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b(t)493 b(graphs.)759 b(Iden)-36 b(tit)g(y)493 b(\()p 0 .5 0 TeXcolorrgb 22443 12408 a SDict begin H.S end 22443 12408 a 0 .5 0 TeXcolorrgb Fg(2)p 0 .5 0 TeXcolorrgb 23093 11552 a SDict begin H.R end 23093 11552 a 23093 12408 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 23093 12408 a Black Fg(\))h(is)g(deriv)-36 b(ed)493 b(from)h(the)f(n)-36 b(um)g(b)36 b(er)492 b(of)j(spanning)e(trees)g(in)h(a)800 14013 y(wheel.)2751 15618 y(Applying)620 b(the)f(same)h(tec)-36 b(hnique)620 b(to)g(a)g(graph)f(kno)-36 b(wn)620 b(as)g(fan)h(giv)-36 b(es)620 b(us)g(a)g(new)g(pro)36 b(of)620 b(for)h(the)800 17223 y(follo)-36 b(wing)436 b(iden)-36 b(tit)g(y:)9787 21744 y SDict begin H.S end 9787 21744 a 9787 21744 a SDict begin 14.5 H.A end 9787 21744 a 9787 21744 a SDict begin [ /View [/XYZ H.V] /Dest (equation.3) cvn H.B /DEST pdfmark end 9787 21744 a Ff(F)10629 21943 y Fp(n)11624 21744 y Fg(=)13004 19855 y Fo(b)13555 19481 y Fk(n)p Fj(\000)p Fi(1)p 13555 19653 1557 40 v 14131 20202 a(2)15245 19855 y Fo(c)13485 20483 y Fd(Y)13476 23311 y Fp(k)24 b Fn(=1)15663 21744 y Fg(\(1)295 b(+)g(4)221 b(sin)20890 21196 y Fn(2)21770 20846 y Ff(k)45 b(\031)p 21770 21439 1507 54 v 22135 22656 a(n)23409 21744 y Fg(\))369 b(=)25665 19855 y Fo(b)26216 19481 y Fk(n)p Fj(\000)p Fi(1)p 26216 19653 1557 40 v 26791 20202 a(2)27905 19855 y Fo(c)26145 20483 y Fd(Y)26136 23311 y Fp(k)24 b Fn(=1)28323 21744 y Fg(\(1)296 b(+)e(4)221 b(cos)33695 21196 y Fn(2)34575 20846 y Ff(k)45 b(\031)p 34575 21439 1507 54 v 34940 22656 a(n)36214 21744 y Fg(\))p Ff(;)960 b(n)369 b Fe(\025)g Fg(4)p Ff(;)p Black 9539 w Fg(\(3\))p Black 800 25880 a(app)36 b(eared)433 b(in)h([)p 0 .5 0 TeXcolorrgb 8279 25880 a SDict begin H.S end 8279 25880 a 0 .5 0 TeXcolorrgb Fg(6)p 0 .5 0 TeXcolorrgb 8929 25024 a SDict begin H.R end 8929 25024 a 8929 25880 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.gar) cvn H.B /ANN pdfmark end 8929 25880 a Black Fg(])g(and)f(its)h(corresp)36 b(onding)433 b(references.)2751 27485 y(Also,)489 b(applying)478 b(this)f(tec)-36 b(hnique)477 b(to)g(the)g(path)f Ff(P)27765 27684 y Fp(n)28869 27485 y Fg(and)h(the)f(cycle)j Ff(C)37916 27684 y Fp(n)39019 27485 y Fg(giv)-36 b(es)478 b(us)f(a)g(new)h(pro)36 b(of)477 b(for)800 29090 y(the)433 b(w)-36 b(ell-kno)g(wn)434 b(iden)-36 b(tities:)17734 32893 y SDict begin H.S end 17734 32893 a 17734 32893 a SDict begin 14.5 H.A end 17734 32893 a 17734 32893 a SDict begin [ /View [/XYZ H.V] /Dest (equation.4) cvn H.B /DEST pdfmark end 17734 32893 a 19948 31232 a Fp(n)p Fo(\000)p Fn(1)19986 31631 y Fd(Y)19976 34460 y Fp(k)24 b Fn(=1)21942 32893 y Fg(sin)23894 31994 y Ff(k)45 b(\031)p 23894 32587 1507 54 v 23934 33804 a Fg(2)p Ff(n)25902 32893 y Fg(=)27713 31037 y Fe(p)p 28820 31037 777 54 v 957 x Ff(n)p 27415 32587 2479 54 v 27415 33804 a Fg(2)28065 33420 y Fp(n)p Fo(\000)p Fn(1)30027 32893 y Ff(;)1698 b(n)369 b Fe(\025)g Fg(2)p Ff(;)p Black 15494 w Fg(\(4\))p Black 19948 35739 a Fp(n)p Fo(\000)p Fn(1)19986 36137 y Fd(Y)19976 38966 y Fp(k)24 b Fn(=1)21942 37399 y Fg(sin)23894 36501 y Ff(k)45 b(\031)p 23894 37094 1507 54 v 24259 38311 a(n)25902 37399 y Fg(=)28267 36501 y Ff(n)p 27415 37094 2479 54 v 27415 38311 a Fg(2)28065 37927 y Fp(n)p Fo(\000)p Fn(1)30027 37399 y Ff(;)1698 b(n)369 b Fe(\025)g Fg(2)p Ff(:)p Black 15494 w Fg(\(5\))p Black 800 41101 a SDict begin H.S end 800 41101 a 800 41101 a SDict begin 14.5 H.A end 800 41101 a 800 41101 a SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn H.B /DEST pdfmark end 800 41101 a 1439 x Fh(2)2152 b(T)-179 b(ec)-60 b(hniques)715 b(and)h(Pro)60 b(ofs)800 45461 y Fg(F)-108 b(or)336 b(a)i(graph)e Ff(G)p Fg(,)356 b(a)338 b Fc(sp)-66 b(anning)374 b(tr)-66 b(e)g(e)335 b Fg(in)i Ff(G)g Fg(is)g(a)g(tree)g (whic)-36 b(h)336 b(has)h(the)f(same)h(v)-36 b(ertex)338 b(set)f(as)g Ff(G)p Fg(.)546 b(The)337 b(n)-36 b(um)g(b)36 b(er)800 47066 y(of)423 b(spanning)f(trees)g(in)g(a)h(graph)f(\(net)-36 b(w)g(ork\))422 b(G,)g(denoted)f(b)-36 b(y)423 b Ff(t)p Fg(\()p Ff(G)p Fg(\))p Ff(;)f Fg(is)g(an)g(imp)36 b(ortan)-36 b(t)422 b(in)-36 b(v)-72 b(arian)-36 b(t)423 b(of)g(the)800 48671 y(graph)459 b(\(net)-36 b(w)g(ork\).)655 b(It)459 b(is)h(also)g(an)f(imp)36 b(ortan)-36 b(t)459 b(measure)g(of)h (reliabilit)-36 b(y)460 b(of)g(a)g(net)-36 b(w)g(ork.)656 b(In)459 b(the)f(sequel,)800 50276 y(w)-36 b(e)434 b(assume)f(our)h (graphs)f(are)h(lo)36 b(opless)434 b(but)f(m)-36 b(ultiple)433 b(edges)h(are)g(allo)-36 b(w)g(ed.)2751 51881 y(A)435 b(famous)g(and)f(classic)i(result)e(on)h(the)f(study)g(of)h Ff(t)p Fg(\()p Ff(G)p Fg(\))f(is)h(the)f(follo)-36 b(wing)437 b(theorem,)e(kno)-36 b(wn)435 b(as)g(the)800 53486 y Fc(Matrix-tr)-66 b(e)g(e)384 b(The)-66 b(or)g(em)p Fg(.)548 b(The)348 b Fc(L)-66 b(aplacian)385 b(matrix)347 b Fg(of)i(a)f(graph)g Ff(G)g Fg(is)g(de\014ned)f(as)h Ff(L)p Fg(\()p Ff(G)p Fg(\))369 b(=)f Ff(D)36 b Fg(\()p Ff(G)p Fg(\))120 b Fe(\000)g Ff(A)p Fg(\()p Ff(G)p Fg(\),)800 55091 y(where)489 b Ff(D)36 b Fg(\()p Ff(G)p Fg(\))489 b(and)g Ff(A)p Fg(\()p Ff(G)p Fg(\))g(are)h(the)e(degree)i(matrix)g(and)e(the)h(adjacency)i (matrix)e(of)i Ff(G)p Fg(,)503 b(resp)36 b(ectiv)-36 b(ely)-108 b(.)800 56696 y(Since)335 b(this)f(theorem)h(is)g(\014rst)f (pro)-36 b(v)g(ed)335 b(b)-36 b(y)334 b(Kirc)-36 b(hho\013)335 b([)p 0 .5 0 TeXcolorrgb 27093 56696 a SDict begin H.S end 27093 56696 a 0 .5 0 TeXcolorrgb Fg(7)p 0 .5 0 TeXcolorrgb 27743 55840 a SDict begin H.R end 27743 55840 a 27743 56696 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.hoff) cvn H.B /ANN pdfmark end 27743 56696 a Black Fg(],)356 b Ff(L)p Fg(\()p Ff(G)p Fg(\))334 b(is)h(also)h(kno)-36 b(wn)335 b(as)h(the)e Fc(Kir)-66 b(chho\013)373 b(matrix)800 58301 y Fg(of)434 b(the)f(graph)h Ff(G)p Fg(.)800 58560 y SDict begin H.S end 800 58560 a 800 58560 a SDict begin 14.5 H.A end 800 58560 a 800 58560 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.1) cvn H.B /DEST pdfmark end 800 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14958 y Fg(No)-36 b(w,)435 b(w)-36 b(e)433 b(state)h(some)g(more)g(de\014nitions)e(and)h(theorems.)800 15217 y SDict begin H.S end 800 15217 a 800 15217 a SDict begin 14.5 H.A end 800 15217 a 800 15217 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3) cvn H.B /DEST pdfmark end 800 15217 a Black 2319 x Fb(De\014nition)476 b(3.)p Black 2252 w Fg(An)413 b Ff(n)254 b Fe(\002)g Ff(n)413 b Fg(matrix)h Ff(C)464 b Fg(=)368 b(\()p Ff(c)24830 17735 y Fp(ij)25637 17536 y Fg(\))413 b(is)h(called)g(a)f Fc(cir)-66 b(culant)445 b(matrix)412 b Fg(if)i(its)f(en)-36 b(tries)413 b(satisfy)800 19141 y Ff(c)1360 19340 y Fp(ij)2536 19141 y Fg(=)369 b Ff(c)4477 19340 y Fn(1)p Fp(;)262 b(j)51 b Fo(\000)p Fp(i)p Fn(+1)8211 19141 y Fg(,)434 b(where)f(subscripts)g (are)h(reduced)e(mo)36 b(dulo)434 b Ff(n)g Fg(and)f(lie)h(in)f(the)g (set)h Fe(f)p Fg(1)p Ff(;)221 b Fg(2)p Ff(;)g(:)g(:)g(:)k(;)c(n)p Fe(g)p Fg(.)800 20485 y SDict begin H.S end 800 20485 a 800 20485 a SDict begin 14.5 H.A end 800 20485 a 800 20485 a 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Fg(=)406 b Fe(f)p Fg(0)p Ff(;)221 b Fg(1)p Ff(;)g(:)g(:)g(:)k(;)c(n)311 b Fe(\000)g Fg(1)p Fe(g)456 b Fg(and)f(edge)g(set)h Ff(E)484 b Fg(=)406 b Fe(ff)p Ff(i;)221 b(i)310 b Fg(+)g Ff(s)30881 25128 y Fp(j)32180 24929 y Fg(\()886 b(mo)36 b(d)442 b Ff(n)p Fg(\))p Fe(g)407 b(j)f Ff(i)g Fg(=)g(0)p Ff(;)221 b Fg(1)p Ff(;)g(:)g(:)g(:)k(;)c(n)312 b Fe(\000)e Fg(1)p Ff(;)628 b(j)481 b Fg(=)800 26534 y(1)p Ff(;)221 b Fg(2)p Ff(;)g(:)g(:)g(:)k(;)c(k)45 b Fe(g)p Fg(.)2751 29112 y(The)379 b(Laplacian)g(matrix)h(of)f Ff(C)17788 29311 y Fp(n)18414 29112 y Fg(\()p Ff(s)19533 29311 y Fn(1)20058 29112 y Ff(;)221 b(s)21253 29311 y Fn(2)21779 29112 y Ff(;)g(:)g(:)g(:)j(;)d(s)25305 29311 y Fp(k)25874 29112 y Fg(\))379 b(is)g(clearly)h(a)f(circulan)-36 b(t)379 b(matrix.)560 b(By)380 b(a)f(direct)f(using)800 30717 y(of)434 b(Theorem)g(4.8)h(of)f([)p 0 .5 0 TeXcolorrgb 11711 30717 a SDict begin H.S end 11711 30717 a 0 .5 0 TeXcolorrgb Fg(12)p 0 .5 0 TeXcolorrgb 13012 29861 a SDict begin H.R end 13012 29861 a 13012 30717 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.zhang) cvn H.B /ANN pdfmark end 13012 30717 a Black Fg(],)g(w)-36 b(e)434 b(obtain)f(the)g (follo)-36 b(wing)436 b(lemma:)800 31049 y SDict begin H.S end 800 31049 a 800 31049 a SDict begin 14.5 H.A end 800 31049 a 800 31049 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5) cvn H.B /DEST pdfmark end 800 31049 a Black 2246 x Fb(Lemma)499 b(5.)p Black 554 w Fc(The)465 b(nonzer)-66 b(o)464 b(eigenvalues)g(of)g Ff(L)p Fg(\()p Ff(C)25990 33494 y Fp(n)26616 33295 y Fg(\()p Ff(s)27735 33494 y Fn(1)28260 33295 y Ff(;)221 b(s)29455 33494 y Fn(2)29981 33295 y Ff(;)g(:)g(:)g(:)j(;)d(s)33507 33494 y Fp(k)34076 33295 y Fg(\)\))465 b Fc(ar)-66 b(e)9665 36072 y Fg(2)p Ff(k)340 b Fe(\000)296 b Ff(!)13518 35523 y Fp(s)13953 35646 y Fi(1)14414 35523 y Fp(j)15196 36072 y Fe(\000)f(\001)221 b(\001)g(\001)296 b(\000)f Ff(!)20555 35523 y Fp(s)20990 35679 y Fk(k)21498 35523 y Fp(j)22280 36072 y Fe(\000)h Ff(!)24467 35523 y Fo(\000)p Fp(s)25634 35646 y Fi(1)26094 35523 y Fp(j)26876 36072 y Fe(\000)g(\001)221 b(\001)g(\001)296 b(\000)f Ff(!)32236 35523 y Fo(\000)p Fp(s)33403 35679 y Fk(k)33911 35523 y Fp(j)34398 36072 y Ff(;)959 b Fg(1)370 b Fe(\024)f Ff(j)444 b Fe(\024)369 b Ff(n)295 b Fe(\000)h Fg(1)p Ff(;)800 38848 y Fc(wher)-66 b(e)465 b Ff(!)416 b Fg(=)369 b Ff(e)7813 37996 y Fi(2)p Fk(\031)24 b(i)p 7812 38157 1209 40 v 8161 38705 a(n)9209 38848 y Fc(.)2751 41426 y Fg(With)434 b(com)-36 b(bining)433 b(Theorem)p 0 .5 0 TeXcolorrgb 17853 41426 a SDict begin H.S end 17853 41426 a 0 .5 0 TeXcolorrgb Fg(2)p 0 .5 0 TeXcolorrgb 18503 40570 a SDict begin H.R end 18503 40570 a 18503 41426 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.2) cvn H.B /ANN pdfmark end 18503 41426 a Black 434 w Fg(and)g(the)g(lemma)h (ab)36 b(o)-36 b(v)g(e,)434 b(w)-36 b(e)434 b(obtain)g(the)f(follo)-36 b(wing)435 b(corollary:)800 41684 y SDict begin H.S end 800 41684 a 800 41684 a SDict begin 14.5 H.A end 800 41684 a 800 41684 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6) cvn H.B /DEST pdfmark end 800 41684 a Black 2320 x Fb(Corollary)521 b(6.)p Black 554 w Fc(The)464 b(numb)-66 b(er)464 b(of)h(sp)-66 b(anning)463 b(tr)-66 b(e)g(es)463 b(in)h Ff(G)369 b Fg(=)g Ff(C)31777 44203 y Fp(n)32403 44004 y Fg(\()p Ff(s)33522 44203 y Fn(1)34047 44004 y Ff(;)221 b(s)35242 44203 y Fn(2)35768 44004 y Ff(;)g(:)g(:)g(:)j(;)d(s)39294 44203 y Fp(k)39863 44004 y Fg(\))465 b Fc(is)g(e)-66 b(qual)464 b(to:)16509 48027 y Ff(t)p Fg(\()p Ff(G)p Fg(\))368 b(=)20961 47128 y(1)p 20898 47721 777 54 v 20898 48938 a Ff(n)22029 46366 y Fp(n)p Fo(\000)p Fn(1)22067 46765 y Fd(Y)22099 49564 y Fp(j)51 b Fn(=1)23802 46951 y Fd(\000)25483 46366 y Fp(k)24780 46765 y Fd(X)24978 49564 y Fp(i)p Fn(=1)26699 48027 y Fg(\(2)295 b Fe(\000)g Fg(2)221 b(cos)32446 47128 y(2)p Ff(j)75 b(s)34322 47327 y Fp(i)34697 47128 y Ff(\031)p 32446 47721 3038 54 v 33576 48938 a(n)35616 48027 y Fg(\))36122 46951 y Fd(\001)36730 48027 y Ff(:)800 52148 y Fc(Pr)-66 b(o)g(of)481 b(of)h(identity)e(\()p 0 .5 0 TeXcolorrgb 11147 52148 a SDict begin H.S end 11147 52148 a 0 .5 0 TeXcolorrgb Fc(1)p 0 .5 0 TeXcolorrgb 11811 51287 a SDict begin H.R end 11811 51287 a 11811 52148 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 11811 52148 a Black Fc(\).)634 b Fg(Consider)452 b(the)g Fc(squar)-66 b(e)482 b(cycle)451 b Ff(C)29270 52347 y Fp(n)29896 52148 y Fg(\(1)p Ff(;)221 b Fg(2\).)635 b(W)-108 b(e)452 b(can)g(use)g(Corollary)p 0 .5 0 TeXcolorrgb 46476 52148 a SDict begin H.S end 46476 52148 a 0 .5 0 TeXcolorrgb Fg(6)p 0 .5 0 TeXcolorrgb 47127 51292 a SDict begin H.R end 47127 51292 a 47127 52148 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 47127 52148 a Black 452 w Fg(to)g(obtain)800 53753 y(the)314 b(n)-36 b(um)g(b)36 b(er)313 b(of)i(spanning)f(trees)g (of)h Ff(C)19565 53952 y Fp(n)20191 53753 y Fg(\(1)p Ff(;)221 b Fg(2\).)540 b(On)314 b(the)f(other)i(hand,)337 b(Kleitman)315 b(and)f(Golden)g([)p 0 .5 0 TeXcolorrgb 47680 53753 a SDict begin H.S end 47680 53753 a 0 .5 0 TeXcolorrgb Fg(8)p 0 .5 0 TeXcolorrgb 48331 52897 a SDict begin H.R end 48331 52897 a 48331 53753 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.kleit) cvn H.B /ANN pdfmark end 48331 53753 a Black Fg(])g(pro)-36 b(v)g(ed)800 55358 y(that)518 b Ff(t)p Fg(\()p Ff(C)5610 55557 y Fp(n)6235 55358 y Fg(\(1)p Ff(;)221 b Fg(2\)\))514 b(=)f Ff(nF)13473 54876 y Fn(2)13292 55686 y Fp(n)13999 55358 y Fg(.)832 b(No)-36 b(w,)541 b(with)518 b(a)g(little)h(additional)g(algebraic)g (manipulation,)540 b(iden)-36 b(tit)g(y)518 b(\()p 0 .5 0 TeXcolorrgb 51644 55358 a SDict begin H.S end 51644 55358 a 0 .5 0 TeXcolorrgb Fg(1)p 0 .5 0 TeXcolorrgb 52294 54502 a SDict begin H.R end 52294 54502 a 52294 55358 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 52294 55358 a Black Fg(\))800 56963 y(follo)-36 b(ws.)46811 b Fa(2)800 59807 y Fc(Pr)-66 b(o)g(of)468 b(of)h(identity)e(\()p 0 .5 0 TeXcolorrgb 11108 59807 a SDict begin H.S end 11108 59807 a 0 .5 0 TeXcolorrgb Fc(5)p 0 .5 0 TeXcolorrgb 11773 58946 a SDict begin H.R end 11773 58946 a 11773 59807 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4) cvn H.B /ANN pdfmark end 11773 59807 a Black Fc(\).)591 b Fg(Lo)36 b(ok)439 b(at)f(the)g(cycle)h Ff(C)24558 60006 y Fp(n)25184 59807 y Fg(\(1\))376 b(=)g Ff(C)29541 60006 y Fp(n)30167 59807 y Fg(.)592 b(W)-108 b(e)438 b(kno)-36 b(w)439 b(that)e Ff(t)p Fg(\()p Ff(C)41494 60006 y Fp(n)42120 59807 y Fg(\))376 b(=)g Ff(n)p Fg(,)440 b(therefore)e(b)-36 b(y)800 61412 y(applying)434 b(Corollary)p 0 .5 0 TeXcolorrgb 11856 61412 a SDict begin H.S end 11856 61412 a 0 .5 0 TeXcolorrgb Fg(6)p 0 .5 0 TeXcolorrgb 12506 60556 a SDict begin H.R end 12506 60556 a 12506 61412 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 12506 61412 a Black 434 w Fg(to)f(it,)h(\()p 0 .5 0 TeXcolorrgb 16697 61412 a SDict begin H.S end 16697 61412 a 0 .5 0 TeXcolorrgb Fg(5)p 0 .5 0 TeXcolorrgb 17347 60556 a SDict begin H.R end 17347 60556 a 17347 61412 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.4) cvn H.B /ANN pdfmark end 17347 61412 a Black Fg(\))g(follo)-36 b(ws.)29324 b Fa(2)800 61744 y SDict begin H.S end 800 61744 a 800 61744 a SDict begin 14.5 H.A end 800 61744 a 800 61744 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.7) cvn H.B /DEST pdfmark end 800 61744 a Black 2246 x Fb(De\014nition)483 b(7.)p Black 2262 w Fg(The)419 b(join)g Ff(W)17619 64189 y Fp(n)18615 63990 y Fg(=)368 b Ff(C)20926 64189 y Fp(n)21773 62993 y Fd(W)23102 63990 y Ff(K)24209 64189 y Fn(1)25153 63990 y Fg(of)420 b(a)f(cycle)h Ff(C)31821 64189 y Fp(n)32865 63990 y Fg(and)f(a)g(single)g(v)-36 b(ertex)420 b(is)f(referred)g(to)g (as)800 65595 y(a)448 b Fc(whe)-66 b(el)448 b Fg(with)f Ff(n)i Fc(sp)-66 b(okes)p Fg(.)621 b(Similarly)-108 b(,)452 b(the)447 b(join)i Fe(F)25625 65794 y Fp(n)26644 65595 y Fg(=)392 b Ff(P)28886 65794 y Fp(n)29734 64599 y Fd(W)31063 65595 y Ff(K)32170 65794 y Fn(1)33143 65595 y Fg(of)448 b(a)g(path)f Ff(P)39623 65794 y Fp(n)40698 65595 y Fg(and)g(a)h(single) g(v)-36 b(ertex)449 b(is)800 67200 y(called)434 b(a)g Fc(fan)p Fg(.)p Black 26475 70521 a(3)p Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest 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end 26007 39285 a 26007 40141 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 26007 40141 a Black Fg(\))h(and)f(\()p 0 .5 0 TeXcolorrgb 29982 40141 a SDict begin H.S end 29982 40141 a 0 .5 0 TeXcolorrgb Fg(3)p 0 .5 0 TeXcolorrgb 30632 39285 a SDict begin H.R end 30632 39285 a 30632 40141 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3) cvn H.B /ANN pdfmark end 30632 40141 a Black Fg(\).)20308 b Fa(2)800 44578 y Fh(References)800 45893 y SDict begin H.S end 800 45893 a 800 45893 a SDict begin 14.5 H.A end 800 45893 a 800 45893 a SDict begin [ /View [/XYZ H.V] /Dest (section*.1) cvn H.B /DEST pdfmark end 800 45893 a 800 45340 a SDict begin H.S end 800 45340 a 800 45340 a SDict begin 14.5 H.A end 800 45340 a 800 45340 a SDict begin [ /View [/XYZ H.V] /Dest (cite.ander) cvn H.B /DEST pdfmark end 800 45340 a Black 1450 47498 a Fg([1])p Black 652 w(W.)564 b(N.)h(Anderson)e(and)g(T.)i 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