ࡱ> z|~z|~  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxy{|}~Root Entry*n׾@MatOST -7%n׾ -7%n׾MMMN0NDH0H"j  H88=/L|F Finding the "Middle" of a Graph and Why it Matters Joseph Malkevitch Mathematics and Computing Dept. York College (CUNY) Jamaica, New York 11451 email: joeyc@cunyvm.cuny.edu What is the public's perception of mathematics and mathematicians? Typical response: Mathematics is a sea of problems involving symbols whose meaning I can not understand! Samples: (x2) 5 = ? x/(x3) + (x4)/(x+7) = ? X2(y  4x)  y(3X2 + 3Y) = ? 3x  2(5x  4) = 8 (Solve for x) Typical response: Mathematicians are aging males who wear beards and glasses and if they have any hair, it's a mess! or Albert Einstein is the greatest living mathematician. Neither the general public nor surprisingly large parts of the scientific community are aware of: a. The dramatic growth in new mathematical tools. b. The role of mathematics in the development of many recent technologies. Examples: Medical imaging Fax and wireless communication Compact disc and CDROM technology Fuel efficient transportation Mapping the human genome c. The emerging role of mathematics in areas outside of the sciences and engineering. Examples: Political science Applications: elections, legislative apportionment, weighted voting, political games Economics Applications: bankruptcy, fair allocation, cost sharing, equilibrium prices, efficiency, games Business Applications: bargaining, optimization, security, fairness, new financial instruments Accounting Applications: internal cost sharing, financial instruments, generational equity Fine arts Applications: fabric design, classification of patterns Mathematics can be viewed as a subject which has concerns with various themes in contrast with various techniques. Techniques: 0. Arithmetic 1. Algebra 2. Geometry 3. Trigonometry 4. Calculus 5. Linear (Matrix) Algebra 6. Probability and Statistics 7. Graph Theory 8. Modern Algebra 9. Coding Theory 10. Knot Theory (Many more!) Themes 1. Optimization 2. Growth and Change 3. Information 4. Fairness and Equity 5. Risk 6. Shape and Space 7. Pattern and Symmetry 8. Order and Disorder 9. Reconstruction 10. Conflict and Cooperation 11. Unintuitive behavior Example: What is the best location for a hospital in this community? How would your analysis differ if the problem involved locating a firehouse a school, or a supermarket? What difference is there if a facility must be at a vertex versus located somewhere along an edge?  Distance: * Two points in space * Insulin molecules in a gorilla and chimpanzee * Two relatives * Preference schedules of two voters * Two images or texts * Two code words Distances between two points in space: Euclidean distance Taxicab distance Max distance Abstract properties of distance: 1. Distance is a nonnegative real number. 2. The distance between two points is zero if and only if they are the same. 3. The distance between p and q is the same as the distance between q and p. 4. The triangle inequality: The distance from p to q plus the distance from q to r is at least as large as the distance from p to r. Eccentricity of a vertex v: Maximum distance any vertex can be from v Central vertex: Vertex with minimum value for its eccentricity Numbers are eccentricities of the vertices.  Center: Subgraph induced by the central vertices of a graph Jordan's Theorem: The center of a tree is either a single vertex or a pair of vertices joined by an edge. Exercise: A tree has one central vertex if and only if the diameter of the tree is twice its radius. Branch weight: Numbers show the branch weights of the vertices.  <>T-B1ysmsgsa[gTgTgTgaX.B$. N1Ae:Goysymgmga[U[U[U[U%%+.&$@o|+<O) 5  N ysmgmga[UOICN@8 @,8 >hA%% N P n ` a c p * X ysmgma[U[U[OI?18 @J00N G_CDFHysymygagy[gagWRAA8 @8@  '<>tttttttttttttggW M$ >RTgobUUUUUH===   B  >$>$  +-@BttgggZTT  :$ /1ttiii^^SSMA w A V   ?yll_RE:   A   ?Atv)Ice8:tt * :EG moz|ttii^^SS    N+xkkkkk`SSF A! y y C$ +-:<>@MO^jwrrb\QQQQQQQQ     ' ) + 3 5 F \ l tttti\iQQQQQQ " =  =      * , > N P ttttttf[M      " P b n p r t v x z | ~ \ ^ tmmmmmmmmmmmmmmm< ^ ` c n p tfXG<1 + +x +x  $  $   & ( * , . V X q``U 5#ttZ ttX l n 4 6 tii^^^WWWWWW< M N  /1G]_pr1x13ACFHxx1oN H!"#$%>?:+ P ^ X 1H&'()*+,-./0123456 II  x 45 IC xx(x#.#.????????|||||~|||~|||~~~~~>>>>|>|>|>|>>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>|>>|>|>|>|?~?~?~?>>>>>>>>>>>>>>>>>?>>>>>>>>>>>>>>>>>>>>>?>>>>>>>>>>>>>>>>>???????????? ?Xp{      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{}~<  ?x` ?Ɂ?`0@8 ??40g0ÀX 05~ ``0 ?w0@` `#0%0D????4??>>>>>>>>>>p?>x>>|??>?>?>?>?>>?>>?|>>>|>>>|>>>|<>>|<~>|<?>|>?>|<>>|<>>|?>>>|<>|> |<>|>|<><>|?<>>>|<>>>||>>>|~>>|??>|>|>|>|>??@>>>~>?>>>?|>|<>|?>||0><>?>>>>>>>>>>>>>??>???????????      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}????? hfh  P( 45 fhC P(P(((P#.#.<<~~|||||||?|??~~>~?|?|?~?~?~~?>???~<>8>~ ??@vπ??~?~~?~~~?????????????????????????????????????x???????|?????~~~~?????|||||????>>~????????????????      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~????????>>|>~~~??~>??>????>|>| >|>|>|?|??|>?~>|>?|>>?>??????????????????`????????|?>??|@||?x?~|???>??~????|?|~~|||@||?|||>|?~?<?<?||~~?~~~|~~~|||?|~||~~~|||?~?~??~~~~~~|||~?~?~|||||~?||~~??~?|>?|>?~>|>~|?~|??|>|>?>>>?|>?|>?|>|>|>~>|>|?|??|>|>>?~>?~>?|>|>|>?~|?|>|>|>|>|>|?|??|>|>|>|>|>|>|>?>>~>~>|>?|>?|>?|>|>|?~~?~?~>~?~??>?>?>~~>?>?>>~~>>>?>?>?>>>>~>?`>?>?>>~?~?>?? <~>~~?|~?||?|~?????~?~?~?>?>>~??>~~0~>>>?>>>????????????<??|???|?|?|??|?~?~~????????~?~?||?||???|??|>?|?|??|???????????????|?????????????? ?????~???>>|<>>>>>>?>?>?????? H`@??!0 N9 ??0ṇ$0?@?p 80 |0@`|ǣ<|sV$  ` <`H@'?B7p @# D>?0@`A`B8g????????????I|||?||?|?|?|??|~?|?|?|||?|?|?|~|~||??|||?|?|p|||||||~>|>>|?>|>||???|<||??~<?|>||>||?|?|?<x|>>|>~|?~|?~?|?~>~>||>||?|p?|>~~>|>?>|??>~?>|??>||?|?|?~1G||?~?????>??????|||??????|?||?||?|?||???|???||||??|??|~~?~|?~|?|????????|      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxy{|}~|??|????|???>>>>~??|???>?>??|>?????x??||?||~x|||xx|xx<|>>?|?|????|~??  ||??>>>>> |>|?>>>>>>?>>>>>>???>>>>>>>|>|>>?>>>>>>~>~>>>?>>>> >?>>|>>>>>?>>>> >>>>?>?>>>>>>>?>>>?>>>>>>>>>>>>>>>>|||~~~~~~~~||||||||||||||||||||||||||||||||||||||||x||x|xx||||||||||xxx||||||||x|????p??Word.Picture.8~ࡱ>     !"#$%&'()*+,-/012345789:;<=Root Entry F %\&n׾ 1TableCompObjhObjInfo [$@$NormalmH <A@<Default Paragraph Font /X2$A/qBYu0)4b$?r6N$^CUU4) @z( S3P B  3 AB S  ?k#t@ G@GTimes New Roman5Symbol3& Arial"h77!r0 DIMACS Center DIMACS Center  FMicrosoft Word Picture MSWordDocWord.Picture.89qWordDocument -@SummaryInformation(.DocumentSummaryInformation86G bjbjَ -@] $^k"  kgԾ   jUmH' 0P2 P2 N N!"#g$h%n3?r6N$^CUUPNG  IHDRhh\gAMAPLTEٟ IDATxM,ə,EخAi$,G@& Gٱy]Qv@A&KigKikIpٲ0 }Aؼ<=#&]m7-ң}K|U%yS~|Ki^j@eS>OE𴄤K?q}-8[xq޺i{Z8Kk nymGm2}{NR- <}0@c< fhw\(ޅ̟$޼Hj6a˯/Uv{ƛiC>2n $٭4[Ӳߥa irث0k~-i#xl/-O6ŰݒmZFv[F{Ne^"Z)p6?CqdD|/ufxl.mS%MCg{4)BhA >q*d:ߒ gG$xnOԒߎmڱEs;ֹҔ\O)6޺{ӎiiӎiiӎiiӎu.4޺{X=WK4۱NFm]i:mu)ҔҔ :WzZhR%O;hK|/5)O;hKW/iiӎu^Ҕd֥Kv,ߒiGj:YK.^^X[OKSvsўO>Zh}K)O;ֹh7OSvsўk=Z)O;)O;)oKu4Wii:mmd-9ں{ӎiiӎiiӎii ­KW ڴI@N6>cKGhzX*CNcBk[2Pmڵwo>}vϐp&/iـM^}x%ͺ]l$m@'ªT}ɕiX#.VgfDv\UKY29'mߒTDO~ƺҖV-=v9nں-vgh߰0Stǧpv5l]\~~mV^Ff\7,,+xF%Җ1w~EEK%3pyR[.8m <*X50h$hrFZ\NZfd6IҦ&4i\z_i|Q7+ IISυWؗ˞v hӎeWVvsѦ!#Z&u)]þMrWSݴ!K@ @ed32גaRĸ6ůq 5sjd,GWpciEs*v^ڟz5U7ۓ6Drң-%PT:=3Smh5̔5Wy}qhmc&uv`ZFZOY|m hk7;-{n{Zy,iZ8"zi:խaZ֒kJ4DrĞvV2D[OĞOfwZGe'm'״9 {z͒ݙatifJ7]1=ӢG m5}+ZrOw3zn̓W6#hoJJ[0# o,chSѴ9wyHe%k9(@OdfS;{("<,5t bL9 :p͜E@? #2ܕ6BrZH#%X[AxQBfJy|&k)-b/J'}WZ$Ys;|jyDx'`^t_{,I*ѱ +0Qi8\o]>oɛRHzZ6˃TǏzs *X,U@iEɽ&h|¥ae_}iRڱ]۞v0Š\׼@Khu+t9F||d@+{lM;0doɳ_fz-UҾ>aV_dL/|3]{=:CAwS0>7 #cJ%Q[ LЂn$hALU(^êʽ5׀LG hޖhiMΪ _ŗGX6F@huVבshr1%^?KXhT9uЪf=),PE˷h.k.PU:mZ͎--CԯŒY m< w-cU|* ڻ{Ļc-Q:j@WpЊEi[ym.09mۇ¯yù" 㭂6|~3U hqZӊc?-jj@vn3" {Z.ա$ܶӦDRϋ$'ֺw)UG[3vx=0M ,_K#mԽh.]*зso[j^廀V3$ 8譓l".ԭ%Kʺ}ы<&c9|ɖZQX|)90Pvdi,)D\OT:nۿ24/>ĿPK3a 77 iq'[ Ұ/d =ĥ?ki;B~jX2_$(SC MvDIk^= `Q! YLYZ5M\ 6!jgCjA/Z1ܭl$e^2' OȃfOTB:(;@[>H)k|"Xm*8t` IHcyhS9 MzyĀS \>(1ڧ/V)MMCx#(1SS!PfWIZ$Գ@R,}D{#q^rH؟$i#e*wLbmE@P~]wb{ N_ (޽uݝ KQG,Y_Js{JG}U(bS^w_R1csTRq~B~?w+76 ?꿺o~_||3h@%i_}?O~/e. ^>-eyڱY])%|oo~ǯ/|;} P_/? ^}3*_oV_?kozQş3GE/ukf7T-듢]R铸>k?~=8B5mޏ,raշ]sy^llp@P:Nϥj=.6i]dz[6?hyM])E}ia ҆i[ Mmlm^PGQ ZÏ0TQLۭb"e"5w(.C}QZ4 qf F+s} }mLFXxPZ co"އ ?@8rgFDhP*hk~m+ez@SJ Jon%Cœ4![ed3HhFbкuiV6ش'v{AZڱ(-ֳ`k-Iυ }*lo:ñ]-b/Fj.Јm:vBNmriM>h lH#8> uUkZJxl F+:Qa(l3H3ՓQu>kVV'r`6YM p{0H Zh-yp vAI+i15ik#s--Zu hznȄ"g^jԷ%u&Pw+h︷lFP@Y nhifcVH{O*ns+}&XӶmThk=F>C:Bޛm> h^^bf(\*)?Tu]v.4EiU4жApG$h-3 VK V-9M'4h!SpJРE*g~C; ZrMi޺Ն[iigJ:l (,#kiDhU<r3h?+Q*$=" ם U䙃׆V1@kv$|UxUrG3´shӰ`uBB5̥mwe_b.ipiZ<zaڹ;@ZţqmYƥ&%MR i 2CrVU 6)VBOp_L̮u˟ZB㳏nh른 qHE[?-y$ @ɀ틂VUmh Ev۔5|Sx7M\JaN;epZwԭbߝ6YD8v}H\VN@P@єvPG8]14i(M7m>E7_Θ`_ToXe'v},Yu(_XVAbS#:{F'swkh=#mhmﱯMٛn@L[@,Ѧ!ho&緦Mel_"߾3m^ nP@0ꄳ65L8X𹍀Dm}YߒcGRr mC>En&Y6bԴD 9KB%Zu掫L; *4uT2]9(}իP1-9]a:Ҥf?ѤMg;5-9PҴS|u;;Ф0͖ptÈJ[u@<DP,6CkMKN8+1Zih;Ҧ!/mZp<j"jp6 xi*Pzn^a6@+ !LV2YK>D2YKAKnU*L/jҷiի!Vp B 9Z(X8Cк@mJ  *u fhc ѦPxI|S D`\*mIWs"3XIP6Y!/)߫Sƒ7ЪsH^T"(/^(-6WI>Q`xaבxnJRZV0TXkp&55Ԕ"[CE&+PS춦nc{"DPZ|jag'|:$A15!Y<~zGfU#U$h5=&ZN-'p[Xki'sjwU{rZu_6 j M2ڙ 0 :mVӧf^=m8y+#M{yVNkn~Caw;ڙP[xƫRe/V= -QG;ShuUKvZN3hKmF̌IW4eF;sPE4"NMtI&icD MV 6KJM93]#,ރ4Iâm uu[h9mCAeUyn]`fN6]sou6_M{X&%fu@꿱j)[}n9CVBr f@m}cm~h ZY9g5mhr D+Єң䡘 4ߋMh3I93-m,ITz+mg+h;Hƈ+Mre?;gV/ܗc rMRڐ>[6Ӱ71Ui3i}?ٔ&$6~ Mڧ; oqBbS=P*n*_'j,{P@ZA.zګ^H1D366ho;{!ͬ/#0skGi M&60GlC2 N;9 x۵t=g>$ nKEJ,bI0[Xm-]b{wUSx6ܞV3mtL+-͛MWfb. ژ mVul9>[r%m?ZW9d?m60MjfbDy i;/&Uūvhh3 yR񨥴fÚiu g3i%?IUvEnq^Y6 VNi¯ۼZ >6v(Uպ.-"oɭd=Yu$R ]Oj}@#:=4VӦ!:߇r:q,~Zd2δ߫jv Vh}/U-捨VMѮ0@F[-L+S5neZV\$Q_sWn׸@VmM mVڙ}K&Rks6B5.uӝbKMk5msNY^;N5-m6__ v1i:C ;QMj"$i7Q+c$mJf2ZPΦ N cs ű 3%PO虎0-(b{hޤ3acz=^E/Wc6DGvѦ!7LGh<|Mo1=9Nuڡm`-kQLoe(Zeh!o8EP ?)d-(mXZ]3?었ҲZ5, ~bl-kHG1Qge(Cͮhgp|{NcW"N&mq C*;Ǧ"6R"ZPc7@PFhƻ 4^"c(ylɬΪ]6?.zCMn:eif&ôq埝+y/y11v9{藯E*[URF0Lf&ceuqisњ.2śsj qwvNuwʅYQmd m{ i ziՈYL;ڴ0!ZvƸȯ9254ҰxLK^u݊j{fB囙.-LWfi/Fcuemxn ؒ[@F1 }m6\CF.XKFtM;FFi{Փ%;5(Cuk2v;m#͛ookΒ`m,v m ݣp+iO^n(B0UV>m zwf+ eiSfVzZEp Uʎ*xeTⴑG9৒wfrm.}ЩQ "%4T&s)-9 'oSMbe26Nz5hy;>uu=hɞ;p6u:ɮmܣc;SK ]aBi]6 >kޔ2YpO´9Hۋ:ނƤis5.+϶*Q /Eٺ(:J JGiog.]}R,=q*Z3# gq=3ӺSђ]h\$l4D]yZwiKe h߾<3yl$yڱM"by"ghZUKVZͩڲj--iWM{W6ށpiEڠk+m;zuR ={YLj}R>w^Դ      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{}~rҾ_PA!eZZSIe-*HVg_LW4mι*ZL>uZdkWVbZNVbZ9E,nki3>Xdo`ziH6jndKC Li/0ʖh+_tiupilⴕV;šR}j^Si?3.,ZKciEF[j+-pJ[zYKozZiVmiݕuW]yZwiݕuW]yZwiݕuW]yZwiݕuW]yZwiݕuW]yZwiݕuW]yZwiݕuW]yZwiݕuW]yZwiqv'=;vIENDB`Oh+'0`   ( 4@HPXssDIMACS CenteroIMANormalCDIMACS Centero2MAMicrosoft Word 8.0@F#@ԀԾ@TRԾ՜.+,D՜.+,< hp  Rutgers Universityj  Title 6> _PID_GUIDAN{BC8A06C1-4050-11D3-B058-00609755CC33} METAFILEPICTh>>h>- k&WordMicrosoft Word     -@Times New Roman(- & C hh (hh???????x`?????????????0x~@???????????????????? ???????p??????????????????????????????????a ???<?????<????????????@????????????`??????????????????????????????????????????|??8~?<??????????????????????????????~???@<5??????????????????????????????????????????????????????????????????      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? `p?_???O?8???????<????????????O ?|83?| N?? ` 2 x :|`}~;<6??>hoOy3zy`' q (|`8`?????????????????????????????????????????????????????????????`??????????????????????????????@p ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????<?????????????????????????????????????? ??????????????????????????????????????????????????>??????x??|????????z????????????????      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~??? ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????0????&-` CH#k#k“ 9"9"Rίb#b#, 02020603050405020304Times New Roman 020b0604020202020204 Arial