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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 258 21 "Graph Theory in Maple" 
}}{PARA 258 "" 0 "" {TEXT 256 47 "Date: Nov 28, 2001\nLast Revision: N
ov 28, 2001\n" }{TEXT 267 7 "Maple 6" }}{PARA 260 "" 0 "" {TEXT 259 
16 "Bent E. Petersen" }}{PARA 259 "" 0 "" {TEXT 260 17 "bent@alum.mit.
edu" }}{PARA 259 "" 0 "" {TEXT 261 22 "petersen@math.orst.edu" }}
{PARA 0 "" 0 "" {TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT 263 32 "Course:
 Mth 355 (a.k.a. Mth 399)" }}{PARA 0 "" 0 "" {TEXT 264 15 "Term: Fall \+
2001" }}{PARA 0 "" 0 "" {TEXT 265 11 "File name: " }{TEXT 257 19 "355f
2001_graphs.mws" }}{PARA 0 "" 0 "" {TEXT 266 94 "Assignment 8 - due \"
never\" (There is an OSU tradition of no new assignments during dead w
eek.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 
"This worksheet contains a few comments on some of Maple's graph suppo
rt. It was composed fairly quickly and is unlikely to be free of error
s. If you find a serious error be sure to let me know." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(networks):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "G
raphs are created by the  graph()  command. If  V  is the set of verti
ces and  E  is a set of edges, each described as a set of two vertices
 (undirected edge) or as a list of two vertices (directed edges), then
 graph(V,E)  constructs the corresponding graph. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "G1:=graph( \+
\{v1,v2,v3,v4\},\{ \{v1,v2\},\{v2,v3\},\{v3,v4\},\{v3,v1\} \} ):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "On
ce we have defined a graph we can draw it:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(G1);" }}
{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6/-%'POINTSG6#7$$
\"\"\"\"\"!$F)F)-%'CURVESG6$7$7$$!\"\"F)$!+:w1-T!#>7$$\"+A95`hF4F0-%'C
OLOURG6&%$RGBGF)$\"#5F1F)-%%TEXTG6$F5Q#v46\"-F?6$F/Q#v3FB-F$6#F5-F$6#F
/-F$6#7$$!+3Q.^?F4F'-F?6$FLQ#v2FB-F?6$F&Q#v1FB-F,6$7$F/FLF8-F,6$7$F/F&
F8-F,6$7$F&FLF8-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 59 "The adjacency matrix is given by the  adjacency()  c
ommand:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "A1:=adjacency(G1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#A1G-%'matrixG6#7&7&\"\"!\"\"\"F+F*7&F+F*F+F*7&F+F+F*F+7&F*F*F+F*" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 
"Graphs may also be specified by using the  addvertex()  and  addedge(
)  commands:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "new(G2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "addvertex(\{1,2,3,4\},G2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "addedge(Cycle(1,2,3),G2):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "addedge(\{3,4\},G2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "draw(G2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6/-%%TEXTG6$7$$\"+A95`h!#>$!\"\"\"\"!Q\"46\"-F$6$7$F*$!+:
w1-TF)Q\"3F.-F$6$7$$\"\"\"F,$F,F,Q\"1F.-F$6$7$$!+3Q.^?F)F8Q\"2F.-%'CUR
VESG6$7$F7F1-%'COLOURG6&%$RGBGF,$\"#5F+F,-FC6$7$F>F1FF-%'POINTSG6#F7-F
C6$7$F1F&FF-FP6#F&-FP6#F1-FP6#F>-FC6$7$F7F>FF-%*AXESSTYLEG6#%%NONEG" 
1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur
ve 9" "Curve 10" "Curve 11" "Curve 12" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Instead of using  new() \+
 and  addvertex()  as above we can use  void()  which creates a graph \+
with no edges:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "G3:=void(\{1,2,3,4\}):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "vertices(G3);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#<&\"\"\"\"\"#\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "edges(G3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "We can no
w use the  addedge()  command to add any desired edges." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "addedg
e( \{\{1,2\},\{2,3\},\{3,4\},\{3,1\}\},G3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&%#e1G%#e2G%#e3G%#e4G" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "draw(G3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
{PLOTDATA 2 "6/-%%TEXTG6$7$$\"+A95`h!#>$!\"\"\"\"!Q\"46\"-F$6$7$F*$!+:
w1-TF)Q\"3F.-F$6$7$$\"\"\"F,$F,F,Q\"1F.-F$6$7$$!+3Q.^?F)F8Q\"2F.-%'CUR
VESG6$7$F7F1-%'COLOURG6&%$RGBGF,$\"#5F+F,-FC6$7$F>F1FF-%'POINTSG6#F7-F
C6$7$F1F&FF-FP6#F&-FP6#F1-FP6#F>-FC6$7$F7F>FF-%*AXESSTYLEG6#%%NONEG" 
1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur
ve 9" "Curve 10" "Curve 11" "Curve 12" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "The characteristic polyn
omial of an undirected graph is just the characteristic polynomial of \+
its adjacency matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p3:=sort(charpoly(G3,z),z);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,**$)%\"zG\"\"%\"\"\"F**&F)F*)F(
\"\"#F*!\"\"*&F-F*F(F*F.F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 81 "Maple has builtin definitions for compl
ete graphs. Here is the complete graph  K5" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "K5:=complete(5):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(K5);" }}{PARA 13 "
" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "67-%%TEXTG6$7$$\"\"\"\"\"!$F
)F)Q\"16\"-%'POINTSG6#F&-F.6#7$$\"+Q*p,4$!#5$\"+l^c5&*F5-F$6$F2Q\"2F,-
%'CURVESG6$7$F27$$!+]*p,4)F5$\"+9D&y(eF5-%'COLOURG6&%$RGBGF)$\"#5!\"\"
F)-F<6$7$F27$$\"+l*p,4$F5$!+c^c5&*F5FD-F<6$7$F27$$!+M*p,4)F5$!+PD&y(eF
5FD-F<6$7$F&FNFD-F<6$7$F&FVFD-F<6$7$F&F2FD-F<6$7$F&F?FD-F$6$FNQ\"5F,-F
.6#FV-F$6$FVQ\"4F,-F.6#FN-F.6#F?-F$6$F?Q\"3F,-F<6$7$FVFNFD-F<6$7$F?FNF
D-F<6$7$F?FVFD-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1
8" "Curve 19" "Curve 20" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 61 "As another example here is the complete b
ipartite graph  K3_3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "K3_3:=complete(3,3):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 11 "draw(K3_3);" }}{PARA 13 "" 1 "" {GLPLOT2D 
400 300 300 {PLOTDATA 2 "68-%'CURVESG6$7$7$$\"\"\"\"\"!$F*F*7$$!+2+++]
!#5$!+MSDg')F/-%'COLOURG6&%$RGBGF*$\"#5!\"\"F*-F$6$7$F'7$$F8F*$\"+&QKz
*e!#>F2-%%TEXTG6$7$$!+(*******\\F/$\"+SSDg')F/Q\"36\"-%'POINTSG6#F<-FB
6$F<Q\"4FJ-FL6#F,-FL6#7$$\"+-+++]F/$\"+PSDg')F/-FB6$FUQ\"2FJ-FB6$F'Q\"
1FJ-FL6#FD-F$6$7$FUF<F2-F$6$7$FU7$$\"+\"*******\\F/$!+VSDg')F/F2-F$6$7
$F'FboF2-FB6$F,Q\"5FJ-FL6#Fbo-FB6$FboQ\"6FJ-F$6$7$FDFboF2-F$6$7$FDF,F2
-F$6$7$FDF<F2-F$6$7$FUF,F2-FL6#F'-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 
2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve \+
10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" 
"Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 402 "Accordin
g to Kuratowski (circa 1930) a graph is planar if and only if it conta
ins no subgraph that is homeomorphic to  K5  or  K3_3. Note the term \+
\"homeomorphic\" is used here in a sense peculiar to graph theory: two
 graphs are homeomorphic if they can be obtained from the same graph b
y inserting vertices of degree 2  on its edges. This is a beautiful re
sult though I have no idea how effective it is." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Maple does have a builtin
 planarity test,  isplane()." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G4:=icosahedron():" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(G4);" }}{PARA 13 "" 1 "
" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6Y-%'POINTSG6#7$$!+*)******\\!#5$
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F0-F$6#7$$\"+<+++]F)$!+GSDg')F)-F-6$F?Q#10F0-F$6#7$$\"+]SDg')F)$!+z***
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\"3F0-F$6#7$$!+0+++]F)$\"+NSDg')F)-F-6$FipQ\"4F0-F$6#7$$!+VSDg')F)$\"+
\"*******\\F)-F-6$FcqQ\"5F0-F$6#7$F7$!+in?59F6-F-6$F]rQ\"6F0-F$6#7$$!+
HSDg')F)$!+;+++]F)-F-6$FerQ\"7F0-FR6$7$FboF\\pFen-FR6$7$FboFipFen-FR6$
7$F]oFerFen-FR6$7$F]oFipFen-FR6$7$F]oFboFen-FR6$7$F&F?Fen-FR6$7$F]rFIF
en-FR6$7$FerF3Fen-FR6$7$FerF?Fen-FR6$7$FboFcqFen-FR6$7$FboF&Fen-FR6$7$
F\\pFUFen-FR6$7$F?FIFen-FR6$7$F]oF\\pFen-FR6$7$FcqFIFen-FR6$7$F]rF3Fen
-FR6$7$FUFerFen-FR6$7$F\\pFcqFen-FR6$7$F&FIFen-FR6$7$F3F?Fen-FR6$7$F3F
IFen-FR6$7$FcqF&Fen-FR6$7$F\\pF]rFen-FR6$7$FUF]rFen-FR6$7$FcqF]rFen-FR
6$7$FipF&Fen-FR6$7$FipF?Fen-FR6$7$FipFerFen-F$6#F]o-F$6#FU-FR6$7$F]oFU
Fen-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+
19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" 
"Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur
ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3
8" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" "
Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curv
e 51" "Curve 52" "Curve 53" "Curve 54" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "isplanar(G4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr
ueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "G5:=cube(5):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "draw(G5);" }}{PARA 13 "" 1 "
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