叶形图的强连通性.pdf

Advances in Applied Mathematics A?,2024,13(3),1099-1115Published Online March 2024 in Hans.https:/www.hanspub.org/journal/aamhttps:/doi.org/10.12677/aam.2024.133103“/?r5?-=“O?vF2024c2?27FF2024c3?21FuF2024c3?28F p?X?G=(V(G),E(G)V(G)“L?:8E(G)“LX?8:O“LXp?n?nm?&3p?n&y?;?3X-:?NU?53p?N55X-D?5#N?n&u)?3d?eJ?r5r5#N?n&u)?D5?)w,u“/k?5?“/?r5?“/?r9rg,c5N5“/The Strong Connectivity of Leaf-SortGraphsHuanhuan Wang,Shiying WangSchool of Mathematics and Computer Science,Shanxi Normal University,Taiyuan Shanxi:?,?-=.“/?r5J.A?,2024,13(3):1099-1115.DOI:10.12677/aam.2024.133103?-=Received:Feb.27th,2024;accepted:Mar.21st,2024;published:Mar.28th,2024AbstractAn interconnection network is usually modeled as an undirected,connected graphG=(V(G),E(G),where V(G)represents vertex set,V(G)represents edge set,andnodes represent processors,edges represent communication links between processors.In the interconnect network,the failure of processors or communication links is un-avoidable.We focus on the fault tolerance of the network.The connectivity playsan important role in measuring the fault tolerance and reliability of interconnectionnetworks.Traditional connectivity only allows processor or communication link tofail,its scope of application is relatively limited.Under this background,the strongconnectivity of the network is proposed.Strong connectivity allows both the processorand the communication link to fail at the same time and is a derivative of traditionalconnectivity.Obviously more applicable to general situations.Leaf graphs have manygood properties.In this paper,we study some strong connectivity of leaf graphs,andthe strong connectivity and strong natural connectivity of leaf graphs are obtained.KeywordsConnectivity,Fault-Tolerance,Leaf-Sort GraphCopyright c?2024 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.p?G=(V(G),E(G)V(G)=v1,v2,.,vn“LG?:8E(G)=e1,e2,.,en“LXG?8:veO“LXp?n?nm?&uG?:vNG(v)5L:v?=v?:8-NGv=NG(v)vEG(v)Lv?8-U V(G)G?:f8NG(U)=SvUNG(v)UL:8U?-V0 V(G)G?:f8dV0p?5?fGV0:8V08G:3V0?kDOI:10.12677/aam.2024.1331031100A?-=8?8dG(v)5L:v3G?(G)=mindG(v)|v V(G)LG?3GXJ?:v?koGk?K13p?du?n&u)?53p?N55X-?25.?NU?3?u)?!:?3?k?53?e?5XZhaoHao?6?/?2Guo?7?UaN?55Lin?8?N?(?f(?GuoLu?N?Strong Menger5Wang?10?/(?r5Feng?11?2-?Wang?5#N&u)?r5w,D5?)u3“/CFn?r5?“/?r109rg,102.?(1)“/Sn5Ld1,2,.,n?kp1p2pn5L 12np1p2pn!(ij)?5?i j?ij=(p1pi pjpn)(ij)=(p1pjpipn)“?k?(1i):i=2,3,.,n+Sn?)8d(1i):i=2,3,.,n(j(j+1):j=2,4,.,n1(n)(1i):i=2,3,.,n(j(j+1):j=2,4,.,n2(n)Sn?)8n“/CFn?Xe2.1?nn“/CFnSn:8(u,v)|u=v(1i)(i 2,n),u=v(i(i+1)(i 2,4,.,n1)8?nn“/CFnSn:8(u,v)|u=v(1i)(i 2,n),u=v(i(i+1)(i 2,4,.,n 2)8?z:?r“/CFnyn?fCF1n1,CF2n1,.,CFnn1XifCFin1pz:?i(i 1,n)w,zCFin1CFn1?11?B?LCFn=CF1n1 CF2n1 CFnn1LCFn?A)?k:r:uf?XJ?(vk?:)?u?fp?:uu+Lu(1n)uLu(n 1)n-nN+u=u+,u1,np?i,jrifjfmkX?Ei,j(CFn)=ECFn(V(CFin1),V(CFjn1)5LCFn?n?CFn?n!:?3n32?K?nDOI:10.12677/aam.2024.1331031101A?-=?CFn?n!:?3n42?K2!3!4?“/CF2CF3CF41Figure 1.The leaf-sort graphs CF2,CF3and CF4 1.“/CF2CF3CF4n2.2(12)CFn?n2.3(12)?n?fCFin1m3X2(n 2)!?n?fCFin1m3X(n 2)!?n2.4(12)-v CFin1(1 i n)Knv+vu?fCFjn10sj 6=inv+uCFjn1j 6=in2.5(13)uifCFin1p?:uvnN+uN+v=nu+6=v+n2.6(13)?nu12fCF1,2n1p?:u?k33?nfCF3,nn1p=u+V(CF3,nn1)u V(CF3,nn1)DOI:10.12677/aam.2024.1331031102A?-=(2)52.7(10)?G=(V,E)F V Ee(G F)gKr?8Frg-?8G F?rg-?8F?G?rg-?rg-?G?rg-?g(G)LeGkrg-?KGrg-?G?r0-?G?rP(G)G?r1-?G?rg,Pn(G)eF EG?rg,G?rg,n(G)5Lb?g0 geGrg0-?Gkrg0-?Fd(GF)g0o(G F)gdGrg-?dd?e?nn2.8(10)-g0 gGrg0-?Kg0(G)g(G)2.9(10)?G=(V,E)F V EeG F?z|?kg:Kr?8Frg-?8G F?rg-?8F?G?rg-?rg-?G?rg-?g(G)LeGkrg-?KGrg-?b?g0 geGrg0-?oGkrg0-?Fd?G F?z|?kg0:oG F?z|?kg:dGrg-?dd?n2.10(10)-g0 gGrg0-?K g0(G)g(G)n2.11(10)-g0 gGrg-?Kg(G)g(G)n2.12(10)-g0 gGrg-?g=0,1kg(G)=g(G)XJG?z?r FU?5:KGr|F|?d?eG Fk|?:KGr;|F|?XJG?z?g,r FU?5KGr|F|g,?d?eG Fk|KGr;|F|g,?n2.13(12)-CFnn“/(n 2)nCFn?(CFn)=3n32nCFn?(CFn)=3n42n2.14(12)-CFnn“/(n 2)nCFn?(CFn)=3n32nCFn?(CFn)=3n42n2.15(13)-CFnn“/(n 4)nCFn;3n32?nCFn;3n42n2.16(14)?n 4-F V(CFn)n|F|3n 6n|F|3n 7eCFn FnKke:(1)CFnk?:DOI:10.12677/aam.2024.1331031103A?-=(2)CFnkn?:n2.17(14)-F E(CF3),|F|3 eCFnKCFnk?:n2.18(14)?n 3-F E(CFn)n|F|3n 6n|F|3n 7eCFnKCFnk?:3.“/CFn?r;5n3.1 u?n 3n(CFn)=3n32n(CFn)=3n42y-FCFn?rXJF V(CFn)dn2.13?n|F|=3n32n|F|=3n42eF E(CFn)dn2.14?n|F|=3n32n|F|=3n42?Tnb?n 3F*V(CFn)-n|F|3n52n|F|3n62?Fv=F V(CFn)Fv6=Fe=F E(CFn)Fe6=dd?n|Fv|3n72n|Fv|3n82odn2.13CFn Fv?duFeCFn Fv?CFn Fv Fek|-|Fe|=in 3?nkn!|Fv|n!(3n52 i)2(i+1)?nkn!|Fv|n!(3n62 i)2(i+1)d3CFnFv Fek|Cv|V(C)|i+1-VCCFn FvFe:?8du|VC|Fe|=i?n|VC|+|Fv|Fe|+|Fv|=|F|3n52n|VC|+|Fv|Fe|+|Fv|=|F|3n62dn2.13CFnFvFe?=CFnFFCFn?rgK?n|F|3n32n|F|3n42-u=(1)v=(12)?n-F=(1,i):3 i n i(i+1):i=246n 1 uv?n-F=(1,i):3 i n i(i+1):i=246n 2 uvwFCFn?r?n(CFn)3n32n(CFn)3n42?yn3.2u?n 4nCFn;3n32nCFn;3n42y-FCFn?n|F|=3n32n|F|=3n42-Fi=F E(CFin)i 1n-|F1|F2|Fn|?CCFnk?8-Fc=F Cena?1?n=4|F|=4ke1|F1|23de?i 14|Fi|2CFi4=CF3dn2.14i 14CFi4 Fi|Fc|4|N(V(CFi4 Fi)|=(4 1)!=6|Fc|?CF4 F2|F1|=33de?i 24|Fi|1|Fc|1CFi4=CF3dn2.14DOI:10.12677/aam.2024.1331031104A?-=CFi4Fiqdn2.3?fk(42)!=2?du(42)!=2|Fc|?CF4V(CF24F2)V(CF34F3)V(CF44F4)Fc?dn2.17CF14F1?k|?:?CF4 F1du(4 2)!=2|Fc|KCF4F?CF4F1k|?:v1ev+=Fii 234|v+(F2 F3 F4)|=1 dCF4 Fk|?:?yKFCFn?g3|F1|=43de?i 24|Fi|=0|Fc|=0dn2.149n2.3CF4F2?n 5keT(Jn=4yn8Bb?T(CFn11 n|F1|3n92n|F1|3n823de?i 1n?n|Fi|3n92|Fc|3n32n|Fi|3n82|Fc|3n42dun 52(n 2)!3n32n 6(n 2)!3n422(n2.149n2.3CF4 F2 n|F1|=3n72n|F1|=3n623de?i 2n?n|Fi|2|Fc|2n|Fi|1|Fc|1duCFin=CF3dn2.14CFin Fi?qdn2.3?n 52(n2)!2|Fc|?n 6(n2)!1|Fc|KCFnV(CF2nF2)V(CF3nF3)V(CFnnFn)Fc?d8Bb?CFnF1?k|?:veCFn F1?dn2.3CFn F?geCFn F1k|?:v1e?n|Fc|=2FcTv|N+v(F2F3Fn)|=2nFcv|v+(F2F3Fn)|=1dCFn(V(CF1n F1 v)V(CF2n F2)V(CFnn Fn)Fc?CFn Fk|?:?yKCFn FF?g3 n3n52|F1|3n32n|F1|=3n423de?i 2n?n|Fi|1|Fc|1n|Fi|0|Fc|0duCFi4=CF3dn2.14CFin Fi?qdn2.3?n 52(n2)!1|Fc|?n 6(n2)!0|Fc|?CFnFF?gnn?yn3.3 u?n 4nCFnr;3n32?nCFnr;3n42y-FCFn?rn|F|=3n32n|F|=3n42-Fi=F V(CFin)Fei=F E(CFin)i 1n-|F1 Fe1|=max|Fi Fei|i=12nDOI:10.12677/aam.2024.1331031105A?-=?CCFnk?8Fc=F Cen?1a?1?n=4|F|=41 0|F1 Fe1|23dedu|F1 Fe1|=max|Fi Fei|i=12ndn3.1i 14CFi4Fi Fei2dn2.4|N(CFin)|=6|F|?CF4 Fg2|F1 Fe1|=33de?|F2Fe2|+|F3Fe3|+|F4Fe4|+|Fc|=1duCFi4=CF3dn3.1 i 24CFi4FiFeiqdn2.3?fk(42)!=2?du(42)!=2 1?CF4V(CF24F2Fe2)V(CF34F3Fe3)V(CF44F4Fe4)Fc?du|F1Fe1|=3wCF14F1Fe1ken(1)CF14F1Fe1?(2)CF14 F1 Fe1k|?:(3)CF14 F1 Fe1n?:?CF14F1Fe1|F2Fe2|+|F3Fe3|+|F4Fe4|+|Fc|=1|N(V(CF14F1Fe1)(V(CF24)V(CF34)V(CF44)|3 1KCF4F?CF14F1Fe1k|?:ub?u+=Fii 234uTFcdCF4 Fk|?:?yKCF4 Fg?CF14 F1 Fe1n?:?:vb?v+=Fii 234vTFcKCF4 Fk|?:?yKCF4 Fg3|F1 Fe1|=43de|Fi Fei|=0|Fc|=0dn3.192.4CF4 Fg2?n 5keT(Jn=4yn8Bb?T(CFn11 n|F1 Fe1|3n92n|F1 Fe1|3n823den|Fi Fei|3n92|Fc|3n32n|Fi Fei|3n82|Fc|3n42dn3.1?i 1nCFinFiFei2dn2.4CFnFg2 n|F1 Fe1|=3n72n|F1 Fe1|=3n623de?n|F2 Fe2|+|F3 Fe3|+|Fn Fen|+|Fc|=2n|F2 Fe2|+|F3 Fe3|+|Fn Fen|+|Fc|=1b?CF1n F1 Fe1dn2.4n 5k2(n 1)!3n72 2n 6k(n 1)!3n72 1?CFn Fgb?CF1n F1 Fe1d8Bb?k|?:v1en|N(v)(F2 F3Fn)|=2|Fc|=2 Fcvn|N(v)(F2F3Fn)|=1|Fc|=1FcvdCFnFk?:?yKg3 n3n52|F1 Fe1|3n32n|F1 Fe1|=3n423den|F2 Fe2|+|F3 Fe3|+|Fn Fen|+|Fc|1nDOI:10.12677/aam.2024.1331031106A?-=|F2Fe2|+|F3Fe3|+|FnFen|+|Fc|=0dn2.49n3.1CFnFg4.“/CFn?rg,5n4.1u?n 5nCFn;3n 5g,nCFn;3n 6g,y-F V(CFn)CFn?g,?n|F|=3n 5n|F|=3n 6-Fi=F V(CFin)i 1n-|F1|F2|Fn|em?(1?n|F3|1?n|F2|1(1?y?n|F3|1e|F3|=0K|F3|=|F4|=|Fn|=0dn2.1392.3CFnV(CF3n)V(CF4n)V(CFnn)2dn2.6CFn Fg?n|F2|1e|F2|=0K|F2|=|F3|=|Fn|=0dn2.139n2.3CFn Fg?yd(1?n1|F1|3n 7d|F|=3n 5?1|F2|3n72K1e|F2|3n52Kk|F|2 3n52+1=3n 4|F|g?y?n1|F1|3n 7d|F|=3n 6?1|F2|3n62K1e|F2|3n42Kk|F|2 3n42=3n 4|F|g?y1 n|F1|3n92n|F1|3n82du|F1|=max|Fi|1 i ndn2.13CFin Fidun 52(n 2)!3n 9=|Fi|+|Fj|n 6(n 2)!3n 8=|Fi|+|Fj|i 6=jij 1n2dn2.3CFn Fg2 n3n72|F1|3n 10n3n62|F1|3n 9f2.1 n|F2|3n92n|F2|3n823de?i 2nn|Fi|3n92n|Fi|3n82aqu1?CFnV(CFin Fi)V(CFjn Fj)?i 6=jij 2n?CFnV(CF2nF2)V(CF3nF3)V(CFnnFn)eCF1nF1?|Fn|=2n 5k2(n1)+3n72=7n112 3n5n 6kK2(n1)+3n62=7n102 3n6?gd|Fn|1qn 52(n2)!3n10+1=3n9|F1|+|Fn|n 6(n2)!3n9+1=3n8|F1|+|Fn|CFnV(CF1nF1)V(CFnnFn)?CFnFeCF1nF1dn2.16CF1nF1k|?:kn|?:-CCF1nF1?|n 52(n1)!(3n10)2 3n53n72n 6(n1)!(3n9)2 3n63n62?CFnV(C)V(CF2 F2)V(CF3 F3)V(CFn Fn)F?g,gf2.2n|F2|=3n72n|F2|=3n623den|F3|+|F4|+|Fn|2n|F3|+|F4|+|Fn|=0dDOI:10.12677/aam.2024.1331031107A?-=n2.139n2.3CFnV(CF3nF3)V(CF4nF4)V(CFnnFn)b?CF2nF2n 52(n1)!3n72 2n 6(n1)!3n62 0dn2.6CFnV(CF2n F2)V(CF3n F3)V(CFnn Fn)1eCF1n F1n 52(n 1)!(3n 10)3n 5 3n72n 6(n 1)!(3n 9)3n 6 3n62dCFnV(CF1n F1)V(CF2n F2)V(CFnn Fn)=CFn Fg1eCF1n F1dn2.16CF1n F1k|?:kn|?:-CCF1n F1?|n 52(n 1)!(3n 10)2 3n 5 3n72n 6(n 1)!(3n 9)2 3n63n62KCFnV(C)V(CF2nF2)V(CFnnFn)F?g,gb?CF2n F2dn2.15CF2n F2k|?:-B|n 52(n 1)!3n72 1 2n 6(n 1)!3n62 1 0?CFnV(B)V(CF3n F3)V(CFnn Fn)1eCF1n F1n 5 2(n1)!(3n10)3n53n72n 6(n1)!(3n9)3n63n62dCFnV(CF1nF1)V(B)V(CFnnFn)F?g,g1eCF1nF1dn2.16CF1nF1k|?:kn|?:-B0|n 52(n1)!(3n10)2 3n53n72n 6(n1)!(3n9)2 3n63n62?CFnV(B)V(B0)V(CF3nF3)V(CFnnFn)deCFn Fke?(1)CFn Fk|?:g(2)CFnFkn|?:g(3)CFnFko|n?:g(4)CFnFk|?y(5)CFnFkn|?:,?g3 n3n 9|F1|3n 7n3n 8|F1|3n 73den|F2|+|F3|+|Fn|4n|F2|+|F3|+|Fn|2dn2.392.13CFnV(CF2nF2)V(CF3nF3)V(CFnnFn)1eCF1nF1dn2.4CFnFg1eCF1nF1?B1B2BkCF1nF1?|e|V(Bj)|3?n|N(V(Bj)|V(CF2n)V(CF3n)V(CFnn)|6 4?n|N(V(Bj)|V(CF2n)V(CF3n)V(CFnn)|3 2dCF1nF1k|?:kn|?:ko|n?:gk|?yn4.2 u?n 3nn(CFn)=3n5;?nn(CFn)=3n6y-u=(1)v=(12)?nF=ux:x (1,i):3 i n j(j+1):j=2,4,n 1 vx:x (1,i):3 i n j(j+1):j=2,4,n 1nF=ux:x (1,i):3 i n j(j+1):j=2,4,n 2 vx:x (1,i):3 i n j(j+1):j=2,4,n 2dn2.2CFn?Kkx V(CFn F)ndCFnF(x)3n52 1ndCFnF(x)3n62 1?FCFn?g,Knn(CFn)3n 5;?nn(CFn)3n 6dn2.18n|F|3n 5n|F|3n 6d?nn(CFn)=3n 5;?nn(CFn)=3n 6?yDOI:10.12677/aam.2024.1331031108A?-=n4.3u?n 3nCFng,3n-5nCFng,3n-6y?FCFn?g,dn4.2n|F|=3n 5n|F|=3n 6-Fi=F E(CFin)i 1,n|F1|F2|Fn|?CCFnk?|?8-Fc=F C?e5na?(1)n=3|F|=4w,dCF3 Fgk?:gk?(2)n=4|F|=6ke1|F1|23dei 1,4|Fi|2|Fc|6duCFi4=CF3Kdn2.14CFi4Fi2dn2.3 e|Fc(V(CF14),V(CF24)V(CF14),V(CF34)V(CF14),V(CF44)|=6dw,CF4 Fk|CF14 F1ge|Fc|KCF4V(CF24F2)V(CF34F3)V(CF44 F4)FcduCF14=CF3dn3.2CF14 F1k|?:d|F2|+|F3|+|F4|+|Fc|=3CF4 Fk|?:gf2.2|F2|=33de|F3|+|F4|+|Fc|=0dn2.39n2.14CF4V(CF34F3)V(CF44F4)Fc1e?i 1,2CFi4Fiw,CF4Fg1eCF14F1?CF24 F2dn3.2CF24 F2k|?:du|Fc|=0w,CF4Fge?i 1,2CFi4FiKdn3.2CFi4Fik|?:vi|Fc|=0CF4V(CF14F1v1)V(CF24F2v2)V(CF34F3)V(CF44F4)Fcyb?v1v2Kdn2.4?|N(vi)(V(CF34)V(CF44)|=0d|F3|+|F4|+|Fc|=0TK?CF4Fk|?v1v2?yKg3|F1|=43e|F2|+|F3|+|F4|+|Fc|=2dn2.392.14CF4V(CF24 F2)V(CF34 F3)V(CF44 F4)FcduCF14=CF3d(1)CF14 F1k|?:k|?uvdn2.4CF4 Fk|?uv?yk|?:gDOI:10.12677/aam.2024.1331031109A?-=45|F1|63e|F2|+|F3|+|F4|+|Fc|1dn2.39n2.14CF4V(CF24 F2)V(CF34 F3)V(CF44 F4)Fcd?CF14 F1kB1,B2,Bk|k?|V(Bk)|2du|Fc|1?CF4V(Bk)V(CF24F2)V(CF34F3)V(CF44F4)Fc?CF4 Fk|?:g(3)n 5ke1?n|F1|3n92?n|F1|3n823dei 1,n?n|Fi|3n92|Fc|3n 5n|Fi|3n82|Fc|3n6dn2.14CFinFin 52(n2)!3n5n 6(n2)!3n62dn2.3CFnV(CF1nF1)V(CF2nF2)V(CFnnFn)Fc?CFn Fg2?n|F1|=3n72?n|F1|=3n62f2.1?n|F2|3n92?n|F2|3n823dei 2,nn|Fi|3n92|Fc|3n32ni 2,n|Fi|3n82|Fc|3n62dn2.14CFinFin 52(n2)!3n32n 6(n 2)!3n622dn2.3CFnV(CF2n F2)V(CF3n F3)V(CFnnFn)Fcdn3.2CF1nF1k|?:-B|dun 52(n 1)!1 3n32n 6(n 1)!1 3n62KCFnV(B)V(CF2n F2)V(CFnn Fn)Fc?CFn Fk|?:gf2.2?n3n72|F2|3n32?n|F2|=3n623dei 3,nn|Fi|3n72|Fc|2n|Fi|3n62|Fc|=0dn2.12CFin Fi2(n 2)!2(n 2)!02dn2.3CFnV(CF3nF3)V(CF4nF4)V(CFnnFn)Fc1e?i 1,2 CFinFiw,CFnFg1eCF1nF1?CF2nF2?kB1,B2,Bk|k?|V(Bk)|2CFnV(CF1n F1)V(B)V(CF3n F3)V(CFnnFn)FcdCFnFk?:ge?i 1,2CFinFiKdn3.2CFin Fik|?:vin2.3CFnV(CF14F1 v1)V(CF24 F2 v2)V(CF34 F3)V(CF44 F4)Fc1ev1,v2?n|Fc|=2Fcvindu|Fc|=0dCFn Fk|?v1v2?yKk|?:g3?n3n52|F1|3n 10?n3n42|F1|3n 9f3.1n|Fi|3n92n|Fi|3n823dei 2,nn|Fi|3n92|Fc|3n52n|Fi|3n82|Fc|3n82dn2.14CFinFin 52(n2)!3n72n 6DOI:10.12677/aam.2024.1331031110A?-=(n2)!3n722dn2.3CFnV(CF2nF2)V(CF3nF3)V(CFnnFn)Fcdn2.18CF1n F1dw,CFn Fgk|?:dun 52(n 1)!1 3n52n 6(n 1)!1 3n82?dCFn Fk|?:gf3.2?n3n72|F2|3n523dei 3,n|Fi|3n72|Fc|1dn2.14CFin Fin 52(n2)!12dn2.3CFnV(CF3nF3)V(CF4nF4)V(CFnnFn)Fc1e?i 1,2CFinFiw,CFnFg1eCF1nF1?CF2nF2du3n52 3n 10dn2.17CF2n F2k|?:q|Fc|1w,CFn Fge?i 1,2 CFin FiKdn2.15CFin Fik|?:vi|Fc|1Kw,CFn Fg4n3n 9|F1|3n 5n3n 8|F1|3n 6dn|F2|+|F3|+|Fc|4n|F2|+|F3|+|Fc|2dn2.392.14CFnV(CF2n F2)V(CF3n F3)V(CFnn Fn)Fcyb?CF1n F1kB1,B2,Bk|k?|V(Bk)|3dun|Fc|4n|Fc|2?CFnV(Bk)V(CF2nF2)V(CF3nF3)V(CF4nFn)Fc?CFn Fk|?:kn|?:k|?yn4.4 u?n 4F V(CFn)E(CFn)?n|F|3n 6?n|F|3n 7KCFn Feve?(i)CFn Fk|?:(ii)CFn Fkn|?:y?n 4eF V(CFn)?n|F|3n 6?n|F|3n 7dn2.16CFn Fve?(i)CFn Fk|?:(ii)CFnFkn|?:eF E(CFn)?n|F|3n6?n|F|3n 7dn2.17CFn Fk|?:db?n 4F V(CFn)E(CFn)?n|F|3n 6n|F|3n7b?CFnF?F V(CFn)6=F E(CFn)6=-Fi=F V(CFin)Fei=FE(CFin)i 1,n|F1Fe1|F2Fe2|FnFen|?CCFnk?8-Fc=F Cd?n1|F1|+|F2|+|Fn|3n71|Fe1|+|Fe2|+|Fen|3n 7?n1|F1|+|F2|+|Fn|3n 81|Fe1|+|Fe2|+|Fen|3n 8?n=4-|F|5CFi4=CF3e|F|3dn3.1CF4 Fge|F|=4dn3.3CF4 Fk|?:?ye|F|=5CF4 Fe1|F1 Fe1|=5DOI:10.12677/aam.2024.1331031111A?-=d|F2 Fe2|+|F3 Fe3|+|F4 Fe4|+|Fc|=0w,CF4 Fg2|F1 Fe1|=4d?CF14 F1ke?6(1)(2)?:(3)n?:(4)k|?:(5)?(6)kn|?:du|F2 Fe2|+|F3 Fe3|+|F4 Fe4|+|Fc|=1(n2.4?CF4 F(1)(5)CF4 Fg?CF4 FCF4 Fk|?:?y3|F1 Fe1|=3d?CF14F1ke?3(1)(2)k|?:(3)n?:du|F2Fe2|+|F3Fe3|+|F4Fe4|+|Fc|=2(n2.4CF4Fk|?:kn|?:?y30|F1 Fe1|2ddn3.1i 1,4CFi4FiFeidu|N(CFin|=6|F|?CF4Fgdd(n=4n8Bb?n 5CFn1ewCFn(2)?n 5n-|F|3n 6n-|F|3n 71n|F1 Fe1|3n92n|F1 Fe1|3n82d?i 1nn|FiFei|3n92|Fc|3n6n|FiFei|3n82|Fc|3n 7dn3.1CFin Fi Fein 52(n 2)!3n 6n 6(n 2)!3n 72(n2.3CFn F2n|F1 Fe1|=3n72n|F1 Fe1|=3n62f2.1n|F2 Fe2|3n92n|F2 Fe2|3n82dn|F3 Fe3|+|F4 Fe4|+|Fn Fen|+|Fc|3n52n|F3Fe3|+|F4 Fe4|+|Fn Fen|+|Fc|3n82dn3.1?i 2nCFin Fi Fein 52(n 2)!3n52n 6(n 2)!3n822(n2.3CFnV(CF2n F2 Fe2)V(CF3n F3 Fe3)V(CFnn Fn Fen)Fcdn3.3CF1n F1 Fe1k|?:?CF1n F1 Fe1w,CFn F?CF1n F1 Fe1k|?:vn 52(n1)!3n721 3n52n 6(n1)!3n621 3n822(n2.3CFnV(CF1nF1Fe1v)V(CF2nF2Fe2)V(CF2nF2Fe2)Fc?CFnFk|?:?yf2.2n|F2 Fe2|=3n72d|F3Fe3|+|F4Fe4|+|FnFen|+|Fc|1aquf2.1CFnV(CF3nDOI:10.12677/aam.2024.1331031112A?-=F3Fe3)V(CF4nF4Fe4)V(CFnnFnFen)Fc?i 1,2dn3.3CFinFiFeik|?:vib?CFinFiFeiw,CFnFb?CF1nF1Fe1?CF2nF2Fe2n 52(n2)!3n721 1?CFnV(CF1nF1Fe1v1)V(CF2nF2Fe2)V(CFnnFnFen)Fc?CFnFk|?:b?CFinFiFeidn3.3CFinFiFeik|?:vidu2(n 1)!3n72 1 1?CFnV(CF1n F1 Fe1v1)V(CF2n F2 Fe2 v2)V(CF3n F3 Fe3)V(CFnn Fn Fen)Fcdu|F3 Fe3|+|F4 Fe4|+|Fn Fen|+|Fc|12(n2.6?CFn Fgk|?:?y3n|F1 Fe1|=3n52f3.1n|F2 Fe2|3n92d|F3Fe3|+|F4Fe4|+|FnFen|+|Fc|3n72dn2.393.1CFnV(CF2nF2Fe2)V(CF3nF3Fe3)V(CFnnFnFen)Fcdu3n52 3n10=3(n1)7(n 5)d8BCF1n F1 Fe1k|?:kn|?:du2(n 1)!3n52 2 3n72?CFn Fk|?:kn|?:?yf3.2n|F2 Fe2|=3n72d|F3 Fe3|+|F4 Fe4|+|Fn Fen|+|Fc|=0dn2.6CFn F4n3n32|F1 Fe1|3n 10n3n42|F1 Fe1|3n 9dn|F2 Fe2|+|F3 Fe3|+|Fn Fen|+|Fc|3n92n|F2Fe2|+|F3 Fe3|+|Fn Fen|+|Fc|3n102dn2.393.1CFnV(CF2n F2 Fe2)V(CF3n F3 Fe3)V(CFnn Fn Fen)Fcd8BCF1n F1 Fe1k|?:kn|?:-|F1Fe1|=in 52(n 1)!i 2 3n92n 6(n 1)!i 2 3n102?CFn Fk|?:kn|?:5n3n 9|F1 Fe1|3n 6n3n 8|F1 Fe1|3n 7dn|F2Fe2|+|F3Fe3|+|FnFen|+|Fc|3n|F2Fe2|+|F3Fe3|+|FnFen|+|Fc|1dn2.393.1CFnV(CF2nF2Fe2)V(CF3nF3Fe3)V(CFnnFnFen)Fcb?CF1nF1Fe1k|B1B2Bk?|V(Bk)|2dn2.4CFnV(Bk)V(CF2nF2Fe2)V(CF3nF3Fe3)V(CFnnFnFen)Fc?CFn Fk|?:n4.5?n 4?nn(CFn)=3n 5?nn(CFn)=3n 7-FCFn?g,rdn4.4?n|F|3n 5?n|F|3n 6-u=(1)v=(12)n-F=ux:x (1i):3 i n (ii+1):i=2,4,n 1 (12)(1,i):3 i n (i(i+1):i=2,4,n 2)nDOI:10.12677/aam.2024.1331031113A?-=-F=uxx (1i):3 i n (i(i+1):i=2,4,n 2)(12)(1,i):3 i n (i(i+1):i=2,4,n 1)duCFn?FCFn?rg,?nn(CFn)3n 5nn(CFn)3n 6(?y5.(?“/CFn?r5?D?5T5#N:u)?u?u?n 4n(CFn)=3n32n(CFn)=3n42=?K?r?“/U?:3r?nn(CFn)=3n 5nn(CFn)=3n 7=?K?r?“/U?“/?:I?Kr?o“/?:$?I?Kr?k6u?78Ig,78(61772010):Oy(202203021221128)z1 Bondy,J.and Murty,U.(2008)Graph Theory.Springer,Berlin.2 Hung,C.-N.,Hsu,H.-C.,Liang,K.-Y.and Hsu,L.-H.(2003)Ring Embedding in FaultyPancake Grap