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阿贝尔群上2度有向Cayley图的研究 33页

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  • 2022-06-16 12:02:10 发布

阿贝尔群上2度有向Cayley图的研究

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漳州师范学院硕士学位论文阿贝尔群上2度有向Cayley图的研究姓名：陈宇申请学位级别：硕士专业：基础数学指导教师：陈宝兴20080501 ii[{Ba#f#U#r{tx=-Gq2y^-ueqGEsSG7"!~~r2e/∈S,!|S|=2.W-G:2WDCayleyhΓ=Cay(G,S)T8qV(Γ)=G,E(Γ)={(x,x+g)|x∈G,g∈S}.u~/*^-:2WDCayleyhsq5~<zT2yedRN(N≥5),DNyR!md^-:2WDCayleyhuj[;Æuz,α=(x1,y1),β=(x2,y2)),uz,α=(x1,y1),β=(x2,y2)),analgorithmisgiventocomputethefourparametersl,h,m,nofitsL−shapedtileinthispaper.Thatis,forgivenintegersx1,y1,x2,y2,thefourparametersofL−shapedtilecanbefiguredoutthroughthealgorithm.AndthenthediameteroftheCayleydigraphcanbecalculatedbyd(Ω)=max{l+h−m−2,l+h−n−2}.Foranygivenintegerk(k≥0),itisprovedthatasigulark-tightoptimalinfinitefamilyof2-degreeCayleydigraphsofabeliangroupscanbeconstructed.KeyWords:Cayleydigraph;Algorithm;L-shapetile;Singulark-tightop-timal. 漳州师范学院学位论文原创性声明本人郑重声明：所呈交的论文是本人在导师的指导下独立进行研究所取得的研究成果。除了文中特别加以标注引用的内容外，本论文不包含任何其他个人或集体已经发表或撰写的成果作品。对本文的研究做出重要贡献的个人和集体，均已在文中以明确方式标明。本人完全意识到本声明的法律后果由本人承担。作者签名：日期：年月日学位论文版权使用授权书本学位论文作者完全了解学校有关保留、使用学位论文的规定，同意学校保留并向国家有关部门或机构送交论文的复印件和电子版，允许论文被查阅和借阅。本人授权漳州师范学院可以将本学位论文的全部或部分内容编入有关数据库进行检索，可以采用影印、缩印或扫描等复制手段保存和汇编本学位论文。本学位论文属于1、保密□，在______年解密后适用本授权书。2、不保密□。（请在以上相应方框内打“√”）作者签名：日期：年月日导师签名：日期：年月日 ℄,92VCCayleyg)11uZ42"b785#*H1.1=V?/Fs9{}V`h=R$*V`A.CayleyU1895aw>gWÆgi.xWÆ"Z5{E(/~3Gm}3_Cayleyh~3m}DmdY6Z^YjXAR^-:2WDCayleyh+ED5:ARD<~/E"R9;`R9E(>iuwh-2ym}jD2yy/&w/&&9Xjm}u~/*^-:2WDCayleyhj6ss1.2s9d/"bSm}2d`^-:2WDCayleyhqm^s[$`*Y^-:2WDCayleyh*Y9;BGuY^-:2WDCayleyh*USm}=:iY5~?GGuY*!BRxRyD"mG!BGy−x∈M.w7MG2y7"5a.Dh!.md$Uu[1]u4GYNyR^-:2WDCayleyhjÆq*Nym≥0,h>n≥0l>n,h≥ma&[d(N).FE(:Wa}Ha&[d(N)9uzmh>nWygndR~aXy:6sEsdxiwL-j"U(0h2.1):0≤xm≥0,h>n≥0,l>n,h≥m.D2,TG5:;/2~DL(N;l,h,m,n)2yDCayleyhD(G,{a,b})EuG=(Z×Z)/K,!K=h(l,−n),(m,−h)i,a=(1,0)+K,b=(0,1)+K.;qGL(N;l,h,m,n)(m>n)ZEwÆL(N;l,h,n,m)0ZE;w+e=m≤n.Bu[12]uTg5,wt+x0+y0−i−k+z≥0,2t+y0≥t+x0+y0−i−k+z≥0.(x+y−i−k)(x+y−i−k+z)−xy+(A+z−2i−2k)t+B=0(1)C!/4k-AL-jL(N;l,h,m,n)ZEBt+x0+y0−i−k+z≥0,2t+y0≥t+x0+y0−i−k+z≥0.;qN=lh−mn,36,f(x0,y0)d8(1)2y=;i4(x0,y0)d8(1)2y=!2t+x0>t+x0+y0−i−k+z≥0,2t+y0≥t+x0+y0−i−k+z≥0,WZvl=2t+x0,h=2t+y0,m=t+x0+y0−i−k,n=t+x0+y0−i−k+z.;qN=lh−mn,Be&!,3$(1)*Sbg:Step1:H:=4×{(i+k)2−(i+k)z+z2+3[(2i+2k−A−z)t−B]};Step2:ifH≥0then√s:=⌊H⌋;dowhiles≥0pq:=⌊(H−s2)/3⌋; ℄,92VCCayleyg)7ifH==s2+3q2thenFindoutintegerpairsinthefourpairsofTheorem2.1.6;endifs:=s−1;enddoendif√7[`1R[D.mSWBHkRT!Hk~/B√12∗(2i+2k−A−z)tRT;[`1R[D.mSWqO(kt),0√O(kN0.25).GBTg2.1.6Z5~Tg2.1.7.!2.1.7:zTdRt,A,B,i,k,z,[`1Æ^<+Td8(1)Dd√kN0.25).R=!^R[D.mSWqO(Y17zTedRN(N≥5),~[`2Z5^2yk-AA^-:2WDCayleyh02:._6+)hfN(N≥5),YCWA=d^5k-;`)/H2-aCayleyTStep1:GivenN,calculatet,A,B,isuchthatN=3t2+At+B∈Ii(t).Letk:=0;Step2:Dowhile.t.z:=0;Dowhilez<=2k+6/*._6+)ft,A,B,i,k,z,01E8d%+2#(1))Raf9*/.+2#(1))^5f9(x0,y0),l:=2t+x0;h:=2t+y0;m:=t+x0+y0−i−k;n:=t+x0+y0−i−k+z;if(h≥m≥0,l>n≥0,h>0)L(N;l,h,m,n)<[)gotoStep3;endifz:=z+1;enddok:=k+1;enddoStep3:WA4"d"?^5<[)L-℄UL(N;l0,h0,m0,n0),K= 8[{Ba#f#U#r{th(l0,−n0),(m0,−h0)i,a=(1,0)+K,b=(0,1)+K.WAi(?aCayleyTD((Z×Z)/K,{a,b})^5k-;`)/H2-aCayleyT!2.1.8:Y17zTedRN(N≥5),[`2Æ^<2yk-AA2.5N0.25).^-:2WDCayleyh!^R[D.mSWqO(kC[`2e+BqO(N),G[`2R[D.mSWqO(N).2~wz2yEkaT[`28vN0=2814.N0=3t+4t−6,ut=30.;DA=4,B=−6,i=2.lkk=0.22#1:vz=0.B[`1DH=4×{(i+k)−(i+k)z+z+3[(2i+2k−A−z)t−B]}=88.B[`1ZhA=4,B=−6,i=2,k=0,t=30,z=0,aD+Td8(1)DdR=#2:G1≤z≤6,DH<0.aD+Td8(1)0DdR=:2814+0-AA5lkk=1.22#1:vz=0.B[`1DH=4×{(i+k)−(i+k)z+z+3[(2i+2k−A−z)t−B]}=4×207.B[`1ZhGA=4,B=−6,i=2,k=1,t=30,z=0,aD+Td8(1)DdR=22#2:vz=1.B[`1DH=4×{(i+k)−(i+k)z+z+3[(2i+2k−A−z)t−B]}=4×115.B[`1ZhGA=4,B=−6,i=2,k=1,t=30,z=1,aD+Td8(1)0DdR=22#3:vz=2.B[`1DH=4×{(i+k)−(i+k)z+z+3[(2i+2k−A−z)t−B]}=100.20BG100=10+3×0,0s=10,q=0.;GA=4,B=−6,i=2,k=1,t=30,z=2,(3,3)d8(1)2ydR=B[`2,vl=63,h=63,m=335#n=35.BG63>3533>0,ZhL-jL(2814;63,63,33,35)ZEvK=h(63,−35),(33,−63)i,a=(1,0)+K, ℄,92VCCayleyg)9b=(0,1)+K.BDCayleyD((Z×Z)/K,{a,b})2y1-AA^-:2WDCayleyh 10[{Ba#f#U#r{t2.2%2#kCayleyXLoZT*R2.2.1t^-:2WDCayleyhk$`%*h|4~=-Gq2y^-ueqGEsSG7"!~~r2e/∈S,|S|=2.W-G:2WDV`hΓ=Cay(G,S)T8qV(Γ)=G,E(Γ)={(x,x+g)|x∈G,g∈S}.2w(S={a,b},G=Z/K,a=(1,0)+K,b=(0,1)+K,uK=<α,β>,α=(x1,y1),β=(x2,y2);x1,y1,x2,y2qdR!α,βqxFpW-G:2WDV`hΩ=Cay(G,{a,b})T8qV(Ω)=G,E(Ω)={(x,x+g)|x∈G,g=ag=b}.^-:2WDCayleyhDeWRY6md6^Y`;>Gm}}f=&iuwhm}jqm}u17oR.eDk*^-:2WDCayleyhj3|Lju[8]Æ`fqYG12ySm}(DoubleLoopNetwork),CUp2LjL=L(N;l,h,m,n),u0≤mSm}0!x1,y1,x2,y2∈Z.22!−x1y1−!|2.2.21xy6q26x2,y1≥0.22!−x1y1−!|2.2.31xy6q36x1,y2~2x1>22−y1≥0,y2≥−x2≥0,x1>−x2,!!y2>−y1!dRx1y1−x1−y1!|2.2.4$xy−→−x−y6q2yF$2222!!x1y1−x2−y2!|2.2.5$xy−→xy6q2yX$2211!!x1y1x2y2!|2.2.6$xy−→−x−y6q2yY$2211!!x1y1x1+kx2y1+ky2!|2.2.7$xy−→xy6qSk$2222(k∈Z)!!x1y1x1y1!|2.2.8$xy−→x+kxy+ky6qTk$222121(k∈Z)!|2.2.9F,X,Y,Sk,Tk$f6qδ$2vα=(x1,y1),β=(x2,y2),WBpα,βUZ:7+`-<α,β>=<−α,−β>==<β,−α>=<α±β,β>=<α,β±α>.x1y1BZ0xyGDAδ$^rHk+$*!+-22<α,β>E(:Dq$$hΩq$qq!d#w"K=<α,β>,uα=(x1,y1),β=(x2,y2)."x1y1A=.x2y22.2.3w3B2/K,a=(1,0)+K,b=(0,1)+K)2.2.1hΩ(uG=ZSRRqN,aiN=|detA|. 12[{Ba#f#U#r{tyDr6Br(x2,y2)rC(x1,y1)r-0xh2.4PCh2.4ayW"o7"w6q"(+CDPBD2").Z:uBx[ya."7W"WySRPY>GR(0,0).,4y"uDoyxRPQUY>GR(x,y),WU℄DW"u0De.#4R(x,y)U:<rz/2DGy℄RaCayleyhDSRUZ:<rz2.PZ;V`hCay(G,{a,b})SRRq"uxRRN.B0.BG6AohΩ+1q$5+ex1y1=>0.~uaSWÆrHDG0,!WyYUqdRx2y2`T!x1y1−2.2.2172×2xyUZ5G1Aδ$7122CZ5i5~7x#℄_|>0,y>0.−n16x12aD4G1<0,y<0.−n26x12aDYÆ2AF$$Z71n36x1y2≤0.(i)−Gx2<0,y1>0DYÆ2AX$$Z7q1(ii)−Gx2>0,y1<0DYÆ2AY$$Z7q1:<|7lB0)/*#3(i)*/x1←→x2;y1←→y2;x1=−x1;y1=−y1;case(x2>0andy1<0)/*#3(ii)*/x1←→x2;y1←→y2;x2=−x2;y2=−y2;endcaseprintf(x1,y1,x2,y2)!−x1y1−2.2.3171xyUÆGDAδ$7222−C1Z5iq5~7x#℄!x1y1−n16x2≥0!y1≥0.aDxy4G222n26x2<0,y1<0.i7x#℄x2(i)6x1+x2>0,!y1+y2>0.aDYÆS1$TÆTt,t=⌈−x1⌉$$Z7#3x1(ii)6x1+x2!>0,!y1+y2≤0.vt=⌈−x2⌉−1,aDYÆSt$′′x1y17.BGx1+x2>0,+0,y1>y1.4y1≥0,q#3;4y1<0,BGx1+x2≤0,;′y1+y2>0.Dq~#(iii).y2(iii)6x1+x2!≤0,!y1+y2>0.vt=⌈−y1⌉−1,aDYTt$x1y17′′.BGy1+y2>0,+0,x2>x2.4x2≥0,q#3;4x2<0,BGy1+y2≤0,;′x1+x2>0.Dq:#(ii).#(ii)uÆlSt(t≥1)$y1srYRkX+#(iii)uÆlTt(t≥1)$x2srYRkX+BGy1,x2DnR 14[{Ba#f#U#r{t5^u2yw$7gnRDq~#3.n36x2y1u2yqn2ygniox#℄y1⌉,−(i)Gx2≥0,y1<0Dvt=⌈−y2ÆSt$$Z72(ii)x2−Gx2<0,y1≥0Dvt=⌈−x1⌉,ÆTt$$Z72:<|7lB0andy1+y2>0)x1=x1+x2;y1=y1+y2;case(x1+x2>0andy1+y2≤0)x1t=⌈−x2⌉−1;x1=x1+tx2;y1=y1+ty2;case(x1+x2≤0andy1+y2>0)y2t=⌈−y1⌉−1;x2=x2+tx1;y2=y2+ty1;endcaseenddodocase/*#3*/case(x2≥0)y1t=⌈−y2⌉;x1=x1+tx2;y1=y1+ty2;case(y1≥0)x2t=⌈−x1⌉;x2=x2+tx1;y2=y2+ty1;endcaseprintf(x1,y1,x2,y2)−−2.2.4172UZ5GDAStTt$73 ℄,92VCCayleyg)15−−C&3$/Y2ÆStTt$5Gx1,y2q~2x1>−y1≥0,y2≥−x2≥0,x1>−x2,!y2>−y1dRStep1:Z5dDAStTt$Gx1>x2≥0,y2>y1≥0.−6Dx2=y1=0,WD3Step2:−Gx2,y1+)q0DB3T8/x1,y2q~2edR#1:6x1+y1>x2+y2,−(i)/G7q3x1!~x1>−y1≥0.;Z5U>0′=x−tx,fx1x1>−y1a~YÆS−t$(t≥1),$x112y′=y−ty.′112/Gx1Ep$t/;iD0t(~+Hx1>0′′>0x1+y1,x1+y1DdR=t=⌈x2+y2⌉−1.BGt≥1,5,}Z′<0.′,y=y′.!y1qqd#"x1=x111(ii)−/G7q3y2!~y2>0,y2≥−x2≥0.;Z5Ufy2>0y2≥−x2≥0a~YÆT−t$(t≥1),$′′=y−ty.x2=x2−tx1,y221BGy1<0,5/Gy2Ept/;i0′′′≤0t(~+Hx2+y2≥0,x2dR=x2⌉,x2x2+y2−−(a)t=⌈x1G⌈x1⌉6=⌈x1+y1⌉DB3T8aD3x2⌉−1,x2x2+y2′′(b)t=⌈x1G⌈x1⌉=⌈x1+y1⌉DaDx2>0,y2>0..Step2,ÆGx1)$−BGx1DedR5!.#1(a),7q3#2:6x1+y1≤x2+y2,−(i)/G7q3y2!~y2>0,y2≥−x2≥0.;Z5′Ufy2>0y2≥−x2a~YÆT−t$(t≥1),$x2=x2−tx1,′′y2=y2−ty1./Gy2Ep$t/;iD0t(~+Hy2>0,x′′x2+y2⌋.2+y2≥0DdR=t=⌊x1+y1BGt≥1,5,}Z!x′<0.=x′′2qqd#"x22,y2=y2.−(ii)/G7q3x1!~x1>−y1≥0.;Z5U′fx1>0x1>−y1a~YÆS−t$(t≥1),$x1=x1−tx2, 16[{Ba#f#U#r{ty′=y−ty.112BGx2<0,5/Gx1Ept/;i0t(~′′′+Hx1+y1>0,y1≤0dR=(a)t=⌈y1⌉,y1x1+y1−−y2G⌈y2⌉6=⌈x2+y2⌉DB3T8aD3(b)t=⌈y1⌉−1,y1x1+y1′′y2G⌈y2⌉=⌈x2+y2⌉DaDx1>0,y1>0..Step2,ÆGy2)$−BGy2DedR5!.#2(a),7q3−−:172UZ5GDAStTt$73Bx2+y2)/*#1*/x1+y1t=⌈x2+y2⌉−1;x1=x1−tx2;y1=y1−ty2;/*(i)*/x2t=⌈x1⌉;x2=x2−tx1;y2=y2−ty1;/*(ii)*/if(x2+y2<0)/*(iii)*/x2=x2+x1;y2=y2+y1;gotoStep2;endif ℄,92VCCayleyg)17case(x1+y1≤x2+y2)/*#2*/x2+y2t=⌊x1+y1⌋;x2=x2−tx1;y2=y2−ty1;/*(i)*/y1t=⌈y2⌉;x1=x1−tx2;y1=y1−ty2;/*(ii)*/if(x1+y1≤0)/*(iii)*/x1=x1+x2;y1=y1+y2;gotoStep2;endifendcaseEND1:printf(x1,y1,x!2,y2).x1y1−a.xy322u[1]Æ`fq^-:2WDCayleyhL-jWy/RZB~}H&[Gwj≥0,k=1,2,...N−1}n=min{j|la=jb,j≥0}h=min{j|ka=jb!j≥k≥0,j=1,2,...N−1}m=min{k|ka=hb,k≥0}−B,α=(x,y),β=(x,y).)!11224x1y1−xy3WΩLjWy/Rl,h,m,nZB5~}H22&[l=x1,n=−y1,m=−x2,h=y2.CBV`hΩT8h2.5uRA,B-GR0,GDx1a=−yb,−xa=yb.−122FB3T8Zhx1,y2~2x1>−y1≥0,y2≥−x2≥0,x1>−x2,!y2>−y1dR{,α=(−5,−8),β=(9,−1)).GA=9−1!58−B[`3,71−91;!58−;B[`4,72117 ℄,92VCCayleyg)19!9−1−B[`5,73−49.B5*+;y/<m}uoGm}}f=&uD[k$`Ym}Dq*/0u[2-21].d(n;s1,s2)a"Sm}G(n;s1,s2)jvd(n)=min{d(n;s1,s2)|1≤s1,s2diam(n),W6hΓ:kAA06n0:kAA7u[2-21]uYAA:AAx*0..-0YG17zTk≥0,k0CU:kAA^-:2WDCayleyhxÆu+u[18]u~Vfd`fq-07l2.3.2(!|*^-:2WDCayleyhj3|Lju[8]Æ`fqYG12ySm}CUp2LjL=L(n;l,h,x,y),u0≤xSm}LjuL=L(n;l,h,x,y),W6LjL=L(n;l,h,x,y)ZE!|2.3.3YGzTLjL=L(n;l,h,x,y),6CU2y^-:2WDCayleyhΓ,GY>LjuL=L(n;l,h,x,y),W6LjL=L(n;l,h,x,y)ZCESS[2]2.3.11zdRn≥4,!CUp2edRt,Gn∈I1I2I3,222222aiI1=[3t+1,3t+2t],I2=[3t+2t+1,3t+4t+1],I3=[3t+4t+2,3t+6t+3]7ydR".*!n∈Ii(t)G!BGlb(n)=3t+i−2,i=1,2,3.[2]22.3.2=i∈1,2,3,n=3t+At+B∈Ii(t),l=2t+a,h=2t+b,x=t+a+b−i−κ,y=x+z,z≥0,L=L(n;l,h,x,y)yLjWjD(L)=lb(n)+κG!BG(a+b−i−κ)(a+b−i−κ+z)−ab+(A+z−2i−2κ)t+B=0.(2)[18]2.3.36Gab+Td8(2)DdR=!2Ht,κ,z=(i+k)(i+k−z)+z+3(2κ+2i−A−z)t−3B.(3)DY;p=2p≡−1(mod6),WUY;i=upARR 22[{Ba#f#U#r{t=YRp~p=2p≡−1(mod6).6pUnYR;i=uDAW6pn;06nD;p.[16]22.3.4=n=3t+At+B∈Ii(t).64Ht,κ,zD;p,W+CUG|x−y|=zκAALjL=L(n;l,h,x,y).[18]2.3.5=a,b,t∈Z,6aYRpYWCUc∈Z,GGt≡c(modp2)2Dat+b≡p(modp).[2]2.3.6=L=L(n;l,h,x,y)2yLjvz=|x−y|,W√d(L)≥3n−0.75z2+0.5z−2.[19]2.3.7L=L(n;l,h,x,y)ZE;/2gcd(l,h,x,y)=1!h≥x,l>y.[1]2.3.8L=L(n;l,h,x,y)ZCE;/2h≥x≥0,l>y≥0,h>0.2.3.3w3B2.3.9+Td82(a+b)−4(a+b)−ab+4=0.(4)D!BD7ydR=(2,2),(2,0)(0,2).k=(a0,b0)q+Td82ydR=vp0=a0+b0,WZha0b0=p2,b220−4p0+4.4a002_Ad8X−p0X+p0−4p0+4=0=p0±m2_Ad8(H!qygndRmd=qX=2,aim2=p2−4(p2−4p+4)=−3p2+16p−16,00000$3p2−16p+m2+16=0.002Gp02_Ad8(H256−12(m+16)≥0,Z&m=0,1,2.i(m7x(k_|6h+Td8(4)D!BD7ydR=(2,2),(2,0)(0,2),f22.3.10=n=3t+4t,ait>1.6CU0AALjL=L(n;l,h,x,y),vz=|x−y|,Wz=0.k,=z≥1, ℄,92VCCayleyg)23222BG3n−0.75=9t+12t−0.75=(3t+1.5)+3t−3>(3t+1.5),√√25d(L)≥3n−0.75z+0.5z−2≥3n−0.75+0.5−2>3t+1.5−1.5=3t=3t+i−2+0(aii=2).aLjL=L(n;l,h,x,y)0AAZf2.3.11Z5~V:0AA^-:2WDCayleyhx22k=n=3t+4t,(t≡0(mod2).WYGn∈{3t+4t|t=2e,t>1,e∈Z+},,=CUSmG(n;s1,s2),GjGlb(n)+0,WBk+k+1.(1)6CUκAALjL=L(n;l,h,x,y),ai0≤κ(3t+1.5)2.√p225d(L)≥3n−0.75z+0.5z−2≥3n−0.75(2κ+1)+0.5(2κ+1)−2>3t+1.5+0.5(2κ+1)−2=3t+κ=3t+i−2+κ(aii=2).aLjL=L(n;l,h,x,y)κAAZf(2),=z≥2k+2,2222Bb73n−0.75(2k+2)=9t+9t−3k=(3t+1)+3(t−k)−1>(3t+1).√p225d(L)≥3n−0.75z+0.5z−2≥3n−0.75(2k+2)+0.5(2k+2)−2>3t+k=3t+i−2+k.aLjL=L(n;l,h,x,y)kAAZf!2.3.11zgndRk,Z5~V:kA^-:2WDCayleyhxkBk+k+1.B=k2+k+1,2WHt,κ,z=(2+κ)(i+κ−z)+z+3(2κ+2i−3−z)t−3B=(2+κ)(2+κ−z)+z2+3(2κ+1−z)t−3B.YGzTdRκ,z,ai0≤z≤2κ,0≤κ≤k.BG2At}R3(2κ+1−z)>0,5>Ht,κ,zUGt2A[2H=a2A[Hi(qajt+bj,1≤j≤d,aid=(k+1).vW+q!~2w≡2(mod3)YRw"6h~2YRDx%[yYGajt+bj(1≤j≤d)UWuCUdyoYdRp1,p2,...pd,GpjajYBk2+k+1,e∈Z+}^-:2WDCayleyhjUDGlb(n)+κ,ai0≤κ3,e∈Z})jq3t+1LjnD2yL(n;2t+1,2t+1,t−1,t+2).Lj+ZEF*ÆCEu[1],ZhB:~V,a=(1,0)+K,b=(0,1)+K,t=5500+36300e,t>3,e∈Z+}. ℄,92VCCayleyg)27:℄e[1]$CayleyDhj[J].5D$(.Y$),1994,11(4):18-24.[2]hT℄vnASm}x[J].uY$A!1993,23(9):979-992.[3]T&[mSm}A=&[J].uY$E!1999,29(3):272-278.[4]Tw2d4AASm}x[J].uY$A!2003,33(1):71-74.[5]F.Aguilo,M.A.Fiol.Anefficientalgorithmtofindoptimaldoubleloopnetworks[J].DiscreteMathematics,1995,138:15-29.[6]P.Erdos,D.F.Hsu.Distributedloopnetworkswithminimumtransmis-siondelay[J].TheoreticalComputerScience,1992,100:223-241.[7]K.Mukhopadhyaya,B.P.Sinha.Fault-tolerantroutingindistributedloopnetworks[J].IEEETrans.onComput.,1995,44(12):1452-1456.[8]G.K.Wong,D.Coppersmith.Acombinatorialproblemrelatedtomu-timodulememoryorganizations[J].J.ofAssociationforComputingMa-chinery,1974,21:392-402.[9]O.J.Rodseth.Weightedmulti-connectedloopnetworks[J].Disc.Math.,1996,148:161-173.[10]5)~VkAASm}xd`[J].uY$A!2006,36(4):438-447.[11]B.X.Chen,J.X.MengandW.J.Xiao.Somenewoptimalandsub-optimalinfinitefamiliesofundirecteddouble-loopnetworks[J].DiscreteMathematicsandTheoreticalComputerScience,2006,8:299-312.[12]B.X.Chen,W.J.Xiao.Optimaldesignsofdirecteddouble-loopnetworks[A].InternationalSymposiumonComputationalandInforma-tionSciences(CIS’04),LectureNotesinComputerScience(LNCS)[C],SpringerVerlag,2004,3314:19-24. 28[{Ba#f#U#r{t[13]s!t5Sm}LjWy/R[J].|Cb$$.Y$2006,52(2):1008-7826.[14]z3"GkAASm}uY%D$[J],2005,35(6):738-742.[15]T2AASm}xvÆ>R$[J],A!2000,15(2):147-151.[16]5uTk-AADSm}~3>R$[J],2006,29(2):362-367.[17]|2z3"GSm}x>R$$[J].2007,30(1):129-137.[18]5)5uT17kAA:kAASm}x~V[J].uY$A!2007,37(6):673-680.[19]HwangF.K.Acomplementarysurveyondouble-loopnetworks[J].The-oreticalComputerScience,2001,263:211-229.[20]z3"k-AASm}#x[J].R$2005,48(6):1213-1220.[21]M.A.Fiol,M.Valero,J.L.A.Yebra,I.AlegreandT.Lang.Opti-maizationofdouble-loopstructuresforlocalnetworks[A].inProc.XIXInt.Symp.MIMI’82[C],Paris,France,pp.1982:37-41.[22]Cayleyh2<[J].5D$.Y$1999,16(3):10-15. ℄,92VCCayleyg)29l|uUHC58zlH~l7℄(Ct$F?#F9&F|Æj>?jw7a5C+BU$1:zw5FlHeDUV::zw5xn+pUC5Cq59o7tb?8℄CYw8-℄%8E8HU_VxzwxEw${xu2b.oUCq59o7twe$7aYw${t|Cb$R$yY$}uHYw${|Æ:g:0tw*0Ywg=g:

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