Discrete mathematics, Study notes of Discrete Mathematics

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Refine Tree? wWhakare diPf typy of Freed erplen 4 Simples+ types of Connected Arephm "Definatien fee, Le | —--A bree! 16 q Simple Connected grap — oe [ewitbout any Circuit . | i f-e. dyee Is a Connected acyelic areph.- 3 I-A Colection OF Set op an Goydlic arap 3 [ {S called a forest se i | A g——« i—™s yt a | Ab trae graph ate tree _ Q AL A veetex of deqree | IN & tree IS called _ | alleaf cra termine! pede .- - i) | fA veetex Or a Adeqree greatee, fhatr one i 15 caled Q branch nede Or inteenal node_ | / Cc | 5 f a a 1 a eS) ~ — ~ f | “Nos —_j | Se £ | ¢-d, @ — ate lea or eam | nel Nostey i LA Geb, & ate brante pede ‘ | Scanned by CamScanner The G bea paced : { 7 the eccentaicity & cfinedl os fod thtae } the ecc ents 5 |b Ea or €CvD a vere jo 7+ ® | distance rom Vv | wm VIR C | with minima ND | A veetex In a graph © aS I| 1 | jE 1s denoted by r C&D €(g)23, € Cb)= ec) =3 [| i E03 23 1 €C4)= 41 et) =3 —}—_—— 7 Salat eccenfrcity (5 2 Rr veered +b A _ re —— Scanned by CamScanner | _& Heic —¥ |) Levet and Hegh! of a Free : mM o * | 1 Pefine Level & Height of A free A Vveeler vy ina foolea free iS Sara te be UU level nif Pheee% a path of length, —t / from the root fo the VEELEK [|p othe highk of free ig Fhe maxinuns of The fevers of cfs yeenees | a —— ——~ —troot, Jewel 6 Scanned by CamScanner define sabre 9 > Subtrees ea. "A vesles x feget pi ie of Ee ° b Ad yi a 3 O ho { x | G is Sub re ew K 4 Poo bat 7% @ Q. Define M- ary tree? —2 ¥ _m-ary tree | A vooted free in @bith tvely Kreror nod& Ted ak mosh m Sore iS callea an M-ory tree. 4 y A mary tree is Scua tohe pequae m-Aby tree of fell m-ary foce ef evety Branch hoe hey txachly M sono. , ee ee = Scanned by CamScanner a a fll ay ) QO. Pefin e by Na oF _@ = binory free! bree Xplan In le beg) 7 Y bree _ a rT M Tay tree Js Knewn ay binary tree if evedy branch Mode Ay Ab Mest 9Q- Sen). g binary dree rs Called as reqalae binacy t veer teee Or yull binaey free rf esCty honest — Mode fray ekacl/y 2 sony CY Qe ds Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner pe Uk . Construct the _fabeted ree, of the goreery calg era, expression ¥ G@Represent the expression a po Coats) /4yd*% Cb Cesta) — ; tsIn 4 nine Fe Scanned by CamScanner eit “a vethiet Cleat d C2eHAlS @) she — — Hof Ae 4 Bossy sens ra — Gey - —— eA AS "il Ye e [a a a — jo _ wheeze Le i = ee As “is the Lenqih oF 3 bof fhe pare ath of - Z _ ak Aff rem fhe Phe Cow} oop of Ae dee Pree, o 2 wcn)s (24+ Canny oa BUSS SDT CGRD OD a ° t8ts— ee = 3 ES BIO Scanned by CamScanner QB. Explain qreph Coloring: SE Selerind 4 Chrermatic number creating of graph Gis an ASSANMENt OF Cofouys fe ity veeHees So thet he p00 adjacent Same Coley. avhiFe 1 white R ES ER areen Yelle / edhife | ey — veehea have tie As shown in £'4 the numbee of cofarrs | esech tn fit qraph, ¢S 5 € [nh C2 grey, » ops ferx . DP , : Cut man & Jotereat TS on Proper cobring Of graph ORIeh WE Minimum numbete o/ | Cd) ows | “Bis pumber is called Okyomeatic pumbee 'Chromakc numbee ‘. toinimum numbe of - Colors meee to pwdure a Pre ppte Colored of Q quph G. - Scanned by CamScanner hie. j den od wef | / ae ah oP 7 nary bee \, or _——— mbe# ch yom? « — _ Hic. MUM || the “chrome oh CG @ hos é Te coc of - sé 3 «14 K- chromate —_—— et we : yebdt ced te tL pet qrepy !? —- Kn eoith p Oe e es re plet& q —— 1 : yee ht lod ~~ A Sn ah Cajorring 9 — b qoaphy — | Expl? qe © Giaay _ LP So LS @ Define Qoeb Preph | _ ® “€ kieb grape _—_ ——— A vetted. F COAL. repay the. weh peg ge. jS _ Cénnectec) fo- veeten B _by cliveck Beat arn A to Bif theée Js — Q fine , age A to _cueh Page R AE from « +e rf thee ISA Meck: eclge ‘ to Vetter B these Orig hy jdotnteg Vettea / PO VERNER the A ef eeeirg fo poqe ® - hype "9 an eeby Scanned by CamScanner | 5 es ical | ae _ _Stepe: : be Veetices b,c, da re Se — pe eS ee) by . ce aber the ee 5S -— fox the veete * b 18 oa a ois (89) i : oe | 20 \ 2,0 cared of a0 4 sah | S \ Fo | loo eink £ coll be (at3) / Accosding to the abe} 6 | ! | | NX NO | C343) | ~ | sink + is adyacenr +o both the veehed ‘e Now the & d, 60 We can choose any vebleA bh Uy e d for _labdling of sink &. we chome d | jor the tevvertead, the Lake} ps Cs™ 3) U minimum of 9 §r IS 3 § thud The fakes - Scanned by CamScanner Scanned by CamScanner es, Here é ; | +h Spanning tree has 4 Brande Oy 6 Po tr Be = ye 4 pundamentey cuss Comrepon- I ng fe €ady ranck of T whith, a — | gY Vth, are iVery I Adcfow o i| oS =P Spranths PEF _ Fandarn enter caer - ey {6,5 Sls 4 fen sf So espe cy 4&7) &s, €y3 “ _ ey | $ eg, 23, Uy 257 &6 } [The Chews of J are, €a, Gy) eh €6 there are G fundamental Chreuif Orrapond TY fo Gach chows OF T MAI, AE given bewe Scanned by CamScanner ee Game Tree cipion § 3D E e positions 7 paced grape What eds 01 Tr cf | x fA dire leel oves, [5 cal Foe game edges are 00 ET ame fee ow i at ee 1 ~ Game / free cu pha ocbtal rm OM me @epratrta 5-00) — dam game GEpret nn 20 et eran a Re 4 "we e clasit y_ game game Into hse - Py of _ Single @& _ playck 23! ete a] Travelin ~salaman, problers b Sliding puzzle Se 4 Hamiltonian pur. Ie conskeurn "Sachs foehen d] ARubikls @ cude prob) ens a i woe Nerf] sadoku a - “2. Two playce game, | iit Fight queery proslen al Chess — Mathematica) pu226_ ; b] Bandminter - Iv] feeb queen prob) ey C]) Tennrs d] checkéss | Scanned by CamScanner ag. Define Fundamenta) caircuiht £ Catsetks bs => __ Fundamental CircecutS and Culsers . let G bea connected areuph Ke “Tbe os Spanning tree of G- An edge of a ius, rs Clee branch. An edqe cf C cohich 15 not r A 17 TO 1S cadled chore oF T. THE Contcuns = a a Unigue Crecurt ce | nelargpentaf Circtusp ~ > L cL cof gil 7 < ; cf G cwitbh respect to T. & fundamental cireut of a Cannezked — qraph G is aways ooth respect +o a spanning _ __ +rec OF G ee “There fore the different Spanning frees. coil] | have different Kicdam ented | Clrewts . a IHL The getioning areph @ eit ec Spanning -+-wees Tr and Tr ta Scanned by CamScanner . TZ amph ¢ V(S)= Lab cod eft a | eo r — ; edge Set en feise2,€9--2- 2S ——— AT) for Spannirp Free Th , branches of T, ave € , eo , &y, C6, € - Chords Ti Ca, Sc) Sy) SB lowing feeble af chores Y Consid ck the 0 x Costs pond/DG fundam enter) ctreedtS vv Chords Cosresponculng Farctarmeni circ és § 21, 25 ,tsf f er § C1, 2, &) esd er ¢ 2, C2, Cb, & e] 24 {2} 22. eu, 4, ear 23 $ -€ Q Refine fundamenta] cuwBtH § x Furdamental Cukseps «- 7 Let T be K Spanning d4yvee of a connecter) graph G. Since evesy edge < of 4 = b 16 an isthmw or bridgé tHe Spl | Injo 400 Components Say Th 4 Jo. Bur L WD =v Cn vCe>= V4) Scanned by CamScanner a 5 b Jo al C 7 v t rusked ‘s step) 2 = ws oO 3 h Scanned by CamScanner spo nhing tree. “fe construck O rainimal | “Use prims algo : thet both free have He same Wl | Vveekt# | ¢ 5. i Scanned by CamScanner Scanned by CamScanner d ; o | e | io | 42 TI oe > a ! | é \ | F Scanned by CamScanner b = C Gop f XN S it ¢ \ Led See | f to | So Wea 1) ‘J Scanned by CamScanner —== ——— 2) suppore dala dems A, B,C, D, B, Fr & OCH FO __- the gellowing_srequenctes . —Dala_items # Be DE whe is he min. ak pats fength. — ve Sequence in increasing arelck 2 ee A D F E& B 5 \ BS {6 ir | Ss 20 8 Oo no i a it 2 "*\ | x’ 5 ! (S) (Ce) _f Cx ce eX ZN 20. ZL \ SS Za f0\ = HY —_& min we - Cio x3) +- (S042) + (5% 4) +S x3) ad footy, (iE) Ceuh ~ 260 mings pah= AS, B 2, C94, DF, mo Fo 3,eS4 Scanned by CamScanner