International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 ISSN The Maximum Eccentricity Energy of a Graph

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1 Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN Th axmm Ecccy Egy of a Gaph Ahmd Na ad N D o Absac I Ths pap w odc h cocp of a maxmm cccy max oba som coffcs of h chaacsc poyoma of a cocd gaph G ad P G of h maxmm cccy max of G W aso odc h maxmm cccy gy E of a cocd gaph axmm cccy gs of som wow gaphs a obad Upp ad ow bods fo E a sabshd I s show ha f G s a sfcd ga gaph wh dam h D s a maxmm cccy ga of G ad E DE oo s aso show ha f h maxmm cccy gy of a gaph s aoa h ms b a Idx Tms: Dsac gaphs maxmm cccy max maxmm cccy gas maxmm cccy gy of a gaph INTRDUCTIN I hs pap a gaphs a assmd o b smp f ad ì cocd A gaph G V E s a smp gaph ha s a í hag o oops o mp ad dcd dgs As sa w î ohws do by V ad m E o h mb of cs Th gas of a max A assmd ad dgs a gaph G spcy o a x Î V h ocasg od a h gas of h gaph G op ghbohood of a gaph G dod N s h fo b h dsc gas of G s of a cs ha a adac o ad h cosd wh mpcy m m m spcy h ms of ghbohood of s N[ ] N È { } Th dg of a gas of A s cad h spcm of G ad x G s d N A gaph G s sad o b dod by ga gaph f d fo y Î V Th dsac d bw ay wo cs ad a gaph G s h gh of a mmm pah cocg hm o a x of G h cccy of a x s max{ d : Î V } Th ads of G s m{ : Î V } ad h dam of G s D max{ : Î V } Hc D fo y Î V A x a cocd gaph G s a ca x f wh a x a cocd gaph G s a ppha x f D A gaph G s cad a sfcd gaph f D fo y x Î V [] W do K C K ad K s h comp cyc sa ad comp bpa gaph spcy o gaph oao ad moogy w f o boos [5 ] Th cocp gy of a gaph odcd by I Gma [6] h ya 97 G b a gaph wh cs ad m dgs ad A a b h adaccy max of G wh p m m f Î E ; m As A s a symmc wh zo ac h gas of G a a wh sm qa o zo Th gy E of a gaph G s dfd o b h sm of h abso as of h gas of G E o mo das o h mahmaca aspcs of h hoy of gaph gy w f o [ 7 ] ad h fcs h Pof C Adga ad mha [] ha dfd h maxmm dg gy E of a gaph G whch dpd o h maxmm dg max of G G b a smp gaph wh cs Th h maxmm dg max of a gaph G dfs as wh 6 hp://wwwsog

2 Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN d ìmax{ d d } í î If ohws Î E ; As s a symmc wh zo ac h h gas of G bg a wh sm qa o zo oad by hs pap w odc h cocp of a maxmm cccy max of a cocd gaph G ad oba som coffcs of h chaacsc poyoma P G of h maxmm cccy max of G W aso odc h maxmm cccy gy E of a cocd gaph G axmm cccy gs of som wow gaphs a obad Upp ad ow bods fo E a sabshd I s show ha f G s a sfcd ga gaph wh dam D h D s a maxmm cccy ga of E DE G ad oo s aso show ha f h maxmm cccy gy of a gaph s aoa h ms b a g THE AXIU ECCENTRICITY Th chaacsc poyoma of ENERGY GRAPH G s P G d I G V E b a smp cocd gaph wh cs ad b h cccy of a x Th maxmm cccy max of G dfg as wh ìmax{ } í î If Î E ; ohws Th chaacsc poyoma of h maxmm cccy max s dfd by P G d I wh I s h max of od Th maxmm cccy gas of G a h gas of c s a ad symmc wh zo ac h s gas a a mbs wh sm qas o zo W ab hm ocasg od Th maxmm cccy gy of a gaph G s dfd as Th h maxmm cccy max of G s G Th h maxmm cccy gas of G a Thfo h maxmm cccy gy of G s E G E To sa hs cocp w sdy h foowg xamps Examp G b h gaph g Examp G b a gaph g 6 hp://wwwsog

3 Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN c x y wh x { Î N : < } ad y { Î N > : } Th h maxmm cccy max of G s D c 5 c d G Th chaacsc poyoma of G s Th maxmm cccy gas of h maxmm cccy gy of E 65 PRPERTIE AXIU ECCENTRICITY ENERGY I hs sco w oba h as of som coffcs of h chaacsc poyoma of h maxmm cccy max ad sga som pops of maxmm cccy gas of a gaph G Thom G b a gaph of od ad P G c c c c b h chaacsc poyoma of h maxmm cccy max of G Th c c G s Poof Th poof s sma sp o h poof of Thom [] Rma I Thom Th sm x y s h mb of dgs h gaph G Nmb of ms h sm D P G G s fag gaph h c If of ad c of P G a G a c x y Thfo s qa o h mb of ags h gaph G Hc If h gaph s sga has zo ga h c Examp o h gaph G g h coffcs c é ê ë [ ] 9 G h c c h s o ag o h gaph G g h coffcs c of c of a c x y ù ú û ad [ ] [ ] 7 6 hp://wwwsog

4 Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN 955 c D [ ] [ ] [] 6 W d o h foowg cs foma Nwo's Idy s [9] o po h x hom B b a max o h fd of a mbs ad P c d I B c c c c b h chaacsc poyoma of B wh I s a max Th b Thom b h maxmm cccy h maxmm cccy gas of G c gas of Th h b h pos gas of G ad h mag a o pos Th x y wh E G x { Î N : < } ad c a agbac mbs so s h sm ad y { Î N > : } hc ms b g f E s aoa 6 D Thom 9 G b a gaph of od sz m ad ads Poof Th poof s cosqcs of Nwo's dy ad If G s h ags maxmm cccy ga of Thom Cooay 5 G b a sfcd gaph wh dam D ad b h maxmm cccy gas of G Th md Examp 6 b h maxmm cccy gas of a gaph G Th G h If K If G C h G K If s ì f s odd; í î f s h s Thom 7 If G s a sfcd ga gaph wh dam D h D s a maxmm cccy ga of G ad E DE Poof G b a sfcd ga gaph wh 6 hp://wwwsog dam Th D fo y Î V By h dfo of h a s qa o D ach aw of d I Th by pacg h fs aw of d I by h sm of a aw w g ha D s a faco of d I Ths D s a maxmm cccy ga of G c G s sfcd ga gaph foows by h dfos of adaccy ad maxmm cccy macs of G ha DA ad hc f s a ga of G h D fo y s a maxmm cccy ga of G s [] Ths E DE Thom If h maxmm cccy gy of a gaph G s aoa h ms b a g Poof G b a gaph of od ad G h m G Poof G b a gaph of od ad b h ags maxmm cccy ga of G Th fom [] w ì X AX ü max X ¹ í ý wh X s ay ozo î X X þ X s s aspos ad A s a max By s X J Th w ha ha co ad J J J J max{ } c ohws foows ha f Î E ad max{ } x y < G fo y Î V ad by Rma c

5 Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN x y m ha m p p K p E K p p foows ha p AXIU ECCENTRICITY ENERGY E TANDARD GRAPH I hs sco w sga h xac as of h maxmm cccy gs of som wow gaphs Thom K b h comp gaph wh cs Th E K Poof c h comp gaph K s sfcd ga gaph wh dam foows fom Thom 7 ha E K DE E K b h sa gaph of od E K Thom Th C b h cyc gaph wh Thom h maxmm cccy gy of C s E C ì p ï csc ï p ícsc ï ï p co ïî Poof c h cyc gaph gaph wh dam ha cs Th f º mod ; f º mod ; f º mod ; C s sfcd ga ë û D foows fom Thom 7 Poof K b h sa gaph wh x s ï wh s h ca x Th ï ï ï ï K ï E K ë ûe í ï ï Th chaacsc poyoma of K s P K Hc h maxmm cccy spcm of K s p K Thfo h maxmm cccy gy of K s E K Cooay Each pos g cccy gy of a sa gaph p p s h maxmm K s Poof c h maxmm cccy spcm of p 6 hp://wwwsog ì ï ï ï ï î ì p ï csc ï p ë ûe ícsc ï ï p co ïî Thom 5 gaph of od gy of K s Poof K fo K p s p s p cos p s f º mod ; f º mod ; f º mod ; f º mod ; f º mod ; f º mod ; b h comp bpa Th h maxmm cccy E K b h comp bpa gaph wh

6 6 hp://wwwsog x s Th K Th chaacsc poyoma of K s K P Hc h maxmm cccy spcm of K s K p Thfo h maxmm cccy gy of K s K E Dfo 6 [5] A dshp gaph fo s h gaph coscd by og cops of K gaph wh a commo x gaph has cs ad dam D Thom 7 b a fdshp gaph Th h maxmm cccy gy of s E Poof b h fdshp gaph wh x s Th Th chaacsc poyoma of s P Hc h maxmm cccy spcm of s p Thfo h maxmm cccy gy of s E Dfo Th cow gaph fo s h gaph wh x s } { ad dg s } : { ¹ Thfo V ad cocds wh h comp bpa gaph K wh h hozoa dgs mod Thom 9 b a cow gaph Th h maxmm cccy gy of s 6 E Poof b h cow gaph wh x s } { Th Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN 955

7 Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN 955 Th chaacsc poyoma of s P Hc h maxmm cccy spcm of p Thfo h maxmm cccy gy of E 6 s 5 BUND R AXIU ECCENTRICITY ENERGY I hs sco w sabshd pp ad ow bods fo h maxmm cccy gy of a gaph ma o ccad's bods fo gaph gy bods fo maxmm cccy gy a g h foowg wo homs Thom 5 G b a cocd gaph of od sz m ad ad dam of G spcy Th m E D m ad D b h ads ad h Poof Cosd h Cachychwaz qay ab By choos a b a ad b ad by Thom w g E G x y D G fo y Î V ad by Rma c x y m foows ha E D m foows ha Now sc E x y fo y Î V h E m x ads Th s E G m G c Thom 5 G b a cocd gaph G of sz m ad Poof c E G d < Usg h qay bw h ahmc ad gomc mas w g /[ ] Õ ¹ ¹ Hc by hs ad Thom w g /[ ] E Õ Õ ¹ Õ / /[ ] 6 hp://wwwsog

8 Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN 955 c x y Õ / m ad fo y cocd gaph foows by sg Rma ha E m m m d d ma o Koo ad ooâ s [] pp bod fo h maxmm cccy gy E of a gaph G s g h foowg hom / Thom 5 G b a cocd gaph of od m ad ads Th m m E D m Poof Cosd h Cachychwaz qay sz a b a b a ad b w ha x y By choosg Hc by Thom ad Rma w ha E md Thfo md E f x x m mo x o dcasg fco f x f x mps ha x md x md x c md m foows ha 9 md md ad hc Ths md f f Tha mas ad fom Thom 6 hp://wwwsog E Thfo by sg ha cocd gaph md f f m E md fo y m m mmd m m D m 6 CNCUIN I hs pap w odcd a w max of a cocd gaph G cad maxmm cccy max I dpds o h dg gaph ad cccy o s cs W obad som coffcs of h chaacsc poyoma of h maxmm cccy max I s show ha f G s a cocd sfcd ga gaph wh dam D h DA wh A s h adaccy max of G W a sd sdyg mahmaca aspcs of h maxmm cccy gy E of a gaph I s show ha f G s a sfcd ga gaph wh dam D h D s a maxmm cccy ga of G Upp ad ow bods fo h maxmm cccy gy of a gaph a fod I s show ha f h maxmm cccy gy s aoa mb h ms b a g mb I s possb ha h maxmm cccy gy ha w a cosdg hs pap may ha som appcaos chmsy as w as oh aas p Pobms Chaacz a gaphs G whch E E Chaacz a gaphs whch a hypgc ad hypogc d h maxmm cccy gy d h maxmm cccy gy fo som oh famy gaphs ad fo som opaos o gaphs d ohs bods fo h maxmm cccy gy REERENCE [] C Adga ad mha axmm Dg Egy of a Gaph I J Comp ah ccs [] J Ayama K Ado ad D As Eccc gaph dsc mah [] R B Bapa Gaphs ad acs Hdsa Boo Agcy [] R B Bapa ad Pa Egy of a gaph s a odd g B of K ah Assoc 9 [5] J A Body ad U R y Gaph Thoy pg B

9 Iaoa Joa of cfc & Egg Rsach Vom 7 Iss 5 ay6 IN 955 [6] I Gma Th gy of a gaph B aha schgsz Gaz 97 [7] I Gma X ad J Zhag Gaph Egy Eds: Dhm Emm b Aayss of Compx Nwos om Boogy o gscs WyVCH Whm 9 57 [] Haay Gaph Thoy Addso Wsy assachss 969 [9] D Kama A max poof of Nwo's ds ah agz 7 5 [] R Kaa R hash B N Dhamda ad R A Ramyash axmm dg gy of ca mash dd wos I J Comp App [] J H Koo ad V oo axma gy gaphs Ad App ah 6 75 [] X Y h ad I Gma Gaph Egy pg Nw Yo [] H üpoh Hadboo of acs Joh Wy ad s 6 hp://wwwsog

Existence of Nonoscillatory Solutions for a Class of N-order Neutral Differential Systems

Existence of Nonoscillatory Solutions for a Class of N-order Neutral Differential Systems Vo 3 No Mod Appd Scc Exsc of Nooscaoy Souos fo a Cass of N-od Nua Dffa Sysms Zhb Ch & Apg Zhag Dpam of Ifomao Egg Hua Uvsy of Tchoogy Hua 4 Cha E-ma: chzhbb@63com Th sach s facd by Hua Povc aua sccs fud