General Results of Local Metric Dimensions of. Edge-Corona of Graphs

Transcription

1 Internatonal Matheatcal Foru, Vol 11, 016, no 16, HIKARI Ltd, www-hkarco htt://dxdoorg/101988/f General Results of Local Metrc Densons of Edge-Corona of Grahs Rnurwat Deartent of Matheatcs, Faculty of Matheatcs and Natural Scence Insttut Teknolog Seuluh Noeber, Surabaya, Indonesa Slan Study Progra of Inforaton Syste, Unverstas Jeber, Indonesa Herry Surajtno Deartent of Matheatcs, Faculty of Scences and Technology, Unverstas Arlangga, Surabaya, Indonesa Coyrght 015 Rnurwat, Slan and Herry Surajtno Ths artcle s dstrbuted under the Creatve Coons Attrbuton Lcense, whch erts unrestrcted use, dstrbuton, and reroducton n any edu, rovded the orgnal work s roerly cted Abstract Let G be any connected grah of order n, X = { x1, x,, xk} be an order set of vertces and y a vertex n G The reresentaton of y wth resect to X s the ordered k-tule r(v X) = (d(y, x1), d(y, x),, d(y, xk)), where d(u, v) reresents the dstance between the vertces u and v X s called a local resolvng set of G f every two adjacent vertces u and v n G satsfy r(u X) r(v X) A nu local resolvng set for G s a local bass of G A local denson for G, dl(g), s the cardnalty of vertces n a local bass for G In ths artcle, we study and deterne general results of local etrc densons of edge-corona of grahs Keywords: Local resolvng set, local bass, local denson, edge-corona 1 Introducton In ths artcle, we assue connected, sle and fnte grahs The vertex set of grah G denote by V(G) and the edge set of grah G by E (G) For the other references, we can see Chartrand et al[4]

2 794 Rnurwat, Slan and Herry Surajtno The lenght of the shortest ath between x dan y s called the dstance d(x, between two vertces x dan y n grah G For an ordered set X = { x1, x,, xk} V(G) of vertces, we use the ordered k-tule r(u X) = (d(u, x1), du, x),, d(u, xk)) as the reresentaton of u wth resect to X If every two adjacent vertces of G have dstnct reresentatons wth resect to X then X s called as a local resolvng set for G A nu local resolvng set s a local bass for G A local etrc denson for G, dl(g), s the nuber of vertces n a local bass for G One of the robles, when soeone studes checal structures of checal coound s how to reresent a checal coound Motvated by ths roble, soe research about etrc denson or local etrc denson of grahs related wth ternology n chestry lke corona and aalgaton have done For exale a research about etrc densons of corona roduct grahs [5], etrc densons of aalgaaton grahs [8], local etrc densons of grahs [], and local etrc densons of corona roduct grahs [3,6,7] Ths aer resents our study about local etrc densons of edge-corona of grahs Frstly, gven soe basc concet and notaton n grahs Two adjacent vertces u, v V( G) wll be denoted wth u~v For the set of neghbors of y V(G) that s oen neghborhood of y V(G) and we use the notaton ( : { x V( G) x ~ y} The set N G ( { y} be called as the closed N G neghborhood of y V(G) N G ( s the degree of vertex y V(G) For set S V(G), oen neghborhood of S, N G (S) s unon of all neghbors of y S The subgrah of G nduced by S s denote by S G We use the notaton x f S {x} Defnton 1 [1,9] Let G and H be any grah of order n1 and n, and of degree 1 and, resectvely and H be the th -coy of grah H Edge-corona grah of G and H, denoted by GH, s a grah obtaned fro a coy-grah of G and 1 coy-grahs of H, and jonng every end-vertex of the th -edge of G by an edge to a vertex n H such that every end-vertex of the th -edge of G adjacent wth all of vertex of H Exale 1 The followng fgure lustrates an edge-corona of grah G = C4 and H = C3

3 General results of local etrc densons 795 Fgure 1 C4C3 The set of vertces of G wll denote by V G) { x, x,, x } and by H ( 1 n1 ( V ( H), E ( H)) for the coy of H such that x adjacent v for every v V Although edge-corona grah KH and jont grah K+H be soorh, to deterne general results of local densons of edge-corona roduct of grahs s used jont grah K1+H as ther subgrahs Here therefore, gven defnton of jont grah The jont G H s the grah obtaned fro a coy-grah of G and a coy-grah of H and jonng th -vertex of G by an edge to a vertex n of H such that every th -vertex of G adjacent wth all of vertex of H Man results General results of etrc densons of edge-corona grahs wll be resent on ths secton Started wth a strctly case If H K n then K H K n, and d (K+H) = n+1 Theore For any connected-grah whch s not bartte grah G of order n1 3 wth E(G) = and any grah coleent of colete grah H K n hold d (GH) = d (G) Proof Let U be any local bass for grah G Snce U be any local resolvng set for grah G, then every ar of adjacent vertces x, yg have dstnct reresentaton wth resect U In GH, reresentaton of two adjacent vertces xg, yh wth resect U are also dfferenttherefore,u s a local resolvng set for GH

4 796 Rnurwat, Slan and Herry Surajtno Suose W be any local bass for grah GH wth W U Because grah H be an ety grah, f v W V (H), for soe, every ar of verttces x, ygh where d(x,v) not the sae as d(y,v) can also d(x,x) dfferent fro d(y,x) ' It eans each vertex v W V (H) can be relace by x, {1,,, n} Let W be a set that ts eleen can be obtaned fro eleen of W such that a vertex v s relaced by x, {1,,, n} So, W be any local resolvng set for grah G and W' W U = d (G)Ths s a contradcton Thus, U be a local bass of GH Followng resent s the an secton of general results of local etrc densons of edge-corona grah GH, where H be a non grah coleent of colete grah Theore 3 For any connected grah whch s not ncolete-bartte grah G of order n1 3 wth E(G) =, H be a non grah coleent of colete grah and z s the vertex of K1 The asserton below hold () If z s not eleen of any local bass of grah K1+H then d ( GH ) d ( K1 H) () If z s eleen of any local bass of grah K1+H then d ( GH ) d (( K1 H) 1) Proof For n1 =, we have obtaned that GH K We assue n1 3 Let H W be a local bass for x H and W ' W { x} Because H non-ety grah and d(u,x) = d(v,x), for two adjacent vertces u, v V H ) then W ' Now we want to see that B W 1 ( ' be a local resolvng set for GH, there s three cases about two adjacent vertces u and v Case (1) u v V( H ) Because the dstance d(u,x) not dfferent wth d(v,x), then, y W ' d GH d x H d x H d GH Case () u V( H ), v x For y W j ', j, we have d G H 1+ d G H Case (3) u x, v x j For y W j ', j, we have d G H = d G H +1 > 1 = d G H So, B s a local resolvng set of GH Furtherore we rove (1) If z s not eleen of any local bass of K1+H, then x W for every { 1,, } and d (GH) B W ' d ( x H ) = d (K1+H) 1 1 l

5 General results of local etrc densons 797 Now, we rove that d (GH) d (K1+H) Let U be any local bass of GH and U V( H ) U Consder u, v belong to V( H ) U and u adjacent v Snce no vertex wu U such that the dstance d(u,w) not dfferent wth d(v,w), then x H d H y U d G dg H d x H Hence, U { x } s a local resolvng set of x H Because x does not belong to any local bass of x H, we obtan that U 1 U { x} d x H ly U d x H Thus, d (GH) = B W ' d ( x H ) = d (K1+H) 1 1 It eans, we have coleted the roof of () Now, we roof () Snce x belongs to any local bass of x H, we consder that v W, {1,, } Consder S S s a local bass for FH by S B In ths case, 1,, } S S V( H ) satsfes B W' { Because no vertex w n S such that d w) not the sae as d w), for the S ars of adjacent vertces u, v V H ), so S x } ust be a local resolvng set of x H Thus ( d ( x H ) S 1 W ' 1 W d ( x H ) Ths s a contradcton So, B s a local bass of GH, therefore d (GH) = B W ' d ( x H ) 1) d ( K1 1 1 { H ) 1) We have coleted the roof of () - Ths result below s obtaned as a drect concequence of Theore 3 Corollary 4 For any connected grah whch s not ncolete-bartte-grah G of order n1 3 by E(G) = hold the asserton below () If be an nteger, then d ( G K ) ( 1) () Let s and t be any nteger Then, G bea star grah, otherwse, G be a star grah d ( G K s, t ), otherwse () For any nteger 5, then (a) d ( GC ) 4, f 1(4)

6 798 Rnurwat, Slan and Herry Surajtno (b) d ( GP ) 4, f 1(4) (v) If 5 be any nteger, then d ( GC ) 4 Corollary 5 For any connected grah whch s not ncolete-bartte grah G of order n1 3 wth E(G) = and any connected-grah H, d ( G H) d ( H) Corollary 6 For any connected grah whch s not ncolete-bartte grah G of order n1 3 wth E(G) = and any connected-grah H wth daeter two, The Lea below refers to [3,6] d ( G H) d ( H) Lea 7 [3,6] Gven a connected grah F wth radus r(f) and z be a vertex of K1 If r(f) 4, then z s not eleen of any local bass of K1+F A drect consequence of Theore 3() and Lea 7 as the followng Theore 8 For any connected grah whch s not ncolete-bartte grah G of order n1 3 wth E(G) = and any connected grah H wth radus r(h) 4, d ( GH ) d ( K1 H) The result below s another consequence of Theore 3 Corollary 9 For any connected grah whch s not ncolete-bartte grah G of order n1 3 wth E(G) = and any non grah coleent of colete grah H of order n, d (GH) (n-1) Acknowledgeents The authors exress ther thanks the anonyous referee for the useful coents whch roved the aer References [1] Shu-Yu Cu and Gu-Xan Tan, The sectru and the sgnless Lalacan sectru of coronae, Lnear Algebra and ts Alcaton, 437 (01), no 7, htt://dxdoorg/101016/jlaa [] F Okaoto, B Phnezy and P Zhang, The Local Metrc Denson of a Grah, Matheatca Boheca, 135 (010) no 3, 39-55

7 General results of local etrc densons 799 [3] G A Barragan-Rarez, C G Goez and J A Rodrguez-Velazquez, Close forula for the local etrc denson of corona roduct grahs, Electronc Notes n Dscrete Matheatcs, 46 (014), 7-34 htt://dxdoorg/101016/jend [4] G Chartrand, L Eroh, M A Johnson and O R Oellerann, Resolvablty n Grahs and The Metrc Denson of a Grah, Dscrete Aled Matheatcs, 105 (000), htt://dxdoorg/101016/s x(00) [5] I G Yero, D Kuzak and J A Rodrguez-Velazquez, On The Metrc Denson of Corona Product Grahs, Couters and Matheatcs wth Alcatons, 61 (011), no 9, htt://dxdoorg/101016/jcawa [6] J A Rodrguez-Velazquez, G A Barragan-Rarez and C G Goez, On The Local Metrc Denson of Corona Product Grahs, Bulletn of the Malaysan Matheatcal Scences, 39 (015), htt://dxdoorg/101007/s [7] J A Rodrguez-Velazquez, C G Goez and G A Barragan-Rarez, Coutng the local etrc denson of a grah fro the local etrc denson of rary subgrahs, Internatonal Journal of Couter Matheatcs, 9 (014), htt://dxdoorg/101080/ [8] R Sanjuntak, S Uttunggadewa and S W Sautro, Metrc Denson for Aalgaatons of Grahs, Chater n Cobnatoral Algorths, Srnger Internatonal Publshng, 015, htt://dxdoorg/101007/ _9 [9] Y Hou and W C Shu, The Sectru of The Edge Corona of Two Grahs, The Electronc Journal of Lnear Algebra, 0 (010), htt://dxdoorg/ / Receved: Noveber 15, 015; Publshed: August 9, 016