IOSR Joural of Mathematc (IOSR-JM) e-issn: 78-578 p-issn: 19 765X PP 7-15 wwworjouralorg Some dtace ad equece a weghted graph Jll K Mathew 1, Sul Mathew Departmet of Mathematc Federal Ittute of Scece ad Techology, Agamaly, Kerala, Ida-68577 Departmet of Mathematc Natoal Ittute of Techology Calcut, Kerala, Ida-6761 ABSTRACT: I a weghted graph, the arc are maly clafed to, ad I th artcle, ome ew dtace ad equece weghted graph are troduced Thee cocept are baed o the aboe clafcato Wth repect to the dtace, the cocept of cetre ad elf cetered graph are troduced ad ther properte are dcued It proed that, oly partal block wth ee umber of ertce ca be elf cetered Ug the equece, a characterzato for partal block ad precely weghted graph (PWG) are obtaed Keyword: dtace, dtace, partal block, partal tree, PWG I Itroducto ad prelmare Graph theory ha ow become a major brach of appled mathematc ad t geerally regarded a a brach of combatorc Graph a wdely ued tool for olg a combatoral problem dfferet area uch a geometry, algebra, umber theory, topology, optmzato ad computer cece Mot mportat thg whch to be oted that, ay problem whch ca be oled by ay graph techque ca oly be modeled a a weghted graph problem Dtace ad cetre cocept play a mportat role applcato related wth graph ad weghted graph Seeral author cludg Body ad Fa [1,, ], Broerma, Zhag ad L [14], Sul Mathew ad M S Sutha [8, 9, 1, 11, 1, 1] troduced may coectty cocept weghted graph followg the work of Drac [4] ad Grotchel [5] I th artcle we troduce three ew dtace cocept weghted graph Thee cocept are dered by ug the oto of coectty weghted graph I a weghted graph model, for example, a formato etwork or a electrc crcut, the reducto of flow betwee par of ode more releat ad may frequetly occur tha the total drupto of the etre etwork [7, 1, 11] Fdg the cetre of a graph ueful faclty locato problem where the goal to mmze the dtace to the faclty For example, placg a hoptal at a cetral pot reduce the loget dtace the ambulace ha to trael Th cocept our motato A weghted graph are geeralzed tructure of graph, the cocept troduced th artcle alo geeralze the clac dea graph theory A weghted graph G : ( V, a graph whch eery arc e aged a oegate umber w (, called the weght of e [1] The dtace betwee two ode u ad G defed ad deoted by d = m{ w( / P a u path G } [1, 6] The eccetrcty of a ode u G defed e P ad deoted by e ( = max{ d( u, / ay other ode of G } [6] The mmum ad maxmum of the eccetrcte of ode are repectely called radu, r (G) ad dameter, d (G) of the graph G [6] A ode u called cetral f e ( = r( G) ad dametral f e ( = d( G) [6] The ubgraph duced by the et of all cetral ode called the cetre of G, c (G) [6] G called elf cetered f t omorphc wth t cetre [6] I a weghted graph G : ( V, the tregth of a path P = e e e 1 1 defed ad deoted by S P) = m{ w( e ), w( e ), w( e ), w( e )} [1] The tregth of coectede of a par of ( 1 ode u ad G defed ad deoted by [11] A u path P called a troget path f ( P) = CONN CONN G = max{ S( P)/ P a u path G } S G [1] A ode w called a partal cut ode ( p - cut od of G f there ext a par of ode u, G uch that u w ad 1 jllkmathew@redffmalcom m@tcac Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala 7 Page
Some dtace ad equece a weghted graph CONNG w CONNG [11] A graph wthout p - cut ode called a partal block( p - block) [11] It alo proed [11] that a ode w a weghted graph G a p -cut ode f ad oly f w a teral ode of eery maxmum pag tree A coected weghted graph G : ( V, called a partal tree f G ha a pag ubgraph F : ( V, E, whch a tree, where for all arc e = of G whch are ot F, we hae CONN G > w( [11] A arc e = called -trog f CONN G e < w( ad -trog f CONN G e = w( ad a arc f CONN > w( A arc called trog f t ether - trog or - trog [11] A precely G e weghted graph (PWG) a complete weghted graph G ( V, W 1, W ), where the weght fucto W 1 :V ad W : E W = W ( W ( ), where deote the mmum[9] ( 1 1 : uch that for eery edge e = of G, we hae II, ad trog dtace I th ecto, we ge the defto of the dtace alog wth a example Defto 1 Let : ( V, G be a weghted graph Let u ad be ay two ode of G The the dtace betwee u ad defed ad deoted by d = m w( e P - trog path betwee u ad, f u =, f there ext o - trog u path Clearly d atfe all the axom of a metrc 1 d for all u ad d f ad oly f u = = d = d (, for all u ad 4 d d w) d ( w, for all u, ad w G Hece V, d ) a metrc pace ( Defto Let : ( V, where P ay G be a weghted graph Let u ad be ay two ode of G The the dtace betwee u ad defed ad deoted by d = m w( where P ay e P - trog path betwee u ad, f u =, f there ext o - trog u path Clearly d atfe all the axom of a metrc 1 d for all u ad u = d = f ad oly f d = d (, for all u ad 4 d d w) d ( w, (, d for all u, ad w G Hece V ) a metrc pace Defto Let : ( V, G be a weghted graph Let u ad be ay two ode of G The the trog dtace betwee u ad defed ad deoted by d = m w( where P e P ay trog path betwee u ad, f u =, f there ext o trog u path Clearly d atfe all the axom of a metrc 1 d for all u ad Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala 8 Page
d f ad oly f u = = d = d (, for all u ad 4 d d w) d ( w, Hece V, d ) a metrc pace ( for all u, ad w G Some dtace ad equece a weghted graph I the followg example (Fgure 1: ), we fd that thee three dtace are geerally dfferet Example 1 Defto 4 The - dtace matrx of a coected weghted graph G : ( V,, V = a quare matrx of order defed ad deoted by D ( d ), where d = - dtace betwee the ertce ad j = j Note that D a ymmetrc matrx I the ame maer D, the - dtace matrx ad j D S, the trog dtace matrx ca be defed The three dtace matrce of the aboe weghted graph (Fgure 1) are ge below 7 7 7 15 4 D = 7 5 11, D 15 5 16 4 11 16 =, Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala 9 Page
Some dtace ad equece a weghted graph D 7 1 = 8 14 7 1 15 7 1 7 1 4 1 7 5 11 8 15 1 5 16 14 7 4 11 16 Remark 1 I ay coected weghted graph G : ( V, there ext a trog path betwee ay two ode [1] Hece all the elemet the trog dtace matrx are fte That d < III Strog cetre ad elf cetered graph I th ecto, we troduce the cocept of cetre wth repect to the dtace whch are defed the aboe ecto Alo we preet a characterzato for a trog elf cetered weghted graph, whch ald for both ad - elf cetered weghted graph Defto 1 The eccetrc ty of a ode u G defed ad deoted by e ( = max{ d / V, d < } Smlarly the eccetrc ty ad trog eccetrc ty are defed below e ( = max{ d / V, d < } e ( = max{ d / V, d < } Defto A ode called the Set of all eccetrc ode of u deoted by Smlarly we ca defe eccetrc ode of u f e ( = d * u eccetrc ode ad trog eccetrc ode Defto Amog the eccetrc ty of all the ode of a graph, the mmum called the radu of G It deoted by r (G) That r = m{ e ( / u V} Alo the radu of G, deoted ad defed by r = m{ e ( / u V} ad the trog radu of G deoted ad defed by r = m{ e ( / u V} A the radu the mmum eccetrcty, the maxmum eccetrcty called the dameter of the graph Defto 4 Amog the eccetrc ty of all the ode of a graph, the maxmum called the dameter of G t deoted by d (G) That d = max{ e ( / u V} Alo the dameter of G,deoted ad defed by d = max{ e ( / u V} ad the trog dameter of G deoted ad defed by d = max{ e ( / u V} Defto 5 A ode u called cetral f e ( = r, ad called cetral e ( = r ad trog cetral f e ( = r f Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala 1 Page
Defto 6 The ubgraph duced by the et of all deoted by < C > Aalogouly the cetre Some dtace ad equece a weghted graph cetral ode called the cetre of G It of G the ubgraph duced by the et of all cetral ode of G, deoted by < > Alo the trog cetre of G deoted by < > C ad defed a the ubgraph duced by the et of all trog cetral ode Example 1 C e (a) =15, e (b) = 8, e (c) = 15, e (d) =, e (a) = 4, e (b) =, e (c) = 4, (d) = e (a) = 7, e (b) = 9, e (c) = 8, e (d) = 9, r (G) = 8, r (G) =, (G) = 7 r b the - cetral ode, d the - cetral ode ad a the trog cetral ode The followg propoto tral ad ald for both ad dtace Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala e, Prepoto 1 I ay coected weghted graph G wth at leat oe trog arc cdet o eery ertex u, the e ( e( for eery ode u G Where e ( the eccetrcty of u the uderlyg graph of G A the clacal cocept of dtace we hae the followg equalte We omt ther proof a they are obou Theorem 1 Let G ( V, be a coected weghted graph, the r d r r d r r d r Defto 7 A weghted graph G called elf cetered f G omorphc wth t cetre ad elf cetered f G omorphc wth t cetre Alo G called trog elf cetered f G omorphc wth t trog cetre The followg theorem true for both ad trog elf cetered graph Theorem Let G : ( V, be a coected weghted graph uch that there ext exactly oe trog arc cdet o eery ode ad that all the trog arc are of equal weght, the G trog elf cetered Proof Ge that all the ode of G are cdet wth exactly oe trog arc, ad all the trog arc are of equal weght That mea f e = a trog, the there wll be o other trog arc cdet o u ad Hece e ( = w( = e ( By th ame argumet we get th ame equalty for ay other trog arc Thu 11 Page
Some dtace ad equece a weghted graph e ( = w( for eery ode u G Th proe that G - elf cetered The ext theorem a characterzato for thee elf cetered graph Theorem A coected weghted graph G : ( V, trog elf cetered f for ay two ode u ad uch that u a trog eccetrc ode of, the hould be oe of the trog eccetrc ode of u Proof Frt aume that G trog elf cetered Alo aume that u a trog eccetrc ode of Th mea e ( = d (, Sce G trog elf cetered, all ode wll be hag the ame trog eccetrcty Therefore e ( = e ( From the aboe two equato we get, e ( = d(, = d( u, Thu e ( = d( u, That a trog eccetrc ode of u Next aume that u a trog eccetrc ode of The a trog eccetrc ode of u Thu e ( = d e ( = d (, u d = d (, u e ( = e ( u, ad ) But ) Therefore ), where u ad are two arbtrary ode of G Thu all ode of G hae the ame trog eccetrcty, ad hece G trog elf cetered Smlarly, we ca proe th reult for ad elf cetered graph IV Self Cetered Partal Block I th ecto, we preet ome eceary codto for the - cetre of partal block Alo we cluded ome characterzato for - elf cetered partal block I the followg two theorem, we characterze partal cut ode of a weghted graph Theorem 41 If a ode commo to more tha - trog arc, the t a partal cut ode Proof I [11] t proed that, f z a commo ode of at leat two partal brdge, the t a partal cut ode Alo [1], we ca ee, a arc e a weghted graph G partal brdge f ad oly f t - trog Hece the proof completed Example 41 I the aboe example (Fgure), the ode a commo to two - trog arc, ad by theorem 41, t a partal cut ode ( CONN ( b, d) = < CONN ( b, d) = ) But the ode c a cut ode ad hece a partal cut G a ode ee though t ot cdet wth ay - trog arc The codto the aboe theorem ot uffcet If we retrct the uderlyg graph (A graph hag o cut od, the th codto wll be uffcet Theorem 4 If G : ( V, a coected weghted graph uch that ode the commo ode of at leat two - trog arc G * G of G to be a block * G a block, the eery partal cut Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala 1 Page
Proof Suppoe that G : ( V, a coected weghted graph uch that Some dtace ad equece a weghted graph * G a block Let u be a partal cut ode of G We hae to proe that u a commo ode of at leat two - trog arc If poble, uppoe the cotrary Cae 1 Whe u cdet wth exactly oe - trog arc Let e = be the - trog arc whch cdet o u Sce G ha o cut ode, u caot be a cut ode, ad hece t le o a cycle, ay, C Let w be the other ode C whch cdet o u By the aumpto, ( u, w) caot be - trog Therefore the weght of the arc ( u, w) ca be at the mot the mmum of all the weght of arc C other tha that of the arc ( u, Hece the deleto of the ode u from G, wll ot reduce the tregth of coectede betwee ay other ode It a cotradcto to the fact that u a partal cut ode Hece our aumpto wrog Cae 11 Whe u cdet wth o - trog arc Sce G ha o cut ode, u caot be a cut ode, ad hece u le o a cycle, ay, C Sce u cdet wth o - trog arc, the two arc whch are cdet wth u wll hae the mmum weght amog the arc C Therefore u caot be a partal cut ode of G a t doe ot reduce the tregth of coectede betwee ay par of ode of G Th a cotradcto to our aumpto that u a partal cut ode of G So our aumpto wrog Hece all the cae, we hae proed that the partal cut ode u cdet wth at leat two - trog arc I the followg theorem, we ca ee oly - trog arc are preet the - cetre of partal block Theorem 4 The - cetre of a partal block G cota all - trog arc wth mmum weght Proof Suppoe that G : ( V, a partal block Therefore G ha o partal cut ode We kow that, f a ode u a coected weghted graph commo to more tha oe - trog arc, the t a partal cut ode [1] A G free from partal cut ode, at mot oe - trog arc ca be cdet o eery ode of G Thu the - eccetrcty, e of a ode u the weght of the - trog arc cdet o u So the - radu of G, that r (G) the weght of the mallet - trog arc Hece the - cetre of G, < > cota all - trog arc of G wth mmum weght Th complete the proof of the theorem I the ext theorem, we preet a characterzato for - elf cetered partal block Theorem 44 Let : ( V, - elf cetered f ad oly f the followg codto are atfed 1 k = ( V )/ All the - trog arc hae equal weght Proof Let : ( V, G be a partal block wth exactly k umber of - trog arc The G G be a partal block wth exactly k umber of - trog arc Frt uppoe that, G - elf cetered We hae to proe codto 1 ad Sce G a partal block, t ha o partal cut ode Alo f a ode a weghted graph commo to more tha - trog arc, the t a partal cut ode [1] So a G depedet of partal cut ode, o ode G ca be a ed ode of more tha oe - trog arcsce G - elf cetered, the eccetrcty of each ode, e ( = w for eery u G Thu there ext exactly oe - trog arc cdet o eery ode of G Moreoer the aboe equalty hold oly whe all the - trog arc are of equal weght Therefore codto 1 ad are true Coerely uppoe that codto 1 ad hold We hae to proe that G - elf cetered That mea we hae to proe that e ( = e ( for eery u, G Sce G a partal block, each ode of G adjacet wth at mot oe - trog arc Alo by codto 1, ( V )/ = k Thee two codto are C Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala 1 Page
Some dtace ad equece a weghted graph multaeouly atfed oly whe exactly oe - trog arc cdet o each ode By codto, all the - trog arc are of equal weght Thu e ( = weght of the - trog arc cdet o u = weght of the - trog arc cdet o = e ( Th true for all ode G Th proe that G - elf cetered The followg reult a corollary of the aboe theorem It alo ued a a eay check for - elf cetered partal block Corollary 41 If a partal block G : ( V, - elf cetered, the V ee That mea there doe ot ext a elf cetered partal block wth odd order Proof Let ( V, the aboe theorem, k = ( V )/, whch mple that V = k, a ee teger Th proe the corollary G be a elf cetered partal block Let there be k umber of - trog arc The by V Sequece a weghted graph I th ecto, we troduce three type of equece a weghted graph ad make a characterzato of PWG ug the - equece Defto 51 Let G : ( V, be a coected weghted graph wth V = p The a fte equece G) = (,,,, ) called the - equece of G f = umber of trog arc cdet o ( 1 p ad =, f o trog arc are cdet o If there o cofuo regardg G, we ue the otato tead of (G) Defto 5 Let G : ( V, be a coected weghted graph wth V = p The a fte equece G) = (,,,, ) called the - equece of G f = umber of trog arc cdet o ( 1 p ad =, f o trog arc are cdet o If there o cofuo regardg G, we ue the otato tead of (G) Defto 5 Let G : ( V, be a coected weghted graph wth V = p The a fte equece (,,,, ) S called the trog equece of G f = umber of or trog arc = 1 p cdet o ad =, f o ad trog arc are cdet o If there o cofuo regardg G, we ue the otato I the followg example (Fgure4), we fd thee equece Example 51 = (1,,,1) = (,1,,1) = (1,,,) S tead of (G) S Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala 14 Page
Some dtace ad equece a weghted graph A a PWG ca hae at mot oe - trog arc ad ha o arc[9], we hae the followg prepoto Prepoto 51 If G ( V, W 1, W ) a PWG, the ether (G) = (,,,,) : (G) = (1,1,,,) ( V, W 1, W ) Prepoto 5 If G : ) a PWG, the ( G ether Prepoto 5 If : ( V, W 1, W ) e ( e ( = e ( or C or ( C 1) G a PWG, the for eery ode u G, we hae = = C Alo, VI Cocluo I th artcle, three ew dtace ad three ew equece weghted graph are troduced A reducto tregth betwee two ode more mportat tha total dcoecto of the graph, the author made ue of the coectty cocept defg the dtace ad equece A pecal focu o elf cetered graph ca be ee a they are appled wdely I ecto 4 ad 5, dcuo about the partal block tructure ad Precely weghted graph tructure hae made a they hae got may practcal applcato Referece [1] J A Body, H J Broerma, J a de Heuel ad H J Veldma, Heay cycle weghted graph, Dcu Math Graph Theory,, pp 7-15 () [] J A Body, G Fa, Cycle weghted graph, Combatorca 11, pp 191-5 (1991) [] J A Body, G Fa, Optmal path ad cycle weghted graph, A Dcrete Mathematc 41, pp 5-69 (1989) [4] G A Drac, Some theorem o abtract graph, Proc Lodo Math Soc (), pp 69-81 (195) [5] M Grotchel, Gaph wth cycle cotag ge path, A Dcrete Mathematc 1, pp -45 (1977) [6] Frak Harary, Graph Theory, Addo Weley, New York (1969) [7] Joh N Mordeo ad Premchad S Nar, Fuzzy Graph ad Fuzy Hypergraph, Phyca-erlag Hedelberg () [8] Sul Mathew, M S Sutha, Bod graph ad fuzzy graph, Adace Fuzzy Set ad Sytem 6(), pp 17-119 (1) [9] Sul Mathew, M S Sutha, O totally weghted tercoecto etwork, Joural of tercoecto etwork 14,(1) [1] Sul Mathew, M S Sutha, Some Coectty cocept weghted graph, Adace ad Applcato Dcrete Mathematc 6(1), pp 45-54 (1) [11] Sul Mathew, M S Sutha, Partal tree weghted graph I, Proyeccoe Joural of Mathematc, 16-174 (11) [1] Sul Mathew, M S Sutha, Cycle coectty Weghted graph, Proyeccoe Joural of Mathematc, 1, pp 1-17 (9) [1] Sul Mathew, M S Sutha, Partal tree weghted graph -I, Proyeccoe Joural of Mathematc,, pp 16-174 (11) [14] S Zag, X L, H Broerma, Heay path ad cycle weghted graph, Dcrete Math, pp 7-6 () Iteratoal Coferece o Emergg Tred Egeerg ad Maagemet 15 Page Sree Narayaa Gurukulam College of Egeerg, Kolechery, Erakulam, Kerala