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Mirrors > Home > MPE Home > Th. List > numclwwlk2 | Structured version Visualization version GIF version |
Description: Statement 10 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v is k^(n-2) - f(n-2)." According to rusgrnumwlkg 29830, we have k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this number, the number of closed walks of length (n-2), which is f(n-2) per definition, must be subtracted, because for these walks v(n-2) =/= v(0) = v would hold. Because of the friendship condition, there is exactly one vertex v(n-1) which is a neighbor of v(n-2) as well as of v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of walks v(0) ... v(n-2) is identical with the number of walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n-2) can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-May-2022.) |
Ref | Expression |
---|---|
numclwwlk.v | β’ π = (VtxβπΊ) |
numclwwlk.q | β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) |
numclwwlk.h | β’ π» = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) |
Ref | Expression |
---|---|
numclwwlk2 | β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ»π)) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelcn 12862 | . . . . . . . 8 β’ (π β (β€β₯β3) β π β β) | |
2 | 2cnd 12318 | . . . . . . . 8 β’ (π β (β€β₯β3) β 2 β β) | |
3 | 1, 2 | npcand 11603 | . . . . . . 7 β’ (π β (β€β₯β3) β ((π β 2) + 2) = π) |
4 | 3 | eqcomd 2731 | . . . . . 6 β’ (π β (β€β₯β3) β π = ((π β 2) + 2)) |
5 | 4 | 3ad2ant3 1132 | . . . . 5 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β π = ((π β 2) + 2)) |
6 | 5 | adantl 480 | . . . 4 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β π = ((π β 2) + 2)) |
7 | 6 | oveq2d 7431 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (ππ»π) = (ππ»((π β 2) + 2))) |
8 | 7 | fveq2d 6895 | . 2 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ»π)) = (β―β(ππ»((π β 2) + 2)))) |
9 | simplr 767 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β πΊ β FriendGraph ) | |
10 | simpr2 1192 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β π β π) | |
11 | uz3m2nn 12903 | . . . . 5 β’ (π β (β€β₯β3) β (π β 2) β β) | |
12 | 11 | 3ad2ant3 1132 | . . . 4 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β (π β 2) β β) |
13 | 12 | adantl 480 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (π β 2) β β) |
14 | numclwwlk.v | . . . 4 β’ π = (VtxβπΊ) | |
15 | numclwwlk.q | . . . 4 β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) | |
16 | numclwwlk.h | . . . 4 β’ π» = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) | |
17 | 14, 15, 16 | numclwwlk2lem3 30232 | . . 3 β’ ((πΊ β FriendGraph β§ π β π β§ (π β 2) β β) β (β―β(ππ(π β 2))) = (β―β(ππ»((π β 2) + 2)))) |
18 | 9, 10, 13, 17 | syl3anc 1368 | . 2 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ(π β 2))) = (β―β(ππ»((π β 2) + 2)))) |
19 | simpl 481 | . . . 4 β’ ((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β πΊ RegUSGraph πΎ) | |
20 | simp1 1133 | . . . 4 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β π β Fin) | |
21 | 19, 20 | anim12i 611 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (πΊ RegUSGraph πΎ β§ π β Fin)) |
22 | 11 | anim2i 615 | . . . . 5 β’ ((π β π β§ π β (β€β₯β3)) β (π β π β§ (π β 2) β β)) |
23 | 22 | 3adant1 1127 | . . . 4 β’ ((π β Fin β§ π β π β§ π β (β€β₯β3)) β (π β π β§ (π β 2) β β)) |
24 | 23 | adantl 480 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (π β π β§ (π β 2) β β)) |
25 | 14, 15 | numclwwlkqhash 30227 | . . 3 β’ (((πΊ RegUSGraph πΎ β§ π β Fin) β§ (π β π β§ (π β 2) β β)) β (β―β(ππ(π β 2))) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
26 | 21, 24, 25 | syl2anc 582 | . 2 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ(π β 2))) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
27 | 8, 18, 26 | 3eqtr2d 2771 | 1 β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3))) β (β―β(ππ»π)) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2930 {crab 3419 class class class wbr 5143 βcfv 6542 (class class class)co 7415 β cmpo 7417 Fincfn 8960 0cc0 11136 + caddc 11139 β cmin 11472 βcn 12240 2c2 12295 3c3 12296 β€β₯cuz 12850 βcexp 14056 β―chash 14319 lastSclsw 14542 Vtxcvtx 28851 RegUSGraph crusgr 29412 WWalksN cwwlksn 29679 ClWWalksNOncclwwlknon 29939 FriendGraph cfrgr 30110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-er 8721 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-oi 9531 df-dju 9922 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-rp 13005 df-xadd 13123 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-word 14495 df-lsw 14543 df-concat 14551 df-s1 14576 df-substr 14621 df-pfx 14651 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-vtx 28853 df-iedg 28854 df-edg 28903 df-uhgr 28913 df-ushgr 28914 df-upgr 28937 df-umgr 28938 df-uspgr 29005 df-usgr 29006 df-fusgr 29172 df-nbgr 29188 df-vtxdg 29322 df-rgr 29413 df-rusgr 29414 df-wwlks 29683 df-wwlksn 29684 df-wwlksnon 29685 df-clwwlk 29834 df-clwwlkn 29877 df-clwwlknon 29940 df-frgr 30111 |
This theorem is referenced by: numclwwlk3 30237 |
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