-indent:.5in;line-height:200%'><span
style=3D'font-family:Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'text-indent:.5in;line-height:200%'><span
style=3D'font-family:Arial'>I&#8217;ve had to use graphs for many different
courses and assignments. Surprisingly, not all of those assignments were fo=
r a
math class. Of course, when thinking of graphs, we assume that it will only=
 be
in the area math. But, I&#8217;ve also had to use graphs in an English cour=
se
and a few of my communications courses. In fact, graphs have popped up in my
studies anytime I had to make comparisons, or connect ideas. Graph Theory is
the study of graphs. Graphs, as defined by Wikipedia, are mathematical
structures used to model pair wise relations between objects from a certain
collection (Wikipedia). This paper will discuss graphs as related to the Gr=
aph
Theory. In this case, a graph is a set of vertices (V) and edges (E), which
connect to make pairs of element V (mathematical atlas). In order for these
vertices and edges to connect, there must also be a rule to connect the two,
forming the graph. Graph Theory is used mainly to illustrate networks, and =
is
extremely in computer science and engineering. We can see instances of Graph
Theory in several different math subjects, as well as in networking problem=
s.
There is even a link between Graph Theory and sociology and biology. </span=
><span
style=3D'mso-bidi-font-size:8.5pt;line-height:200%;font-family:Arial'>A pap=
er written
by Amol Deshpande at the University of Maryland and Sunita Sarawagi of </sp=
an><st1:stockticker><span
 style=3D'mso-bidi-font-size:8.5pt;line-height:200%;font-family:Arial'>IIT<=
/span></st1:stockticker><span
style=3D'mso-bidi-font-size:8.5pt;line-height:200%;font-family:Arial'> Bomb=
ay,
stated that graphical models provide a compact representation of the full j=
oint
distribution of a set of variables as a graph whose nodes are the variables=
 and
whose edges connect variables who interact directly (Deshpande, Sarawagi).<=
/span><span
style=3D'font-family:Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'>Below
is an image illustrating a simple graph in graph theory.</span><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p>&nbsp;</o:p></span></p>

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src=3D"gutierrez_files/image005.gif" align=3Dleft hspace=3D12 v:shapes=3D"_=
x0000_s1030"><![endif]><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><!--[if gte vml 1]><v:shape=
type
 id=3D"_x0000_t66" coordsize=3D"21600,21600" o:spt=3D"66" adj=3D"5400,5400"=
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 <v:stroke joinstyle=3D"miter"/>
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  <v:f eqn=3D"val #0"/>
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  <v:f eqn=3D"sum 21600 0 #1"/>
  <v:f eqn=3D"prod #0 #1 10800"/>
  <v:f eqn=3D"sum #0 0 @3"/>
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 <v:path o:connecttype=3D"custom" o:connectlocs=3D"@0,0;0,10800;@0,21600;21=
600,10800"
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  <v:h position=3D"#0,#1" xrange=3D"0,21600" yrange=3D"0,10800"/>
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</v:shapetype><v:shape id=3D"_x0000_s1032" type=3D"#_x0000_t66" style=3D'po=
sition:absolute;
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  <![if !mso]>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div>
    <p class=3DMsoNormal><span style=3D'font-size:10.0pt;mso-bidi-font-size=
:12.0pt;
    font-family:"Accord Heavy SF"'>Vertices (V)<o:p></o:p></span></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  <![endif]></v:textbox>
 <w:wrap type=3D"square"/>
</v:shape><![endif]--><![if !vml]><img width=3D141 height=3D59
src=3D"gutierrez_files/image006.gif" align=3Dleft hspace=3D12
alt=3D"Left Arrow: Vertices (V)" v:shapes=3D"_x0000_s1032"><![endif]><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><!--[if gte vml 1]><v:shape=
type
 id=3D"_x0000_t13" coordsize=3D"21600,21600" o:spt=3D"13" adj=3D"16200,5400=
" path=3D"m@0,l@0@1,0@1,0@2@0@2@0,21600,21600,10800xe">
 <v:stroke joinstyle=3D"miter"/>
 <v:formulas>
  <v:f eqn=3D"val #0"/>
  <v:f eqn=3D"val #1"/>
  <v:f eqn=3D"sum height 0 #1"/>
  <v:f eqn=3D"sum 10800 0 #1"/>
  <v:f eqn=3D"sum width 0 #0"/>
  <v:f eqn=3D"prod @4 @3 10800"/>
  <v:f eqn=3D"sum width 0 @5"/>
 </v:formulas>
 <v:path o:connecttype=3D"custom" o:connectlocs=3D"@0,0;0,10800;@0,21600;21=
600,10800"
  o:connectangles=3D"270,180,90,0" textboxrect=3D"0,@1,@6,@2"/>
 <v:handles>
  <v:h position=3D"#0,#1" xrange=3D"0,21600" yrange=3D"0,10800"/>
 </v:handles>
</v:shapetype><v:shape id=3D"_x0000_s1031" type=3D"#_x0000_t13" style=3D'po=
sition:absolute;
 margin-left:0;margin-top:9.9pt;width:79.8pt;height:27pt;z-index:2;
 mso-position-horizontal:left' strokecolor=3D"#936" strokeweight=3D"1.5pt">
 <v:textbox style=3D'mso-next-textbox:#_x0000_s1031'>
  <![if !mso]>
  <table cellpadding=3D0 cellspacing=3D0 width=3D"100%">
   <tr>
    <td><![endif]>
    <div>
    <p class=3DMsoNormal><span style=3D'font-size:10.0pt;mso-bidi-font-size=
:12.0pt;
    font-family:"Accord Heavy SF"'>Edge (E)<o:p></o:p></span></p>
    </div>
    <![if !mso]></td>
   </tr>
  </table>
  <![endif]></v:textbox>
 <w:wrap type=3D"square"/>
</v:shape><![endif]--><![if !vml]><img width=3D115 height=3D47
src=3D"gutierrez_files/image007.gif" align=3Dleft hspace=3D12
alt=3D"Right Arrow: Edge (E)" v:shapes=3D"_x0000_s1031"><![endif]><span
style=3D'font-size:9.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-fa=
mily:
Arial'>http://en.wikipedia.org/wiki/File:6n-graf.svg<o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:11=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%;font-family:Arial'><o:p>&nbsp;</=
o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:11=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%;font-family:Arial'><o:p>&nbsp;</=
o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:11=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%;font-family:Arial'><o:p>&nbsp;</=
o:p></span></p>

<p class=3DMsoNormal style=3D'text-indent:.5in;line-height:200%'><span
style=3D'font-family:Arial'>The history of Graph Theory begins with the tow=
n of </span><st1:City><st1:place><span
  style=3D'font-family:Arial'>Konigsberg</span></st1:place></st1:City><span
style=3D'font-family:Arial'>, </span><st1:country-region><st1:place><span
  style=3D'font-family:Arial'>Germany</span></st1:place></st1:country-regio=
n><span
style=3D'font-family:Arial'>. This town was divided into two by the </span>=
<st1:place><st1:PlaceName><span
  style=3D'font-family:Arial'>Preger</span></st1:PlaceName><span
 style=3D'font-family:Arial'> </span><st1:PlaceType><span style=3D'font-fam=
ily:
  Arial'>River</span></st1:PlaceType></st1:place><span style=3D'font-family=
:Arial'>,
connected by 7 different bridges. The people of </span><st1:place><span
 style=3D'font-family:Arial'>Konigsberg</span></st1:place><span style=3D'fo=
nt-family:
Arial'> frequently wondered if there was a walking route where they would be
able to cross each bridge exactly once. This is when Leonhard Euler (1707
&#8211; 1783), a Swiss mathematician, began what is now called the Graph Th=
eory
(Timmons). In order to solve this problem, Euler first took the map of </sp=
an><st1:place><span
 style=3D'font-family:Arial'>Konigsberg</span></st1:place><span style=3D'fo=
nt-family:
Arial'> and the bridges, and made a graph similar to the graph shown above.=
 In
his graph, the vertices illustrated the parts of the land of the town, and =
the
edges illustrated the bridges of the town with the chain becoming the path =
of
the graph. With this graph, Euler was set out to prove that there was no su=
ch
path where a person would be able to take a path and cross each bridge once,
and from this situation the Eulerian path and Eulerian graph was born.<span
style=3D'mso-spacerun:yes'>&nbsp; </span>An Eulerian path is when a path is=
 found
where you follow, or &#8220;walk&#8221; along each path, or edge, exactly o=
nce
and the graph for this is called an Eulerian graph. A good example of a
Eulerian path is the childhood game of drawing a house without lifting your
pencil, or duplicating a line. <o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;text-indent:=
.5in;
line-height:200%'><!--[if gte vml 1]><v:shape id=3D"_x0000_s1033" type=3D"#=
_x0000_t75"
 style=3D'position:absolute;left:0;text-align:left;margin-left:0;margin-top=
:-18pt;
 width:153.75pt;height:113.25pt;z-index:4;mso-wrap-edited:f;
 mso-position-horizontal:left' wrapcoords=3D"-105 0 -105 21457 21600 21457 =
21600 0 -105 0">
 <v:imagedata src=3D"gutierrez_files/image008.png" o:title=3D"eulerian hous=
e"/>
 <w:wrap type=3D"square"/>
</v:shape><![endif]--><![if !vml]><img width=3D205 height=3D151
src=3D"gutierrez_files/image009.jpg" align=3Dleft hspace=3D12 v:shapes=3D"_=
x0000_s1033"><![endif]><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-indent:.5in;line-height:200%'><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:9.=
0pt;
mso-bidi-font-size:12.0pt;line-height:200%;font-family:Arial'>http://www.ma=
thematische-basteleien.de/haus10.gif</span><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:11=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%;font-family:Arial'><o:p>&nbsp;</=
o:p></span></p>

<p class=3DMsoNormal style=3D'text-indent:.5in;line-height:200%'><span
style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;line-height:200%;font-fa=
mily:
Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoBodyTextIndent><span style=3D'font-size:12.0pt;mso-bidi-font-=
size:
8.5pt;line-height:200%;font-family:Arial'>In graph theory, there are two ma=
in
data structures &#8211; list structure and matrix structure. List structure
matrices are also broken down into two &#8211; an incidence list matrix and=
 an
adjacency list matrix. An incidence list is made up of edges represented by
pairs of vertices, with vertices connected by an edge being adjacent (Wiki)=
. </span><span
style=3D'font-size:12.0pt;line-height:200%;font-family:Arial'>An adjacency =
list
is similar to an incident list, as it also lists the adjacent edges and
vertices, but with an adjacency list there could be instances of redundancy
because each adjacent vertex can be counted each time it occurs. If vertice=
s A
and B are adjacent, then the A&#8217;s list will contain B, and B&#8217;s l=
ist
will contain A (Wikipedia). The adjacency list the ideal list for maximum
storage capacity in network systems.<span style=3D'mso-spacerun:yes'>&nbsp;
</span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-indent:.5in;line-height:200%'><span
style=3D'font-family:Arial'>Matrix structures of graph theory include four
separate matrices, which includes incident and adjacency as well. The other=
 two
are the distance matrix and the Laplacian Matrix, which is also known as the
Kirchhoff matrix, or Admittance matrix. </span><span style=3D'mso-bidi-font=
-size:
8.5pt;line-height:200%;font-family:Arial'>An incidence matrix is a graph th=
at
is made up of the number of vertices and the number of edges. Incidence
matrices are (0,1) matrices where there are to be no zero rows columns
(Cameron, Prellberg, Stark). A (0,1) matrix is where the elements are either
zero or one.<span style=3D'mso-spacerun:yes'>&nbsp; </span>In an incidence
matrix, the point where the edge and vertex meets will contain information
about where the end of the matrix will be. The example below shows an incid=
ence
list and matrix.<span style=3D'mso-spacerun:yes'>&nbsp; </span></span><span
style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;line-height:200%;font-fa=
mily:
Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;text-indent:=
.5in;
line-height:200%'><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;
line-height:200%;font-family:Arial'><!--[if gte vml 1]><v:shape id=3D"_x000=
0_i1025"
 type=3D"#_x0000_t75" style=3D'width:338.25pt;height:222.75pt'>
 <v:imagedata src=3D"gutierrez_files/image010.png" o:title=3D"IncidenceMatr=
ix_999"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D451 height=3D297
src=3D"gutierrez_files/image011.gif" v:shapes=3D"_x0000_i1025"><![endif]><s=
pan
style=3D'mso-spacerun:yes'>&nbsp;</span></span><span style=3D'font-size:9.0=
pt;
mso-bidi-font-size:8.5pt;line-height:200%;font-family:Arial'>http://mathwor=
ld.wolfram.com/images/eps-gif/IncidenceMatrix_999.gif</span><span
style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;line-height:200%;font-fa=
mily:
Arial'><o:p></o:p></span></p>

<p class=3DMsoBodyTextIndent><span style=3D'font-size:12.0pt;line-height:20=
0%;
font-family:Arial'>In the first example above, the vertex 4, which is the l=
ast
row, has an edge to each of the other vertices. An adjacency matrix is an <=
i>n</i>x<i>n</i>
matrix A. In this matrix, <i>n</i> is the number of vertices in the graph. =
If
vertex x and vertex y and connected by an edge, then that is labeled as a<s=
ub>xy</sub>
is 1 (one), it is not connected by an edge then it is a zero (Wikipedia). T=
he
Laplacian Matrix, take A (adjacency matrix) and subtracts the degree matrix.
The definition for the Laplacian matrix is D-A. This type of matrix includes
the information not only for the degree, but also for the adjacency as well.
And finally, the distance matrix is a degree matrix shown by an <i>n</i> by=
 <i>n</i>
matrix, which is shows as d<sub>xy</sub>.<span style=3D'mso-spacerun:yes'>&=
nbsp;
</span>In network systems and computing, the adjacency matrix would be the
matrix to use if you were looking for any subgraphs, and the distance matri=
x is
what you would want to use to get the shortest path between an edge and ver=
tex.
Here are examples of what each of these matrices looks like when applied&#8=
211;
<o:p></o:p></span></p>

<p class=3DMsoBodyTextIndent style=3D'text-indent:0in'><!--[if gte vml 1]><=
v:shape
 id=3D"_x0000_s1043" type=3D"#_x0000_t75" style=3D'position:absolute;margin=
-left:0;
 margin-top:14.4pt;width:203.25pt;height:99.75pt;z-index:5;
 mso-position-horizontal:left'>
 <v:imagedata src=3D"gutierrez_files/image012.gif" o:title=3D"incidence"/>
 <w:wrap type=3D"square"/>
</v:shape><![endif]--><![if !vml]><img width=3D271 height=3D133
src=3D"gutierrez_files/image012.gif" align=3Dleft hspace=3D12 v:shapes=3D"_=
x0000_s1043"><![endif]><b><span
style=3D'font-family:Arial'>Incidence Matrix &#8211; </span></b><span
style=3D'font-size:9.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-fa=
mily:
Arial'>http://www8.cs.umu.se/kurser/TDBAfl/VT06/algorithms/LEC/LECTUR14/</s=
pan><st1:stockticker><span
 style=3D'font-size:9.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
 Arial'>IMG</span></st1:stockticker><span style=3D'font-size:9.0pt;mso-bidi=
-font-size:
12.0pt;line-height:200%;font-family:Arial'>887.GIF<o:p></o:p></span></p>

<p class=3DMsoBodyTextIndent style=3D'text-indent:0in'><b><span style=3D'fo=
nt-family:
Arial'>Adjacency Matrix &#8211; <o:p></o:p></span></b></p>

<p class=3DMsoBodyTextIndent align=3Dcenter style=3D'text-align:center;text=
-indent:
0in'><b><span style=3D'font-family:Arial'><!--[if gte vml 1]><v:shape id=3D=
"_x0000_i1026"
 type=3D"#_x0000_t75" style=3D'width:132.75pt;height:2in'>
 <v:imagedata src=3D"gutierrez_files/image013.png" o:title=3D"200309_068_2"=
/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D177 height=3D192
src=3D"gutierrez_files/image014.jpg" v:shapes=3D"_x0000_i1026"><![endif]><o=
:p></o:p></span></b></p>

<p class=3DMsoBodyTextIndent><span style=3D'font-size:9.0pt;mso-bidi-font-s=
ize:
12.0pt;line-height:200%;font-family:Arial'>http://www8.cs.umu.se/kurser/TDB=
Afl/VT06/algorithms/LEC/LECTUR14/</span><st1:stockticker><span
 style=3D'font-size:9.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
 Arial'>IMG</span></st1:stockticker><span style=3D'font-size:9.0pt;mso-bidi=
-font-size:
12.0pt;line-height:200%;font-family:Arial'>887.GIF<o:p></o:p></span></p>

<p class=3DMsoBodyTextIndent style=3D'text-indent:0in'><b><span style=3D'fo=
nt-family:
Arial'><o:p>&nbsp;</o:p></span></b></p>

<p class=3DMsoBodyTextIndent style=3D'text-indent:0in'><b><span style=3D'fo=
nt-family:
Arial'>Laplacian Matrix &#8211; <o:p></o:p></span></b></p>

<p class=3DMsoBodyTextIndent align=3Dcenter style=3D'text-align:center;text=
-indent:
0in'><!--[if gte vml 1]><v:shape id=3D"_x0000_s1044" type=3D"#_x0000_t75" s=
tyle=3D'position:absolute;
 left:0;text-align:left;margin-left:0;margin-top:3.7pt;width:133.95pt;heigh=
t:88.5pt;
 z-index:6;mso-wrap-edited:f;mso-position-horizontal:left' wrapcoords=3D"19=
46 393 1492 491 389 1669 389 2062 130 3535 389 5105 389 5498 1751 6676 2205=
 6676 5254 8247 5449 10015 6357 11389 7784 12960 7070 16102 6616 17575 6811=
 19440 7654 20815 8238 21109 8497 21109 9146 21109 16216 19538 17124 17869 =
17319 16102 17059 15120 16995 14531 20303 12960 21146 11684 21405 9720 2108=
1 8640 20951 7462 19459 6971 16022 6676 15632 5498 15568 4124 13751 3829 45=
41 3535 4346 1669 3243 589 2724 393 1946 393">
 <v:imagedata src=3D"gutierrez_files/image004.png" o:title=3D"graph theory"=
/>
 <w:wrap type=3D"square"/>
</v:shape><![endif]--><![if !vml]><img width=3D179 height=3D118
src=3D"gutierrez_files/image015.gif" align=3Dleft hspace=3D12 v:shapes=3D"_=
x0000_s1044"><![endif]><span
style=3D'font-size:9.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-fa=
mily:
Arial'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1027" type=3D"#_x0000_t75"
 style=3D'width:177pt;height:92.25pt'>
 <v:imagedata src=3D"gutierrez_files/image016.png" o:title=3D"fd90b1415cc54=
f26a57c327f238b49cb"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D236 height=3D123
src=3D"gutierrez_files/image017.jpg" v:shapes=3D"_x0000_i1027"><![endif]><o=
:p></o:p></span></p>

<p class=3DMsoBodyTextIndent align=3Dcenter style=3D'text-align:center;text=
-indent:
0in'><span style=3D'font-size:9.0pt;mso-bidi-font-size:12.0pt;line-height:2=
00%;
font-family:Arial'>http://en.wikipedia.org/wiki/Laplacian_matrix<span
style=3D'mso-tab-count:2'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbs=
p;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span><o:p>=
</o:p></span></p>

<p class=3DMsoBodyTextIndent style=3D'text-indent:0in'><span style=3D'font-=
size:9.0pt;
mso-bidi-font-size:12.0pt;line-height:200%;font-family:Arial'><o:p>&nbsp;</=
o:p></span></p>

<table class=3DMsoNormalTable border=3D0 cellpadding=3D0 align=3Dleft width=
=3D187
 style=3D'width:140.55pt;mso-cellspacing:1.5pt;mso-table-lspace:9.0pt;margi=
n-left:
 6.75pt;mso-table-rspace:9.0pt;margin-right:6.75pt;mso-table-anchor-vertica=
l:
 paragraph;mso-table-anchor-horizontal:margin;mso-table-left:left;mso-table=
-top:
 9.05pt'>
 <tr style=3D'height:11.7pt'>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.7pt'>
  <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;mso-elemen=
t:frame;
  mso-element-frame-hspace:9.0pt;mso-element-wrap:around;mso-element-anchor=
-vertical:
  paragraph;mso-element-anchor-horizontal:margin;mso-element-top:9.05pt;
  mso-height-rule:exactly'><b><span style=3D'font-family:Arial'><o:p>&nbsp;=
</o:p></span></b></p>
  </td>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.7pt'>
  <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;mso-elemen=
t:frame;
  mso-element-frame-hspace:9.0pt;mso-element-wrap:around;mso-element-anchor=
-vertical:
  paragraph;mso-element-anchor-horizontal:margin;mso-element-top:9.05pt;
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-size:
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  <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;mso-elemen=
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-size:
  12.0pt;font-family:Arial'>e<o:p></o:p></span></b></p>
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t:frame;
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9.0pt;
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9.0pt;
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2<o:p></o:p></span></p>
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9.0pt;
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9.0pt;
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9.0pt;
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4<o:p></o:p></span></p>
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9.0pt;
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9.0pt;
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3<o:p></o:p></span></p>
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9.0pt;
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 <tr style=3D'mso-yfti-lastrow:yes;height:11.1pt'>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.1pt'>
  <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;mso-elemen=
t:frame;
  mso-element-frame-hspace:9.0pt;mso-element-wrap:around;mso-element-anchor=
-vertical:
  paragraph;mso-element-anchor-horizontal:margin;mso-element-top:9.05pt;
  mso-height-rule:exactly'><b><span style=3D'font-family:Arial'>f<o:p></o:p=
></span></b></p>
  </td>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.1pt'>
  <p class=3DMsoNormal style=3D'mso-element:frame;mso-element-frame-hspace:=
9.0pt;
  mso-element-wrap:around;mso-element-anchor-vertical:paragraph;mso-element=
-anchor-horizontal:
  margin;mso-element-top:9.05pt;mso-height-rule:exactly'><span
  style=3D'font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:Arial'>23=
1<o:p></o:p></span></p>
  </td>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.1pt'>
  <p class=3DMsoNormal style=3D'mso-element:frame;mso-element-frame-hspace:=
9.0pt;
  mso-element-wrap:around;mso-element-anchor-vertical:paragraph;mso-element=
-anchor-horizontal:
  margin;mso-element-top:9.05pt;mso-height-rule:exactly'><span
  style=3D'font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:Arial'>20=
0<o:p></o:p></span></p>
  </td>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.1pt'>
  <p class=3DMsoNormal style=3D'mso-element:frame;mso-element-frame-hspace:=
9.0pt;
  mso-element-wrap:around;mso-element-anchor-vertical:paragraph;mso-element=
-anchor-horizontal:
  margin;mso-element-top:9.05pt;mso-height-rule:exactly'><span
  style=3D'font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:Arial'>20=
3<o:p></o:p></span></p>
  </td>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.1pt'>
  <p class=3DMsoNormal style=3D'mso-element:frame;mso-element-frame-hspace:=
9.0pt;
  mso-element-wrap:around;mso-element-anchor-vertical:paragraph;mso-element=
-anchor-horizontal:
  margin;mso-element-top:9.05pt;mso-height-rule:exactly'><span
  style=3D'font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:Arial'>83=
<o:p></o:p></span></p>
  </td>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.1pt'>
  <p class=3DMsoNormal style=3D'mso-element:frame;mso-element-frame-hspace:=
9.0pt;
  mso-element-wrap:around;mso-element-anchor-vertical:paragraph;mso-element=
-anchor-horizontal:
  margin;mso-element-top:9.05pt;mso-height-rule:exactly'><span
  style=3D'font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:Arial'>83=
<o:p></o:p></span></p>
  </td>
  <td style=3D'padding:.75pt .75pt .75pt .75pt;height:11.1pt'>
  <p class=3DMsoNormal style=3D'mso-element:frame;mso-element-frame-hspace:=
9.0pt;
  mso-element-wrap:around;mso-element-anchor-vertical:paragraph;mso-element=
-anchor-horizontal:
  margin;mso-element-top:9.05pt;mso-height-rule:exactly'><span
  style=3D'font-size:10.0pt;mso-bidi-font-size:12.0pt;font-family:Arial'>0<=
o:p></o:p></span></p>
  </td>
 </tr>
</table>

<p class=3DMsoBodyTextIndent style=3D'text-indent:0in'><b><span style=3D'fo=
nt-family:
Arial'>Distance Matrix &#8211; <o:p></o:p></span></b></p>

<p class=3DMsoBodyTextIndent style=3D'text-indent:0in'><b><span style=3D'fo=
nt-family:
Arial'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1028" type=3D"#_x0000_t75"
 style=3D'width:102pt;height:100.5pt'>
 <v:imagedata src=3D"gutierrez_files/image018.png" o:title=3D"Clusters"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D136 height=3D134
src=3D"gutierrez_files/image019.jpg" v:shapes=3D"_x0000_i1028"><![endif]><o=
:p></o:p></span></b></p>

<p class=3DMsoNormal style=3D'text-indent:.5in;line-height:200%'><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoBodyTextIndent2><span style=3D'font-family:Arial'>If you&#821=
7;ve
ever wondered how web search engines work, the answer lies in linear algebr=
a,
which is an element of algebraic graph theory. How exactly does it work? We=
ll,
let&#8217;s take a look at Google&#8217;s PageRank algorithm. A webpage is
given a ranking based on how many high ranked websites link to the website.=
 The
more links directing you to a certain website, the better your rank will be=
. A
simple graph for a set of 8 web pages would look like this &#8211; <o:p></o=
:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1029" type=3D"#_x0000_t75"
 style=3D'width:270pt;height:129pt'>
 <v:imagedata src=3D"gutierrez_files/image020.jpg" o:title=3D"goodnet"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D360 height=3D172
src=3D"gutierrez_files/image021.jpg" v:shapes=3D"_x0000_i1029"><![endif]><o=
:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><span
style=3D'font-size:9.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-fa=
mily:
Arial'>http://www.ams.org/samplings/feature-column/fcarc-pagerank<o:p></o:p=
></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-size:11=
.0pt;
mso-bidi-font-size:12.0pt;line-height:200%;font-family:Arial'><o:p>&nbsp;</=
o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'>In
the picture above, the webpage represented by vertex 8, would have the high=
est
page ranking because it has the most links directing to it. Though, this is=
 a very
simple illustration, web search engines use this model as a way to rank the
millions of websites out there in cyberspace. <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%;mso-layout-grid-align:none;
text-autospace:none'><span style=3D'font-family:Arial'><span style=3D'mso-t=
ab-count:
1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </spa=
n>Graphing
Theory is also a part of geometry with Geometric Graph Theory. While
researching geometric graph theory, I found that there is a graphing theory=
 for
a collection of sets. This is called an intersection graph. An intersection
graph is defined as an undirected graph formed from a family of sets
(Wikipedia). For example, of we had V(A) and E(A), we would know that we ar=
e illustrating
the vertices and edge of set A. If set A {1,2,3,4,5}, and set B{5,6,7,8,9},
then we would create a vertex for both of the sets &#8211; V<sub>a, </sub>V=
<sub>b</sub>.
If<span style=3D'mso-spacerun:yes'>&nbsp; </span>I=3D the intersection grap=
h, then
the formula for an intersection graph would be &#8211; <o:p></o:p></span></=
p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;mso-layout-g=
rid-align:
none;text-autospace:none'><b><span style=3D'font-family:Arial'>E(I) =3D {{ =
V<sub>a,
</sub>V<sub>b</sub>} </span></b><b><span style=3D'mso-bidi-font-size:13.0pt;
font-family:Arial'>| A</span></b><span style=3D'mso-bidi-font-size:13.0pt;
font-family:Arial'>|<b>B}<o:p></o:p></b></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;mso-layout-g=
rid-align:
none;text-autospace:none'><span style=3D'mso-bidi-font-size:13.0pt;font-fam=
ily:
Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;mso-layout-g=
rid-align:
none;text-autospace:none'><span style=3D'mso-bidi-font-size:13.0pt;font-fam=
ily:
Arial'><o:p>&nbsp;</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%;mso-layout-grid-align:none;
text-autospace:none'><span style=3D'mso-bidi-font-size:13.0pt;line-height:2=
00%;
font-family:Arial'>As discussed in the engineering journal IIE Transactions=
, an
intersection graph for rectangles on a plane can be used to solve design
problems of the Flexible Manufacturing Systems (</span><st1:stockticker><sp=
an
 style=3D'mso-bidi-font-size:13.0pt;line-height:200%;font-family:Arial'>FMS=
</span></st1:stockticker><span
style=3D'mso-bidi-font-size:13.0pt;line-height:200%;font-family:Arial'>) pu=
nch
press in the production of flat sheet metal. Because the advantages of </sp=
an><st1:stockticker><span
 style=3D'mso-bidi-font-size:13.0pt;line-height:200%;font-family:Arial'>FMS=
</span></st1:stockticker><span
style=3D'mso-bidi-font-size:13.0pt;line-height:200%;font-family:Arial'> inc=
lude
time-savings, it is important to make sure that a solution is found for tho=
se
in this field. This is something that you and I may not find important, but
engineers or machinists who work with sheet metal frequently, would be extr=
emely
interested in the results of this study.<span style=3D'mso-spacerun:yes'>&n=
bsp;
</span>Here is an example of what an intersection graph can be illustrated =
as
&#8211; </span><span style=3D'font-size:11.0pt;mso-bidi-font-size:13.0pt;
line-height:200%;font-family:Arial'><o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%;
mso-layout-grid-align:none;text-autospace:none'><span style=3D'font-size:11=
.0pt;
mso-bidi-font-size:13.0pt;line-height:200%;font-family:Arial'><!--[if gte v=
ml 1]><v:shape
 id=3D"_x0000_i1030" type=3D"#_x0000_t75" style=3D'width:179.25pt;height:13=
5pt'>
 <v:imagedata src=3D"gutierrez_files/image022.gif" o:title=3D"Intersection_=
graph"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D239 height=3D180
src=3D"gutierrez_files/image022.gif" v:shapes=3D"_x0000_i1030"><![endif]><s=
pan
style=3D'mso-spacerun:yes'>&nbsp;</span></span><span style=3D'font-size:9.0=
pt;
mso-bidi-font-size:13.0pt;line-height:200%;font-family:Arial'>http://en.wik=
ipedia.org/wiki/Intersection_graph<o:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%;
mso-layout-grid-align:none;text-autospace:none'><span style=3D'font-size:11=
.0pt;
mso-bidi-font-size:13.0pt;line-height:200%;font-family:Arial'><o:p>&nbsp;</=
o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%;mso-layout-grid-align:none;
text-autospace:none'><span style=3D'mso-bidi-font-size:8.5pt;line-height:20=
0%;
font-family:Arial'>Notice how, although the vertices are in a single set, t=
he
edge must pass through one or more other sets to get to the next vertex. <o=
:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-bidi-fon=
t-size:
8.5pt;line-height:200%;font-family:Arial'><span style=3D'mso-tab-count:1'>&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Th=
is
semester, we also discussed probabilities, which also happens to be an elem=
ent
in graph theory as the Probabilistic graph theory. Graphs that fall under
probabilistic graph theory are graphs that are obtained by adding random ed=
ges
in between vertices at random. Paul Erdos and Alfred Renyi first introduced
random graphs to the world in the paper &#8220;On Random Graphs&#8221;
(Wikipedia). Erdos frequently challenged his followers to solve a problem h=
e created
with a monetary prize offered, coming out of his own pocket.<span
style=3D'mso-spacerun:yes'>&nbsp; </span>The problem was this &#8211; Prove=
 that
among any six people at a gathering, there will always be three mutual
acquaintances and three mutual non-acquaintances (Erdos). In order to solve=
 a
probabilistic problem like this, first you would have to figure out which
graphs do not solve this problem, because Erdos stated that if it can be sh=
own
that this number of graphs that do not work is definitely less than the tot=
al
number of graphs with a given number <i>n</i> of points, then there must ex=
ist
a graph with the property in question (Erdos). Ramsey, an English philosoph=
er
and mathematician, created a theorem from which the probabilistic methods w=
ere
discovered. Ramsey claimed that every infinite graph also contained a compl=
ete
infinite subgroup, or independent set of points (Erdos).<span
style=3D'mso-spacerun:yes'>&nbsp; </span><o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'mso-bidi-fon=
t-size:
8.5pt;line-height:200%;font-family:Arial'><span style=3D'mso-tab-count:1'>&=
nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span>Gr=
aph
Theory plays a role in biology. Because our bodies are made up of so many
different part working together, biologists have had to use graph theory
applications to learn about the network systems of our bodies, much like as
seen in analyzing computer science. In May of 2009, the International Journ=
al
of Health Geographics published a report titled, &#8220;</span><span
style=3D'font-family:Arial'>A graph-theory method for pattern identificatio=
n in
geographical epidemiology - a preliminary application to deprivation and
mortality&#8221; where it discussed using graph theory as an alternative wa=
y to
research the clusters of disease. Usually, geographic information systems a=
re
used to detect clusters of disease and used to identify a pattern, but the =
more
complex the disease is, the more complex it is to track the pattern of the
cluster. The approach graph theory uses in computer science, may be a great
method to use in researching the patterns of the more complex disease clust=
ers.<span
style=3D'mso-spacerun:yes'>&nbsp; </span>In this experiment, the vertices
represented census enumeration districts (CEDs) and the edges were determin=
ed
by whether or not CEDs where neighbors (Maheswaran, Craigs, Read, </span><s=
t1:City><st1:place><span
  style=3D'font-family:Arial'>Bath</span></st1:place></st1:City><span
style=3D'font-family:Arial'>, and Willett) If the CEDs where neighbors, the=
n they
were adjacent, meaning that they were part of the same cluster. Researchers
were able to determine whether or not the clusters were associated with each
other or not. This is just one example of how the graph theory has applied =
to
biological research. Just imagine with all of the different components of a
human body and all the potential connections, how different graphs of diffe=
rent
parts of our bodies would look. <o:p></o:p></span></p>

<p class=3DMsoNormal style=3D'text-indent:.5in;line-height:200%'><span
style=3D'font-family:Arial'>Graph theory is not only present in biology, bu=
t also
in sociology, specifically in social network analysis. Social network analy=
sis
studies the importance of relationships in comparison with other relationsh=
ips.
Think of how Facebook is able to give you friend suggestions. They take the
friends that you have, and compare friends of your friends to decide whether
you may know that person or not. This explanation is very simple, and just a
tiny tip of the iceberg. This idea has been around for many years, but may =
have
been popularized with the Six Degrees of Separation theory. The idea is that
everyone on the planet is linked to another person, but six degrees at the
most. There is even a map to illustrate this idea &#8211; <o:p></o:p></span=
></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><span
style=3D'font-family:Arial'><!--[if gte vml 1]><v:shape id=3D"_x0000_i1031"=
 type=3D"#_x0000_t75"
 style=3D'width:322.5pt;height:242.25pt'>
 <v:imagedata src=3D"gutierrez_files/image023.png" o:title=3D"800px-Six_deg=
rees_of_separation_svg"/>
</v:shape><![endif]--><![if !vml]><img border=3D0 width=3D430 height=3D323
src=3D"gutierrez_files/image024.gif" v:shapes=3D"_x0000_i1031"><![endif]><o=
:p></o:p></span></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><span
style=3D'font-size:9.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-fa=
mily:
Arial'>http://en.wikipedia.org/wiki/File:Six_degrees_of_separation.svg<o:p>=
</o:p></span></p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'>Each
vertex in the middle (V<sub>1 </sub>- V<sub>6</sub>) has an edge connecting=
 to
other people (vertices), all of these vertices at some point intersect with
another but no more than 6 degrees, making a connection. <o:p></o:p></span>=
</p>

<p class=3DMsoNormal style=3D'line-height:200%'><span style=3D'font-family:=
Arial'><span
style=3D'mso-tab-count:1'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&=
nbsp;&nbsp;&nbsp; </span>As
you can see, the ideas of graph theory are applied to many different areas
other than directly mathematics. I have come to notice that my day-to-day f=
unctions
have some sort of link to graph theory. Although the actual formulas are a
little bit scary to look at, if you now what it is used for, it is a lot ea=
sier
to understand. While researching this topic, I was able to link graph theor=
y to
games I&#8217;ve played. Graph theory is a great resource for medical
researches, so you can say that graph theory can even save lives. </span><s=
pan
style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;line-height:200%;font-fa=
mily:
Arial'><o:p></o:p></span></p>

<span style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;
font-family:Arial;mso-fareast-font-family:"Times New Roman";mso-ansi-langua=
ge:
EN-US;mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><br clear=3Dall
style=3D'page-break-before:always'>
</span>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><u><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'>Works Cited<o:p></o:p></span></u></p>

<p class=3DMsoNormal align=3Dcenter style=3D'text-align:center;line-height:=
200%'><u><span
style=3D'font-size:11.0pt;mso-bidi-font-size:12.0pt;line-height:200%;font-f=
amily:
Arial'><o:p><span style=3D'text-decoration:none'>&nbsp;</span></o:p></span>=
</u></p>

<p class=3DMsoNormal style=3D'margin-left:.5in;text-indent:-.5in;line-heigh=
t:200%'><span
class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;
line-height:200%;font-family:Arial'>Timmons, Todd. &quot;The Birth of Graph
Theory: Leonhard Euler and the </span></span><st1:place><st1:PlaceName><span
  class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5p=
t;
  line-height:200%;font-family:Arial'>K&ouml;nigsberg</span></span></st1:Pl=
aceName><span
 class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;
 line-height:200%;font-family:Arial'> </span></span><st1:PlaceType><span
  class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5p=
t;
  line-height:200%;font-family:Arial'>Bridge</span></span></st1:PlaceType><=
/st1:place><span
class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;
line-height:200%;font-family:Arial'> Problem.&quot;&nbsp;Science and Its Ti=
mes:
Understanding the Social Significance of Scientific
Discovery.&nbsp;Eds.&nbsp;Josh Lauer&nbsp;and&nbsp;Neil
Schlager.&nbsp;Vol.&nbsp;4.&nbsp;</span></span><st1:City><st1:place><span
  class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5p=
t;
  line-height:200%;font-family:Arial'>Detroit</span></span></st1:place></st=
1:City><span
class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;
line-height:200%;font-family:Arial'>:&nbsp;Gale,&nbsp;2000.&nbsp;227-229.&n=
bsp;8&nbsp;vols.&nbsp;Gale
Virtual Reference Library.&nbsp;Gale.&nbsp;</span></span><st1:place><st1:Pl=
aceName><span
  class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5p=
t;
  line-height:200%;font-family:Arial'>Mission</span></span></st1:PlaceName>=
<span
 class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;
 line-height:200%;font-family:Arial'> </span></span><st1:PlaceType><span
  class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5p=
t;
  line-height:200%;font-family:Arial'>College</span></span></st1:PlaceType>=
</st1:place><span
class=3Dcitation1><span style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;
line-height:200%;font-family:Arial'> Library.&nbsp;8 May. 2010&nbsp;</span>=
</span><span
style=3D'font-size:11.0pt;mso-bidi-font-size:8.5pt;line-height:200%;font-fa=
mily:
Arial'><br>
<span class=3Dcitation1><span style=3D'mso-ansi-font-size:11.0pt;line-heigh=
t:200%;
font-family:Arial'>&lt;http://0-find.galegroup.com.library.wvmccd.cc.ca.us/=
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