Antipodal Interval-Valued Fuzzy Graphs

Transcription

1 Internatonal Journal of pplcatons of uzzy ets and rtfcal Intellgence IN 4-40), Vol 3 03), ntpodal Interval-Valued uzzy Graphs Hossen Rashmanlou and Madhumangal Pal Department of Mathematcs, Islamc zad Unversty, Central Tehran ranch, Tehran, Iran Emal:hrashmanlou@yahoocom Department of ppled Mathematcs wth Oceanology and Computer Programmng, Vdyasagar Unversty, Mdnapore-70, Inda Emal: mmpalvu@gmalcom bstract Concepts of graph theory have applcatons n many areas of computer scence ncludng data mnng, mage segmentaton, clusterng, mage capturng, networks, etc n nterval-valued fuzzy set s a generalzaton of the noton of a fuzzy set Interval-valued fuzzy models gve more precson, flexblty and compatblty to the system as compared to the fuzzy models In ths paper, we ntroduce the concept of antpodal nterval - valued fuzzy graph and self medan nterval-valued fuzzy graph of the gven nterval-valued fuzzy graph We nvestgate somorphsm propertes of antpodal nterval - valued fuzzy graphs Keywords: ntpodal nterval - valued fuzzy graph, Medan nterval- valued fuzzy graph, - tatus, - tatus Introducton The maor role of graph theory n computer applcatons s the development of graph algorthms number of algorthms are used to solve problems that are modeled n the form of graphs These algorthms are used to solve the graph theoretcal concepts, whch n turn are used to solve the correspondng computer scence applcaton problems everal computer programmng languages support the graph theory concepts The man goal of such languages s to enable the user to formulate operatons on graphs n a compact and natural manner ome of these

2 H Rashmanlou & M Pal languages are ) PNTREE: to fnd a spannng tree n the gven graph, ) GTPL: graph theoretc language, 3) GP: graph algorthm software package, 4) HINT: an extenson of LIP In 975, Zadeh [3] ntroduced the noton of nterval- valued fuzzy sets as an extenson of fuzzy sets [3] n whch the values of the membershp degrees are ntervals of numbers nstead of the numbers Interval- valued fuzzy sets provde a more adequate descrpton of uncertanty than tradtonal fuzzy sets It s therefore mportant to use nterval-valued fuzzy sets n applcatons, such as fuzzy control One of the computatonally most ntensve parts of fuzzy control s defuzzfcaton [8] The fuzzy graph theory as a generalzaton of Euler s graph theory was frst ntroduced by Rosenfeld [] n 975 The fuzzy relatons between fuzzy sets were frst consdered by Rosenfeld, who developed the structure of fuzzy graphs obtanng analogues to several graph theoretcal concepts Later, hattacharya [7] gave some remarks on fuzzy graphs, and some operatons on fuzzy graphs were ntroduced by Modeson and Peng [9] The complement of a fuzzy graph was defned by Mordeson [0] and future studed by untha and Kumar [0] Recently, kram et al ntroduced the concepts of bpolar fuzzy graphs, nterval - valued fuzzy graphs, strong ntutonstc fuzzy graphs n [-5] Taleb and Rashmanlou [5] studed propertes of somorphsm and complement on nterval-valued fuzzy graphs Lkewse, they defned somorphsm and some new operatons on vague graphs [6-7] Rashmanlou and Jun defned complete nterval - valued fuzzy graphs [3] Taleb, Rashmanlou and Davvaz n [8] nvestgated some propertes of nterval- valued fuzzy graphs such as regular nterval- valued fuzzy graph, totally regular nterval-valued fuzzy graph and complement of nterval-valued fuzzy graph Taleb and Rashmanlou defned product bpolar fuzzy graphs [9] and somorphsm and complement on bpolar fuzzy graphs [30] Recently Rashmanlou and Pal defned rregular nterval - valued fuzzy graphs [] More results on nterval - valued fuzzy graphs [4], product nterval- valued fuzzy graphs and ther degrees [5], ntutonstc fuzzy graphs wth categorcal 08

3 ntpodal Interval-Valued uzzy Graphs propertes [7], some propertes of hghly rregular nterval - valued fuzzy graphs [8], study on bpolar fuzzy graphs [9] and nvestgated several propertes They defned sometry on nterval-valued fuzzy graphs [4] amanta and Pal ntroduced fuzzy tolerance graph [], rregular bpolar fuzzy graphs [3], fuzzy k- competton graphs and P-competton fuzzy graphs [4], bpolar fuzzy hypergraphs [] nce nterval-valued fuzzy set theory s an ncreasngly popular extenson of fuzzy set theory where tradtonal [0, ]-valued membershp degrees are replaced by ntervals n [0, ] that approxmate the unknown) membershp degrees; specfc types of nterval valued fuzzy graphs have been ntroduced and nvestgated In ths paper we ntroduce the concept of an antpodal ntervalvalued fuzzy graph and self medan nterval- valued fuzzy graph of the gven nterval-valued fuzzy graph We also nvestgate somorphsm propertes of antpodal nterval - valued fuzzy graphs The natural extenson of ths research work s the applcaton of nterval-valued fuzzy graphs n the area of soft computng ncludng neural networks, expert systems, database theory, and geographcal nformaton systems Prelmnares In ths secton we recall some basc concepts that are necessary for subsequent dscusson y a graph, we mean a par G V, E), where V s the set and E s a relaton on V The elements of V are vertces of We wrte G and the elements of E are edges of G x, y E to mean{ x, y} E, and f e xy E, we say x and y are adacent ormally, gven a graph G V, E), two vertces x, y V are sad to be neghbors, or adacent nodes, f x, y) E The antpodal graph of a graph G, denoted by G ), has the same vertex set as G wth an edge onng vertces u and v f u, v) d s equal to the dameter of 09

4 H Rashmanlou & M Pal G or a graph G of order P, the antpodal graph ) G G f and only f G K p If G s a non-complete graph of order P, then ) G G, for a graph G, the antpodal graph ) G G f and only f a) G s of dameter or b) graph G s dsconnected and the components of G are complete graphs G s an antpodal graph f and only f t s the antpodal graph of ts complement The self medan fuzzy graph were ntroduced by hmed and Gan n [6] The medan of a graph * value d G v) s mnmzed graph * d G v) s constant over all vertces v of G s the set of all vertces v of G for whch the G s self-medan f and only f the value G The status, or dstance sum, of a gven vertex v n a graph s defned by v) d u, v), where d u, v) s the dstance from a vertex u to v In other words, a self medan graph v u whch all the nodes have the same status v) The graphs C n K n, n G s one n, and K n are self medan The status of a vertex v s denoted by v ) and s defned as v ) δ v, v ) The total status of a fuzzy graph v V G s denoted by t [ G )] and s defned as t G )] v ) The medan of a fuzzy graph [ v V s the set of nodes wth mnmum status fuzzy graph G, denoted, G s sad to be self - medan f all the vertces have the same status y a fuzzy subset on a set X s mean a map : X [0,] map : X X [0,] s called a fuzzy relaton on X f x, y) mn x), y) ) for all x, y X fuzzy relaton s symmetrc f x, y) y, x) for all x, y X Defnton : The nterval - valued fuzzy set n V s defned by { x,[ x), x)]) x V }, 0

5 ntpodal Interval-Valued uzzy Graphs where x) and x) are fuzzy subsets of V such that 0 x) x) for all x V If G V, E) s a graph, then by an nterval - valued fuzzy relaton on a set E we mean an nterval - valued fuzzy set such that for all xy E x), )) xy) mn y, x), )) xy) mn y Defnton : y an nterval - valued fuzzy graph of a graph G V, E) we mean a par G, ), where, ] s an nterval-valued fuzzy set on V [ and, ] s an nterval - valued fuzzy relaton on E such that [ x), )) xy) mn y, xy) mn x), y) ) Defnton 3: The complement of an nterval - valued fuzzy graph G, ) s an nterval - valued fuzzy graph, where ) V V, ) ) ) and ) ) for all v V, ) v v ) v ) v ) v v ), v v ) v ) v ) v v ), for all v, v V Defnton 4: n nterval - valued fuzzy graph G s called complete f xy) mn x), y) ) and xy) mn x), y) ) xy E, for each edge Defnton 5: path P n an nterval - valued fuzzy graph G s a sequence of dstnct vertces v,v,, v n such that ether one of the followng condtons s satsfed: ) xy) > 0 and xy) 0 for some x, y ) xy) > 0 and xy) > 0 for some x,y path P v,v,, v n+ n G s called a cycle f v v n + and n 3

6 H Rashmanlou & M Pal Defnton 6: Let P v,, 0, v, v vn be a path n nterval- valued fuzzy graph G The - strength of the paths connectng any two vertces as max v, v )) and s denoted by v v )) connectng any two vertces v, v )) v, v s defned The - strength of the paths v, v s defned as max v, v )) and s denoted by If same edge possesses both the - strength and - strength value, then t s the strength of the strongest path P and s denoted by p v v )), v v )) ] [ for all,,,, n Defnton 7: n nterval - valued fuzzy graph G s connected f any two vertces are oned by a path That s, an nterval - valued fuzzy graph G s connected f v v )) > 0 and v v )) > 0 Defnton 8: Let G be a connected nterval - valued fuzzy graph The - length of a path P: v,v,, v n n G, L p), s defned as L p) v, v ) n + The - length of a path P: v,v,, v n n G, L p), s defned as n L p) v, v+ ) The - length of a path P: v,v,, v n n G, L p), s defned as L p) [ L, ] L Defnton 9: Let G be a connected nterval - valued fuzzy graph The - dstance, δ v, v ), s the smallest - length of any v - v path P n G, where v, v V That s, δ v, v ) mn L P) ) The - dstance, δ v, v ), s the largest - length of any v -v path P n G, where L )) v, v V That s, δ v, v ) max P The dstance, δ v, v ), s defned as δ v, v ) [ δ v, v ), δ v, v )] Defnton 0: Let G be a connected nterval - valued fuzzy graph for each v V, the - eccentrcty of v, denoted by v ), s defned as e

7 e { v, v ) v V, v v } v ) max δ or each V v, denoted by v ) ntpodal Interval-Valued uzzy Graphs v, the - eccentrcty of e, s defned as e v ) max{ v, v ) v V, v v } δ or each v V, the eccentrcty of v, denoted by e v ), s defned as e v ) [ e v ), e v )] Defnton : Let G be a connected nterval - valued fuzzy graph The - radus of G s denoted by r G) and s defned as r G) mn { e v ) v V} The -radus of G s denoted by r G) and s defned by r G) mn { e v ) v V} The radus of G s denoted by rg) and s defned as r G) [ r G), r G)] Defnton : Let G be a connected nterval - valued fuzzy graph The - dameter of G s denoted by d G) and s defned as d G) max { e v ) v V} The -dameter of G s denoted by d G) and s defned as d G) max { e v ) v V} The dameter of G s denoted by d G) and s defned as d G) [ d G), d G)] Example 3: Consder a connected nterval - valued fuzzy graph G such that V { u, v, x, w}, E { w, x), w, v), w,, x, v), u, v)} u [0,04] [0,03] v [03,06] [00,3] [003] [003] [004] [04,06] [0,03] w G x g : Interval - valued fuzzy graph G 3

8 H Rashmanlou & M Pal y routne computatons, t s easy to see that: ) δ wu ) 0, δ w, x) 0, δ w, v) 0 δ, x) 0, δ x, 04, v, 0 v δ δ w, 09, δ w, x) 06, δ w, v) 06, δ v, x) 09, δ x, 09, δ v, 09 Dstance δ v, v ) s δ w, [0, 09], δ w, x) [0, 06], δ w, v) [0, 06], δ v, x) [0, 09], δ x, [04, 09], δ v, [0, 09] ) -eccentrcty and -eccentrcty of the vertces are e w) 0, e x) 04, e v) 0, e 04, e w) 09, e x) 09, e v) 09, e 09 The eccentrctes of the vertces are e w) [0, 09], e x) [04, 09], e v) [0, 09], e [04, 09] ) Radus of G s [0, 09], dameter of G s [04, 09] 3 ntpodal nterval - valued fuzzy graphs Defnton 3: Let G, ) be an nterval - valued fuzzy graph n antpodal nterval-valued fuzzy graph G) E, ) s an nterval - valued fuzzy graph G, ) n whch: ) n nterval - valued fuzzy vertex set of G s taken as nterval - valued fuzzy vertex set of G), that s, x) x) and x) x) for all x V, E ) f δ x, y) d G), then xy) xy) f x and y are neghbors n G, E 4

9 ntpodal Interval-Valued uzzy Graphs x), )) xy) mn y f x and y are not neghbors n G, xy) xy) f x and y are neghbors n G, xy) mn x), y) ) neghbors n G f x and y are not Example 3: Consder an nterval - valued fuzzy graph G such that v, v, v, }, v v, v v, v v, v } { 3 v4 y routne calculatons, we have { v δ v, v ) 0, δ v, v ) 0, δ v, v ) 0, 3 4 δ v, v ) 0, δ v, v ) 0, δ v, v ) 0, δ v, v ) 06, δ v, v ) 0 4, δ v, v ) 06, 3 4 δ v, v ) 06, δ v, v ) 0 4, δ v, v ) e v ) 0, e v ) 0, e v 3 ) 0, e v 4 ) 0, e v ) 06, e v ) 0 6, e v 3 ) 0 6, e v 4 ) 0 6 d G) 0, 06) v v [0,0] [0,03] [0,05] [0,0] [0,0] [0,03] [03,05] [0,0] v 4 v 3 g a): Interval - valued fuzzy graph G 5

10 H Rashmanlou & M Pal v v [0,03] [0,05] [0,03] [03,05] v 4 v 3 g b): ntpodal nterval - valued fuzzy graph G Hence G) E, ) such that E v, v, v, } and φ { 3 v4 Example 33: Consder an nterval - valued fuzzy graph G such that v, v, }, v v, v v, v } { v3 { 3 v3 v [03,06] [03,04] [0,05] [03,05] [03,07] [0,05] v 3 v g 3 a): Interval - valued fuzzy graph G 6

11 ntpodal Interval-Valued uzzy Graphs v [03,06] [03,04] [03,05] [03,07] v 3 v g 3b): ntpodal nterval - valued fuzzy graph y routne calculatons, we have: δ v, v ) 0, δ v, v ) 0 3, δ v, v ) 0, δ v, v ) 0 9, δ v, v ), δ v, v ) 09, e v ) 0 3, e v ) 0, e v 3 ) 0 3, e v ), 3 e v ) 09, e v 3 ), d G) 03,) δ v, v ) 3 Hence G) E, ), such that E v, v, } and v v } { v3 { 3 Theorem 34: Let G, ) be a complete nterval - valued fuzzy graph where, ) s constant functon then G s somorphc to G) Proof: Gven that G, ) be a complete nterval - valued fuzzy graph wth, ) k, k ), where k and k are constants, whch mples that δ v, v ) L, L ), v, v V Therefore, eccentrcty e v ) L, L ) v V, whch mples that d G) L, L ) Hence δ v, v ) L, L ) d ), v, v V G Hence every par of vertces are made as neghbors n G) such that 7

12 H Rashmanlou & M Pal ) n nterval - valued fuzzy vertex set of G s taken as nterval - valued fuzzy vertex set of G), that s, v ) v ) and v ) v ) for all v V, E ) v v ) vv ), snce v and v are neghbors n G and vv ) vv ), snce v and v are neghbors n G It has same number of vertces, edges and t preserves degrees of the vertces Hence G G) Theorem 35: Let G, ) s a connected nterval- valued fuzzy graph Every antpodal nterval - valued fuzzy graph s spannng subgraph of G Proof: y the defnton of an antpodal nterval - valued fuzzy graph, G) contans all the vertces of G That s, ) x) x) and x) x) for all x V and E E ) If δ x, y) d G), then xy) xy) f x and y are neghbors n G, x), )) xy) mn y f x and y are not neghbors n G, xy) xy) f x and y are neghbors n G, x), )) xy) mn y f x and y are not neghbors n G Hence G) s spannng subgraph of G Defnton 36: homomorphsm between two nterval- valued fuzzy graphs G, ) and G, ) s defned h : V V s a map whch satsfes a) h )), h )) for all u V u u b) uv) h h )), uv) h h )) v for all uv E v Defnton 37: Consder two nterval- valued fuzzy graphs G, ) and G, ) n somorphsm between two nterval - valued fuzzy graphs G and G, denoted by G G, s a bectve map h : V V whch satsfes c) h )), h )) for all u V u u d) uv) h h )), uv) h h )) v It s denoted by G G for all uv E v 8

13 ntpodal Interval-Valued uzzy Graphs Defnton 38: co - weak somorphsm between two nterval - valued fuzzy graphs G, ) and G, ) s defned as h : V V s a bectve homomorphsm that satsfes uv) h h )) h h )) uv) v v and Defnton 39: n nterval - valued fuzzy graphs H, ) s sad to be an nterval - valued fuzzy subgraph of a connected nterval - valued fuzzy graph G, ), f v ) v ), v ) v ), v V and v v ) v v ) and v v ) v v ), v, v ) E Example 30: In the followng fgure, we show the nterval - valued fuzzy graph G, ) and ts subgraph H, ), such that V v, v, v, }, { 3 v4 E { v, v ), v, v3 ), v3, v4 ), v4, v ), v4, v5 ), v5, v )}, V { v, v3, v4}, E { v, v3), v3, v4), v4, v)}, where v ) v ), v ) v ), v V and v v ) v v ) and v v ) v v ), v, v ) E v [0,03] [05,08] [0,03] v 5 v v 5 [0,06] [04,08] [0,06] [0,04] [0,04] [0,0] [0,04] [0,04] [03,07] [0,05] [0,0] [03,07] [0,05] [0,0] v 4 v 3 v 4 v 3 g 4: Interval - valued fuzzy graph and ts subgraph 9

14 H Rashmanlou & M Pal Theorem 3: If G and G are somorphc to each other, then G ) and G) are also somorphc Proof: s G and G are somorphc, the somorphsm h, between them preserves the edge weghts, so the -length and -dstance wll also be preserved Hence f the vertex v has the maxmum -eccentrcty and max -eccentrcty, n G, then h v ) has the maxmum -eccentrcty and maxmum -eccentrcty, n G o G and G wll have the same dameter If the -dstance between v and v s k, k) n G, then h v ) and h v ) wll also have ther -dstance as k, ) The same mappng h tself s a becton k between G ) and G ) satsfyng the somorphsm condton ) E v ) v ) h v )) E h v )), v G ) E v ) v ) h v )) E h v )), v G ) v v ) vv ), f v and v are neghbors n G v ), v )) v ) mn v E E, f v and v are not neghbors n G v) vv ) vv ) f v and v are neghbors n G v ), v )) v ) mn v E E, f v and v are not neghbors n G s : G G h s an somorphsm, v ) ) )) v h v h v G v v ) mn v ), v )) neghbors n neghbors n, f v and v are, f v and h v ) h v )) G Hence, v v ) h v ) h v )) v are not and v ) v o, the same h s an somorphsm between G ) and G ) Theorem 3: If G and G are complete nterval - valued fuzzy graph such that G s co-weak somorphc to G then G ) s co-weak somorphc to G ) 0

15 ntpodal Interval-Valued uzzy Graphs Proof: s G s co-weak somorphsm to G, there exsts a becton h : G G satsfyng, v ) h v )), v v ) h v ) h v )), v, v V If G has n vertces, arrange the vertces of G n such a way that v ) v ) v ) vn ) s G and G are complete, 3 co-weak somorphc nterval- valued fuzzy graph, v v ) h v ) h v )), v, v V y Theorem 34 and the defnton of antpodal nterval - valued fuzzy graph, we have G ) contans all the vertces of G, where, That s, x) x) and x) x) for all V E x and v v ) h v ) h v )), v, v V o, the same becton h s co-weak somorphsm between G ) and G ) Theorem 33: If G and G are complete nterval - valued fuzzy graphs such that G s co-weak somorphc to G, then G ) s homomorphsm to G) Proof: s G s co-weak somorphc to G, there exsts a becton h : G G satsfyng, v ) h v )), v v ) h v ) h v )) v ) h v )), v v ) h v ) h v )), v, v V and, v, v V o the - dstance and hence dameter wll be preserved Let d G ) d G ) k, k) If v, v V are at a dstance k, k) n G, then they are made as neghbors n G ) o, h v ), h v ) n G are also at a - dstance k, k) n G and h v ), h v ) are made as neghbors n G ) and v are neghbors then h v ) )) ) )) h v h v h v h v ) h v )) h v ) h v )) v ) ) v v v, vv ) v ) v If v and v are not neghbors n G, then If v v ), v )) mn h v ), h v ))) h v ) h v )), v v ) mn

16 H Rashmanlou & M Pal v ), v )) mn h v )), h v ))) h v ) h v )) vv ) mn Hence G ) s homongmorphsm to G ) Defnton 34: Let G be a connected nterval - valued fuzzy graph The - status of a vertex v s denoted by v ) and s defned as ) v ) δ v, v v V Defnton 35: Let G be a connected nterval - valued fuzzy graph The - status of a vertex v s denoted by v ) and s defned as ) v ) δ v, v v V Defnton 36: Let G be a connected nterval - valued fuzzy graph The - status of a vertex v s denoted by v ) and s defned as v ), v )) v ) Example 37: Consder the followng nterval - valued fuzzy graph G, ) : v v [0,04] [003] [03,06] [0,03] [0,03] [0,03] [04,06] [0,03] v 4 G v 3 g 5: Interval - valued fuzzy graph G δ v, v ) 0, δ v, v ) 0 3, δ v, v ) 0, δ v, v ) 0, δ v, v ) 03, δ v, v ) 0, δ v, v ) 0 9, δ v, v ) 0 6, δ v, v ) 09, δ v, v ) 0 9, δ v, v ) 0 6, δ v, v )

17 ntpodal Interval-Valued uzzy Graphs δ v ) 05, δ v ) 0 6, δ v 3 ) 0 7, δ v 4 ) 0 7, δ v ) 4, δ v ) 4, δ v 3 ) 4, δ v 4 ) 4 Therefore, v ) 05, 4), v ) 06, 4), v 3 ) 07, 4) v 4 ) 07, 4), Defnton 38: Let G be a connected nterval- valued fuzzy graph The mnmum -status of G s denoted by m [ G)] and s defned as m [ G)] mn v ), v V) Defnton 39: Let G be a connected nterval - valued fuzzy graph The mnmum -status of G s denoted by m [ G)] and s defned as m G)] mn v ), v V) [ Defnton 30 Let G be a connected nterval - valued fuzzy graph The mnmum -status of G s denoted by m [ G)] and s defned as m [ G)], m [ )]) m [ G)] G Example 3: Consder the followng nterval - valued fuzzy graph G, ) : v [04,07] [04,05] [03,06] [04,06] [04,08] [03,06] v 3 v g 6: Interval - valued fuzzy graph G y routne calculatons, we have δ v, v ) 0 3, δ v, v ) 0 4, δ v, v ) 0 3, 3 3 δ v, v ), δ v, v ), δ v, v ) v ) 0 7, v ) 0 6, 3 3 3

18 H Rashmanlou & M Pal v 3 ) 07, v ) 3, v ), v 3 ) 3 Therefore, v ) 07, 3), v ) 06, ), v 3 ) 07, 3) m [ G)] 06, ) Defnton 3: Let G be a connected nterval - valued fuzzy graph The maxmum -status of G s denoted by M [ G)] and s defned as M [ G)] max v ), v V) Defnton 33: Let G be a connected nterval - valued fuzzy graph The maxmum -status of G s denoted by M [ G)] and s defned as G)] max v ), v V) M [ Defnton 34: Let G be a connected nterval - valued fuzzy graph The maxmum status of G s denoted by M [ G)] and s defned as M [ G)] M[ G)], M[ G)] ) Example 35: Consder the followng nterval - valued fuzzy graph G, ) : v [03,06] [03,04] [0,05] [03,05] [03,07] [0,05] v 3 G g 7: Interval - valued fuzzy graph G v y routne calculatons, we have 4

19 ntpodal Interval-Valued uzzy Graphs δ v, v ) 0, δ v, v ) 0 3, δ v, v ) 0, 3 3 δ v, v ) 09, δ v, v ), δ v, v ) 0 9, v ) 0 5, 3 3 v ) 04, v 3 ) 0 5, v ) 9, v ) 8, v 3 ) 9 Therefore, v ) 05,9), v ) 04,8), v 3 ) 05,9) M [ G)] 05,9) Defnton 36: The total -status of an nterval - valued fuzzy graph G s denoted by t [ G)] and s defned as t [ G)] v ) Defnton 37: The total -status of an nterval - valued fuzzy graph G s denoted by t [ G)] defned as t [ G)] v ) Defnton 38: The total -status of an nterval - valued fuzzy graph G s denoted by t [ G)] v V v V and s defned as t [ G)] t [ G)], t [ G)] ) Defnton 39: The medan of an nterval - valued fuzzy graph G s denoted by MG) and s defned as the set of nodes wth mnmum status Defnton 330: n nterval - valued fuzzy graph G s sad to be self - medan f all the vertces have the same status In order words, G s self - medan f and only f m [ G)] M[ G)] Example 33: Let G, ) be an nterval - valued fuzzy graph defned as follows: v v [0,03] [0,04] [03,06] [0,0] [0,0] [0,04] [04,06] [0,03] v 4 G v 3 g 8: elf -medan nterval - valued fuzzy graph G 5

20 H Rashmanlou & M Pal y routne calculatons, we have δ v, v ) 0, δ v, v ) 0, δ v, v ) 0 3, δ v, v ) 0, δ v, v ) 03, δ v, v ) 0, δ v, v ) 0 7, δ v, v ) 0 5, δ v, v ) 08, δ v, v ) 0 8, δ v, v ) 0 5, δ v, v ) v ) 06, v ) 0 6, v 3 ) 0 6, v 4 ) 0 6, v ), v ), v 3 ), v 4 ) Therefore, v ) 06, ), v ) 06, ), 3 4 v 3 ) 06, ), v 4 ) 06, ) and t [ G)] 4, 8) Here, v ) 06, ), v V Hence G s self medan nterval - valued fuzzy graph Theorem 33: Let G be an nterval - valued fuzzy graph, where crsp graph s an even cycle If alternate edges have same membershp values and nonmembershp values, then G s self medan nterval valued fuzzy graph Proof Gven that G s an nterval - valued fuzzy graph nce crsp graph * G G s an even cycle lso, alternate edges of G have same membershp values and nonmemebershp values, we have δ v, v ) δ v, v ) δ v n, v ) and smlarly, 3 4 n δ v, v ) δ v, v ) δ vn, v ), δ v, v ) δ v, v ) v, v ) L δ 3 5 Hence, v ) k and v ) m, v V o, G s a self medan nterval - valued fuzzy graph Remark 333: Let G be an nterval - valued fuzzy graph, where crsp graph * G s an odd cycle If alternate edges have same membershp values and nonmembershp values, then G may not be self medan nterval - valued fuzzy graph 6

21 ntpodal Interval-Valued uzzy Graphs v 4 v 3 [0,05] [0,03] [0,05] [0,04] [0,04] v 5 [0,05] [0,06] v [0,03] [0,03] [0,03] G v g 9: Interval - valued fuzzy graph G y routne calculatons, we have δ v, v ) 0, δ v, v ) 0 4, δ v, v ) 0, δ v, v ) 0, δ v, v ) 07, δ v, v ) 0 4, δ v, v ) 0 4, δ v, v ) 0, δ v, v ) 04, δ v, v ) 0, δ v, v ) 4, δ v, v ), δ v, v ), δ v, v ) 4, δ v, v ) 3, δ v, v ), δ v, v ), δ v, v ) 4, δ v, v ), δ v, v ) That s, δ v, v ) 0,4), δ v, v ) 07,), δ v, v ) 04,), 3 δ v, v ) 0,4), δ v, v ) 0,3) 5 3 4, δ v, v ) 04,), δ v, v ) 04,), δ v, v ) 0,4) , δ v, v ) 04,), δ v, v ) 0,3) v ) 5, 48) , v ), 48), v 3 ) 5, 47), v 4 ), 47), v 5 ), 48) 7

22 H Rashmanlou & M Pal Here, v ) v3 ), v ) v4 ) and v) v3), v) v4) That s, all the vertces does not have the same status Hence G s not a self medan nterval - valued fuzzy graph 4 Conclusons It s known that fuzzy graph theory has numerous applcatons n modern scence and engneerng, especally n the feld of nformaton theory, neural networks, expert systems, cluster analyss, medcal dagnoss, traffc engneerng, network routng, town plannng, and control theory In ths paper, we have ntroduced the concept of antpodal nterval- valued fuzzy graphs and self medan nterval - valued fuzzy graphs of the gven nterval - valued fuzzy graphs We nvestgated somorphsm propertes on antpodal nterval-valued fuzzy graphs In our future work, we wll focus on drect sum of two nterval - valued fuzzy graphs and wll study the truncatons of the drect sum of two nterval - valued fuzzy graphs cknowledgement: The authors are grateful to the revewers for the suggestons to mprovement of the presentaton of the paper References [] M kram 0), polar fuzzy graphs, Informaton c, 8, [] M kram, Interval-valued fuzzy lne graphs, Neural Computng & pplcatons, DOI: 0007/s [3] M kram & W Dudek 0), Interval-valued fuzzy graphs, Computers Math ppl, 6, [4] M kram 03), polar fuzzy graphs wth applcatons, Knowledge ased ystems, 39, 8 [5] M kram & Davvaz 0), trong ntutonstc fuzzy graphs, lomat, 6),

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