LABELING THE VERTEX AMALGAMATION OF

Download PDF
0 downloads 0 Views 169KB Size Report
Comment
several results involving the vertex amalgamation of graceful, felicitous and ... that is, for which m and n the vertex amalgamation of n copies of the cycle. Cm at a ...
     

!" #$ %

&('*)+'-,/.1032546&8789;:1846=37@?-&('A.1I93=;4J&8:LK69M>8NO4 PQSRUTWVSXUYLQSRZYL[@\8]1[_^`QbaSac[U]ed$TgfUhAQjilk3mf Qji*n6TW[Ui oFpAqSrAs*tlu-rOv 9wUxSy{zA|~}€{I‚„ƒg…`†F‡3ƒg‚6wUxSy{ˆ`†cŠ‰-‹Œƒ Œ„|A†`x_†F†cŽ-ƒg‚1„‚8y{ƒg{‘1†`ˆc„ƒ ’ †FW‰AAˆc_ƒgU “•”(–˜— }(™šœ›ž-Ÿj UŸ`¡b¡b¡`Ÿ„‡ ¢‚„‰-ˆS|_|*yž$†by{ˆS|†`£AwU†¥¤§¦{}¨ƒg‚~‹©y{ªl†`‹g†`£¬« “­— ¤Œ™F® “­— ¦-™`«Iy{A£¥„|A†x_†b‚_‰A‹ _ƒgAw†`£AwU†‹©y{ªl†`‹g‚@y{x_†£Aƒg‚1„ƒgAˆcb'9;‹g‚„A?Oy — ¯ Ÿ_‡U™3wUxSy{z-|°}¬±5ƒ _| ‡² ¯ ƒg‚|*y{x_³˜UAƒgU‰A‚Zƒ („|-†bx_††`Žƒg‚__‚y{ƒg{‘1†bˆc_ƒ ’ †$Š‰A-ˆ`_ƒgU “´”3–— }(™$š¶µC· ‚_‰AˆS|_|*yž;†jy{ˆ„|†`£AwU†e¤§¦˜{J}¸ƒg‚8‹©y{ªl†`‹g†`£ “­— ¤Œ™O¹ “­— ¦-™ — ³˜º£‡U™Cy{A£~„|A†(x_†`‚„‰A‹ _ƒgAw †`£AwU†¥‹©y{ªl†b‹g‚~y{x„†£Aƒg‚1„ƒg-ˆ`b?5±;|-†bx_†by{‚$}»ƒg‚Š†`‹gƒgˆ`ƒ _U‰A‚~ƒ (_|A†`x„††cŽƒg‚__‚y{€ƒg{‘1†`ˆc„ƒ ’ † W‰AAˆc_ƒgU “”º–˜— }(™/š¨µ ·b¼J½ ‚„‰Aˆ„|„|Ayž;†by{ˆS|†b£-wU†¤l¦˜{M}¸ƒg‚8‹©y{ªl†`‹g†`£ “­— ¤Œ™Œ¹ “­— ¦™ — ³˜£‡U™Fy{A£$_|A†¾x_†`‚„‰A‹ _ƒgAwZ†b£-wU†¾‹©y{ªl†`‹g‚ey{x„†£Aƒg‚__ƒgAˆcj'.L$_|Aƒg‚3z*y{zl†`xj?Œ±C†¾zAx_†`‚„†`º ‚_†`’ †`x„y{‹5x_†`‚„‰A‹ _‚¾ƒg ’ U‹ ’-ƒg-w_|A†Z’ †bx1„†cŽ¿y{³y{‹gw y{³yž„ƒgU¿{;wUxSy{ˆ`†cW‰A‹À?MW†`‹gƒgˆ`ƒ „U‰A‚y{A£ |*y{x_³˜UAƒgU‰A‚CwUx„y{zA|A‚j'/,*‰Ax1„|-†bxb?-±C†Fz*y{x__ƒ©y{‹g‹ Á˜‚„U‹ ’ †3y{~Uzl†`~z-x„UªA‹g†`³´{INO†`†3†c5y{‹À'g? _|*yžeƒg‚j?§WUxF±5|AƒgˆS|ÂÃy{A£Ä+_|A†¾’ †`x__†cŽ¥y{³y{‹gw y{³yž„ƒgU${JÄ¿ˆ`UzAƒg†`‚e{J_|A†ˆcÁ-ˆ`‹g† Å6Æ yž˜y$ÇAŽ†`£’ †`x1„†cŽ°¦È –˜— Å6Æ ™c?J9F³y{‹ — Å6Æ Ÿ1¦*Ÿ„ÄO™c?Jƒg‚@W†`‹gƒgˆbƒ _U‰A‚SÉ»)Ux„†`b’ †bxb? ±C†ezAx_j’ƒg£A†e‚„U³˜†(zAx_UwUx„†`‚„‚8_j±8y{x_£A‚F‚_U‹ ’-ƒgAw_|A†eˆ`U{‘1†bˆc_‰Ax„†{MÊ3U|†c3y{‹À'g?-±;|-ƒgˆS| ‚1Syž_†`‚@_|*yž@_|A†wUxSy{z-|+9F³y{‹ — Å ÆZŸ1¦*Ÿ_ÄI™3ƒg‚ewUxSy{ˆ`†cŠ‰A‹Jƒ 5y{A£+UA‹ Á¥ƒ 8ÂZÄ˕UxÌ — ³˜£~ͺ™c'/,΃g*y{‹g‹ Á ?-±6†(zAx„UzlU‚„†3L±6˜ˆbUž‘_†`ˆc„‰-x„†`‚j' §Ï АÑ{ÒCÓCÔ6Õ5Ö;×5ØMÓ6ÙEÕ;Ò ÚÜÛÝEާߊߊÞ*àá@âMãläjåjä{ãlæ/ç€ãlæMçHè8Û-ébæÎêÀãlëÜì Ϻí ÝEތäZî¿ÞŒébåZÞ§Ý(åjâÎÛï§ä{ãlðÎâåjâ/ÛºÞ§äžñåjÛºäžî¿êWæ/ާߊާï§ñ ã§æ/ç°æÎÞ§åžã*åžêWތæ°ò/éjÛç°êŠæåžâÎêŠé@ð/ã§ðMÛä Ð3ó æð/ã§äbåžêŠôºòÎßÀãlä-õ§à¾ÛZà˜êŠßWßCô ތæ/éjêŠçIÛä@ã$ïŒäžã§ðÎâ¿åjÞöMÛ~÷/æÎêWåjÛ ã§æ/çà˜êWåjâÎތòIå(ßWÞOތð/é5ތä3î¥òÎßWåjêŠðÎߊ۾Û-çIï§Û-é ÐFø òÎäjåjâ/Ûºä-õAåjâÎÛù§ÛäbåžÛ ú¥éjÛ å3ÞlÝMã~ï§ä{ãlð/â¥ûüêÀé3çIÛºæÎÞ§åjÛ-ç öOñýþ1ûÿUõ/à˜âÎÛºäžÛãŒé¾åjâÎÛ$Û-çIï§Û$éjÛ å˜ÞlÝ(û êÀé˜çIÛæÎÞlåžÛçöO ñ þ1ûÿ Ð ;ÞÝEތäjî°ãlߊêºÛ+Þ§òÎäðÎäžÛéjÛºæ­åžãlåjêŠÞ§æHÞ§ÝåžâÎêŠéðMãlðJÛºä-õCà@Û+à˜êWߊß(êŠæŒåžäjÞIçIòMô Û¿ébތî+Û¿çIÛº÷/æÎêWåjêŠÞ§æ/é æÎÞ§åÝEÞ§òÎæ/çêŠæ ì Ϻí„Ð Ý  „ù*ãlߊò/ãlåjêŠÞ§æ/éõ8à˜âÎêÀô{â à@ÛäjÛ°éjòÎö/éjÛ ­òÎÛºæ­åžßWñ ó æ Ï   õ ÞŒéžãÜì *í êWæÎêWåjêÀã*åžÛç¸åjâ/Ûé`åžò/çIñ Þ§ æ/ã§î¿ÛçÜï§ä{ã§ôºÛ ÝEòÎß(ßÀãlöJۺߊêWæ/ïŒéöO ñ ތßWތî¥ö ì  í„Ð âÎÛ¿åjÛäjî€õFï§ä{ã§ô ÛºÝEòÎß(ßÀãlöJۺߊêWæÎïMõ5êÀéæÎÞ*à åjâÎÛ ðJÞ§ðÎò/ߊã§ä+ތæÎÛêWæ•åjâÎۀߊê åžÛºä{ã*åžòÎäjÛÞlÝï§ä{ãlðÎâ´ßŠã§öMÛßWêŠæÎï  Û HêÀ é !#"!$&% Ð  ï§ä{ãlðÎâ û ÞlÝébê º êWÝ@åžâÎÛºäžÛ¿Û úIêÀé`å{éãlæÜêW(æ '`ÛôUåžêWùŒÛ¿ÝEòÎæ/ô åjêŠÞ§* æ ),+8ýþ_û. ÿ -0/2143 Ï 356565738 9ébòMô{â åžâ/ã*åÛ-ã§ô{âÜÛçIïŒÛ ç = )3þ : @ÿ ?*)3þ ; 6ÿ =JãlæMçåjâÎÛ+äjÛ-ébò/ß åžêWæÎïÛçIïŒÛߊã§öMÛߊéãläžÛ¥çIêÀébåjêŠæ/ôUå B Ð A ò/ô{â¸ã : ÝEòÎæMôUåjêŠÞ§æ êÀé¥éžãlêÀçHåžÞöMÛC ã ( !D"!$&%E%FG!6%IHKJ  Ðó æ ì *í Lõ ÞŒéžãã§ßŠéjހêWæ­åžäjÞIçIò/ôºÛç¸åjâ/ÛæÎÞlåžêWތæ ާݘã§* æ M Sù*ãlߊò/ã*åžêWތæ C ï )´Þ§ÝZã€ï§ä{ãlðÎâ¬û êÀéã§* æ M !NOO%I$P(Q#HDR(J êWݘåjâ/ÛºäžÛ Ð  ïŒäžãŒô ÛºÝEòÎßeßÀãlöJۺߊêWæÎ ÛºúOêÀébåžé$ãlæÜêŠæŒåžÛºïŒÛº. ä S6õ;ôãlߊßWÛ-çHåžâÎC Û 8TPO UQV67HXW!QYHD°Þ§Z Ý )Fõ;ébÞåjâ/ãlåî¿êW[æ /\)3þ : 7ÿ 3!)3þ ; !ÿ 9]^S`_ î°ã*È@HKQKTsW7Q#R(J %Iª²"U6Í %IHD6HKQVRO$kW%F G86%IHKJkÁ)(íE"UR(. Ï(ï jÎÛð•T46JfQÊT4´8­PTC¯ñR GUQVOHKJ•!̬"!8ROh û ‰ (J ã§æ/ç ý Š þ1û ‰ ÿ õ€ý Š þ1û Š ÿô£Ò/ ;· ýþD¯¸ÿ+(ÕCþ ; ÿÓÑS ‰ rfS Š 9 Ð âÎÛäjÛºÝEÞ§äžÛ§õ¶Õ Û úOåjÛæ/çÎéåjÞ ã ébåjäžÞ§æ/ï§ßŠñ¸ÝEÛßWêÀô êWåjތò/é+ߊã§öMÛßWêŠæÎïÜÞlÝp¯ à˜ê åžâ•ô{â/ã§äžãŒôUåžÛºäžêŠébåjêÀô S ‰ rfS Š Ð  â/ÛºÞ§äžÛºîÿ ÐWÏ õIà@ÛæÎÞ*à»Þ§öIå{ãlêŠæåjâÎÛ$æÎÛºúOå˜ôºÞ§äžÞ§ßŠßŠã§äjñ Ð ó æ€ßWêŠï§â­å˜ÞlÝ(è8Ûî¿î¿ã όÐWÏ ãlæMçÁ éjê\é #Ä&ê  ë“ÆKë“ƬǤW8W7$&hiiQÊT4(¾Q ûËí²HXW¦W7Q#R(J %Iª."U6%IHDUHKQVRO$kW¬!­4T*È@HKQKT>W7Q#8ROJ %Iª."U6Í %IHD6HKQVRO$kW%F G!U%IHKJkC)OíŸ"URO¬î²£ Ï 38jÎóðµ Ð  â/ÛºæåjâÎÛ$ùŒÛºäjåjÛºúߊã§öMÛßWêŠæÎï´Õ­+ ýþD ¯ ÿl- m  o!ö÷[ù éjò/ô{âåjâMã*å £ ;

‰ ãlæ/ç Ï  ] ]fµ Õ6þD×Jÿl£ Ø /13) õ þ í ÿdr*I þ_ûÿ þ²? Ï ÿUõ êê ݶݶ×Á Á × £ P õ ±   ] ô î ]€ c ã§æ/ç Ï  ] ]fµ í ; ; Ûºú­åžÛºæ/ç/é¾åjÞã+â/ãläžî¿Þ§æÎêŠÞ§òMé@ßÀãlöJۺߊêŠæÎï¿ÞlÝ@¯ õIà˜âÎÛºäžÛËI þ1ûÿêÀé¾åjâ/Ûébêº Û$Þ§ÝFû Ð  âÎÛ ðÎäžÛºùOêŠÞ§ò/éåjâ/ÛºÞ§äžÛºî Ûã§éjêŠßWñ Ûºú­åžÛºæ/ç/éåjޕåjâÎÛ ïŒäžã§ðÎâ/éà˜ê åžâ»ÝEÛßWêÀô êWåjތò/é€ßÀãlöJۺߊêWæÎï­é6¥  

âÎÛæ/ô یõIà@Ûã§æ/ãlߊާïŒÞ§ò/éjߊñ+ތöIåžã§êWæåjâ/Û$æÎÛ úOåäžÛéjòÎß å Ð . èLÂkéPêOÂkà ë ÆdƴǤW8W7$&hiÁQKTPOQZÖ û HXW´"U6%IHD6HKQ«R($kWi!­4TÈ@HKQÊTÑ´"UU%¨HDUHKQVRO$kW%FG!6%IHKJ ¸) ï (J
)3þ ; ‰ ÿ[rf)3þ ; 0/ § Š ÿô£ a SËr`¿

)3þ ; í ÿ r>)3þ ; íF§ ‰ ÿô£ a SËr*î•r>bÎõIêWÝ@±]îô]¿ SËr Ï õ à˜âÎêÀô{âð/äjÞ*ùOêÀçIÛé@åjâ/ÛébÛºåÞlÝ;êŠæ­åjÛï§Ûºä{éB/ a SBr*î•r>b´+ Ï f ] îl]>¿kSËr Ï 9 Ð

ø ތäåjâÎÛ$äžÛºî°ã§êWæÎêŠæÎï+ÛçIïŒÛéÞ§Ý3û¿õÎà¾Ûâ/ãAù§Û~åjâMã*å )3þ : í„ ÿ r>)3þ : í§ ‰ ñ ÿ £ ¿kS¤r*î{õÎêWÝ Ï ]îô]Ñ SËr>4¥ )3þ : ‰ dÿ r>)3þ : 0/ § Š ñ ÿ £  S¤r>b4¥ )3þ ; ‰ [ÿ rf)3þ ; Š ñ ÿ £  S¤r*¿P¥ )3þ : í ÿ r>)3þ : í§ ‰ ñ ÿ £  S¤r*î{õÎêW@Ý  SBr>b´]fîl]f¿kS¤r ϧР©Ûæ/ô یõOåjâÎÛ$êŠæ/çIòMô ÛçÛ-çIï§Û$ßÀãlöJÛºßÀé˜ãläžÛ$çIêŠébåjêŠæ/ô åêW¦ æ m1 / §  Ð ø êWæMãlߊßWñŒõIæÎÞlåžêŠôºÛåjâ/ãlåZû êÀéöÎêŠð/ãläjåjêWåjÛ$à˜êWåjâ€ð/ã§äbåžê åžÛéjÛ å{é ý ‰ £Ò/ : Š í. ‰ 3 ; Š í. ‰ + : ‰ £ ; ‰ ãlæMç Ï ]fîl] SËr Ï 9 ãlæMç ý Š £Ò/ : Š íV3 ; Š í@+ Ï ]fîô],SËr Ï 9 Ð âOò/éPõ )êÀéã¿é`åžäjތæÎï§ßŠñ¿ÝEۺߊêŠôºê åžÞ§ò/éßŠã§öMÛßWêŠæÎï¿ÞlÝ(û à˜êWåjâô{â/ã§äžãŒôUåžÛºäžêŠébåjêÀôt¿kS Ð á¾ã§éjÛËb4+  éžéjòÎî¿B Û ¹ £,¿kS¤r*¿/õIà˜â/ÛºäžË Û SêÀéã+ðMÞ­ébêWåjêŠù§ÛêŠæŒåžÛºïŒÛºä-õOåjâÎÛæßŠÛ å î[? Ï õ ê @Ý ×q£ : Š í. ‰ ãlæMç Ï ]fîl],S¤r>4¥ ¿ Ë S ` r “ î * r / b õ ê @Ý ×q£ : Š í ãlæMç Ï ]fîl],S•¥ ûüü ®SËr>®Îõ ê @Ý ×q£ : § ¥ ç SËrf±]fîô],SËr Ï ¥ )3Êþ ×6lÿ £ ý üü ¿ SËr`î“r>®Îõ ê @Ý ×q£ : ŠŠ í/ ãlŠ æMÁ ¿ SËr*b/õ ê @Ý ×q£ : 0/ §  ¥ üü SËr`î{õ ê @Ý ×q£ ; í. ã§æ/¦ ç ¬]fîE]SËrf&¥ üü  SËr`î“r  õ ê @Ý ×q£ ; ŠŠ í 㧉 æ/ç Ï ]fîE]SËrf Ð þ á@Þ§æMéb2Û Œò/Ûºæ­åjߊñ§õ§åjâÎÛù§ÛäbåžÛ ú¿ßŠã§öMÛߊé¾ãläžÛZçIêÀé`åžêWæMôUåà˜ê åžâÎÞ§òIå¾ò/éjêWæÎï¥åjâÎÛêŠæŒåžÛºïŒÛºä{é ®S¤r*¿ãlæ/ ç ®SËr  õIà˜âÎêÀô{âébâÎÞ*àé@åžâ/ã*p å )HêŠé˜êŠOæ '`Û-ôUåjêŠù§Û Ð ;Þ°ù§ÛºäžêWÝEñ°åjâ/ãlBå ) êŠé˜êŠæ/çIÛÛçã¿ébåjäžÞ§æÎïŒßWñ°ÝEÛßWêÀô êWåjތò/éßÀãlöJۺߊêŠæÎï°ÞlÝ(û¿õ/à¾Ûô ތ¬î  ðÎòIåžÛåjâÎÛ$êŠæ/çIòMô ÛçÛ-çIï§ÛßÀãlöJÛºßÀé Ð å / ; ‰ ; 0/ §  9dõ/ ; í ; íF§ ‰ + ±]>îE]f¿kSrsb&9­õŒà¾ÛZâ/ãAùŒÛ ø ތäeåžâÎÛZÛ-çIï§Û-éêŠæ°åjâ/ÛéjÛ p åjâMã*å )3þ ; ‰ [ÿ rf)3þ ; 0/ §  ôÿ £ a SËr a

‹ ŒOŽŒ2 K‘“’•”&‹P–b&9 ø ތäåjâÎÛ$äžÛºî°ã§êWæÎêŠæÎï+ÛçIïŒÛéÞ§Ý3û¿õÎà¾Ûâ/ãAù§Û~åjâMã*å

Ð

)3þ : ‰ ÿ[rf)3þ : 0/ §  ÿ £ ¿ SËr>b4¥ )3þ : í ÿdr>)3þ : í§ ‰ ÿ £ SËr*î•rf®IõIêWÝ Ï ]îô],S•¥ )3þ : 3/ § á ÿ[rf)3þ : 0/ §  ÿ £  SËr*¿P¥ 3) þ : í ÿdr>3) þ : í§ ‰ ÿ £ ¿ SËr*î•r*¿/õIêWÝ@ SËr Ï ]îô],SËr>4¥ 3) þ ; ‰ ÿ[rf3) þ ; Š ÿ £  SËr ¥ 3) þ : _í ÿdr>3) þ : í§ ‰ ÿ £ ¿ SËr*î•rf®IõIêWÝ@ SËr*b´]îô]¿ SËr> Ð ©Ûæ/ô یõOåjâÎÛã§öMÞ*ùŒÛêWæ/çÎò/ô Û-çÛ-çIï§ÛßÀãlöJÛºßÀé˜ãläžÛ$çIêŠébåjêŠæ/ô åêWæ¦m1 / § 1 Ð ø êWæMãlߊßWñŒõIÞ§ö/éjÛºäžù§Û~åjâ/ãlåZû êÀéöÎêŠð/ãläjåjêWåjÛ$à˜êWåjâ€ð/ã§äbåžê åžÛ$ébÛºåžé ãlæMç

ý

‰ £Ò/ : Š í . ‰ 3 ; Š í. ‰ + : ‰ £ ; ‰

ãlæMç Ï

]fîl] SËr>&9

ý Š £Ò/ : Š í 3 ; Š í + Ï ]fîô],SËrfk9 Ð âOò/éõ )êŠéeã$ébåjäžÞ§æ/ï§ßŠñÝEۺߊêÀô êWåjÞ§òMé(ߊã§öMÛßWêŠæÎï$Þ§ÝCû¨à˜êWåjâô{â/ã§äžãŒôUåžÛºäžêŠébåjêÀôl¿kSlr Ð âÎÛäjÛºÝEÞ§äžÛ§õIà˜êWåjâåjâÎÛ-ébÛåžâÎäjÛÛôºãŒébÛ-é@êŠæ€â/ãlæMçCõÎÞ§ò/ä˜ðÎäjÞOÞ§ÝFô ތæ/ô ߊò/çIÛ-é Ð ç b ÐŠÏ õAà¾Ûâ/ãAùŒÛ(åjâÎÛô ތäjތßWßÀãläžñðÎäžÞ§î¿êÀébÛ-ç$êŠæ$åjâÎÛ¾ð/ã§äžã§ï§ä{ãlðÎâ ó æ¥ßŠêŠï§â­å;Þ§ÝPâÎۺތäjÛî¿Lé  Ð ¿ZãlæM. ðÎäžÛôºÛçIêŠæÎïåjâÎÛã§öMÞ*ùŒÛåjâÎÛÞ§äžÛºî Ð éjê\é#Ä&ê [ÆXë Æ4,Ê"Ÿ¹ e¿sHXW²6N\6J ï QKTPUJ¸QKTPB!­4TÁÇph´(% #þ º » 3 ; 3x¿OËÿ HXW¤W7Q#R(J %IªŽ"U6%IHD6HKÍ

QVRO$kW6Î

 âÎÛÞ§äžÛºîòb ÐŠÏ åjÞ±âÎۺތäjÛîò Ð ¿Mõ§åžâÎÛºæà@Û~Þ§öIå{ãlêŠæ°åjâÎÛZÝEާߊßWÞ*à˜êŠæÎï¥ôºÞ§äžÞ§ßŠßŠã§äjñ Ð ó Ý5à¾ÛZãlð/ðÎßWñi jé ê\é#Ä&ê5[Æ[Æ6,Ê" ¹ ef¿±HXW6N\6J´(J