Minimum Rank of Graphs Powers Family

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1 Oe Joural of Discrete Mathematics htt://dxdoiorg/046/odm00 Published Olie Aril 0 (htt://wwwscirporg/oural/odm) Miimum Rak of rahs Powers Family Alimohammad Nazari Marzieh Karimi Radoor Deartmet of Mathematics Uiversity of Arak Arak Ira Istitute Scietific ad Research Bahar Hameda Ira a-azari@arakuacir karimiradoor@yahoocom Received Jauary 5 0; revised February 7 0; acceted March 5 0 ABSTRACT I this aer we study the relatioshi betwee miimum rak of grah ad the miimum rak of grah some families of secial grah where is the th ower of grah for Keywords: Miimum Rak; Power of rahs; Zero Forcig Set Itroductio A grah is a air = V E where V is the set of vertices (usually or a subset thereof) ad E is the set of edges (a edge is a two-elemet subset of vertices); what we call a grah is sometimes called a simle udirected grah I this aer each grah is fiite ad has oemty vertex set The order of a grah deoted is the umber of vertices of A ath is a grah P = v v E such that E = vi vi : i = A cycle is a grah C = v v E E = vi vi : i = v v such that The legth of a ath or cycle is the umber of its edges A comlete K = v v E such that grah is a grah i E = v v : i < A grah is biartite V E if the vertex set V ca be artitioed ito two oemty subsets U ad W such that every edge of E has oe edoit i U ad oe i W A comlete biartite grah is a biartite grah K q = U W E such that U = V = q ad E = u w : u U ww = V E deoted L The lie grah of a grah is the grah havig vertex set E with two vertices i L adacet if ad oly if the corresodig edges share a edoit i Sice we require a grah to have a oemty set of vertices the lie grah L is defied oly for a grah that has at least oe edge The coroa of with H deoted H is the grah of order H obtaied by takig oe coy of ad coies of H ad oiig all the vertices i the ith coy of H to the ith vertex of See Figures 6 7 for a icture of K6 K Note that H ad H are usually ot isomorhic (i fact if H the H H ) Defiitio The th ower of a grah is a grah with the same set of vertices as ad a edge betwee two vertices if there is a ath of legth at most betwee them Defiitio For such a matrix the grah of A deoted A is the grah with vertices ad edges i : ai 0 i < Note that the diagoal of A is igored i determiig A The set of symmetric matrices of grah (over R) is defied to be S = A S R : A = The miimum rak of a grah (over R) is defied to be mr =mi rak A : A S For A R the corak of A is the ullity of A ad the maximum ullity (or maximum corak) of a grah (over R) is defied to be Clearly = maxcorak : A S M A mr M = More geerally the miimum rak of a simle grah is defied to be the smallest ossible rak o ver all symmetric real matrices whose ith etry (for i ) is ozero wheever i is a edge i ad is zero otherwise [] The solutio to the miimum rak roblem is equivalet to the determiatio of the maximum multilicity of a eigevalue amog the same family of matrices [] Coyright 0 SciRes

2 66 A NAZARI ET AL Zero Forcig Sets ad the rah Z Parameter Here we itroduce the grah arameter Z as the miimum size of a zero forcig set from [] The zero forcig umber is a useful tool for determiig the miimum rak of structured families of grahs ad small grahs [4] Defiitio Color-chage rule: If is a grah with each vertex colored either white or black u is a black vertex of ad exactly oe eighbor v of u is white the chage the color of v to black ive a colorig of the derived colorig is the result of alyig the color-chage rule util o more chages are ossible A zero forcig set for a grah is a subset of vertices Z such that if iitially the vertices i Z are colored black ad the remaiig vertices are colored white the derived colorig of is all black Z is the miimum of Z over all zero forcig sets Z Z For examle a edoit of a ath is a zero forcig set for the ath I a cycle ay set of two adacet vertices is a zero forcig set V E be a zero forcig set The ad thus Corollary [5] Let = be a grah ad let Z V M Z M Z The Coli deverdiere-tye arameter ca be useful i comutig miimum rak or maximum ullity (over the real umbers) A symmetric real matrix M is said to satisfy the Strog Arold Hyothesis rovided there does ot exist a ozero symmetric matrix X satisfyig: MX 0 M X =0 I X =0 where deotes the Hadamard (etrywise) roduct ad I is the idetity matrix For a grah is the maximum ullity amog matrices A S that satisfy the Strog Arold Hyothesis It follows that M A cotractio of is obtaied by idetifyig two adacet vertices of ad suressig ay loos or multile edges that arise i this rocess A mior of arises by erformig a series of deletios of edges deletios of isolated vertices ad/or cotractios of edges A grah arameter is mior mootoe if for ay mior of The arameter was itroduced i [6] where it was show that is mior mootoe It was also established that K = ad Kq = (uder the assumtios that q q ) (see [57]) Corollary [6] Let be a grah ) If K is a mior of the M K ) If q q = ad K q is a mior of the K q M = Other ossible bouds for miimum rak derived from certai easy to comute arameters of the grah were cosidered leadig to a ivestigatio of the coectio betwee miimum degree of a vertex ad miimum rak [8] Corollary 4 For ay grah ad ifiite field F F mr Mai Results I here we calculate the miimum rak of grah For this urose we obtaied Zero forcig set ad the grah arameter Z ad the arameter ad determied the uer bo ud ad lower boud for maxi- mum ullity M Sice we have M Z The we ca achieve miimum rak of grahs (see Ta- For all ad for all grah such ble ) Theorem that Z = M we have mr mr Proof It is clear that mr = M = Z ad Z Z Z the Z Z mr mr Proositio [] For each of the followig families of grahs Z = M : ) Ay grah such that 6 (The miimum raks of all grahs of order at most 7 are available i the sreadsheet [9]) ) K C P ) Ay tree T 4) Some families of secial grahs By this Proositio we have followig Theorem Theorem For each of the followig families of grahs Z = M : ) Ay grah such that 6 ) Comlete grah ad comlete multiartite grah ) Peterse grah 4) Wheel grah 5) Clebsch rah for > 6) Comlemet of cycle grah C where 5 Proof The roof is trivial Proositio 4 If P be a ath grah P is homomorhic to the comlete grah K Coyright 0 SciRes

3 A NAZARI ET AL 67 Table Summary of miimum rak results established i this aer Z mr : ~ Comlete rah reamble K K W Peterso rah 0 9 C Clebsch rah 6 5 K C C P > K K = & 5 L - - & Theorem 5 For all < we hav mr P Proof (Figure ) I grah P we have ZP = = M P because if we start colorig from the ed oit vertex u this vertex at least is adacet with vertices The vertex u with its adacet vertices are colorig The other vertices also are colorig sice they are adacet to colorig vertices ad the umber of colorig vertices is therefore we have Z P = From Corollary we have M P ZP = O the other had with cotractio of the vertices of we reach to the comlete grah K ad we kow that K is a mior for P The we have = K P M P thus Cosequetly (see also [0]) Proositio 6 e Z P = = M P mr P = C is homomorhic to = K Theorem 7 mr C = for all Proof (Figure ) I C ay vertex u is adacet to vertices the M C ZC = O the other had C = The ad fially = C M C Figure rah P 9 mr C = Figure rah C 8 Coyright 0 SciRes

4 68 A NAZARI ET AL Proositio 8 For all Theorem 9 For all = we have = C K K we have Z C K = Proof (Figures 4 5) I the th ower of grah C K a exteral vertex u is adacet to 4 vertices If we start to colorig of exteral vertex the 4 vertices are colorig which colored vertices are i exteral cycle ad the remaiig vertices are located i the ier cycle For more colorig we use the earest adacet vertex to u o the set of exteral vertices We call thi s vertex by u 4 adacet vertices to u are same adacet vertices to u ad oly two of them is differet Oe of these which is located o the ier cycle has colored ad the aother vertex that is located o the exteral c ycle colored from color-chage rule We cotiue the rocess util all vertices are colored o the iteral cycle Fially Z C K = Theorem 0 For all < ad 5 we have mr C K = Proof (Figures 4 5) whe we make ower of C K the its iteral cycle reach to comlete grah With cotractio of vertices to vertices of this grah we reach to comlete grah of order the K is a mior of C K O the other had we have = K C K M C K Figure C K 8 Figure 4 C K 8 Figure 5 C K 8 Also accordig to the revious Theo rem we have Z C K = the M C K = ZC K = mr C K = ad hece Proositio K K Theorem is homomorhic to K mr K K = Proof (Figures 6 7) With the cotractio of exteral vertices of K K o the vertices which is located i ite ral cycle we have the comlete grah K the the mior of K K is K So we have = the = = K K K M K K O the other had it is trivial that Ad we have Z K K mr K K Z K K = Coyright 0 SciRes

5 A NAZARI ET AL 69 Figure 6 C 6 K Figure 7 K K 6 Proositio [] If has vertices ad cotais a Hamiltoia ath the mr L= Theorem 4 For all ad vertices we have mr L = Proof By usig the i ductio o the it ca be show that cotais a Hamiltoia ath So we have mr L = REFERENCES [] AIM Miimum Rak-Secial rahs Work rou Zero Forcig Sets ad the Miimum Rak of rahs Liear Algebra ad Its Alicatios Vol 48 No doi:006/laa [] S Fallat ad L Hogbe The Miimum Rak of Symmetric Matrices Described by a rah: A Survey Liear Algebra ad Its Alicatios Vol 46 No doi:006/laa [] L Hogbe Sectral rah Theory ad the Iverse Eiad H va der gevalue Problem of a rah Chamchuri Joural of Mathematics Vol No [4] F Barioli W Barrett S M Fallat H Tracy Hall L Hogbe B Shader P va de Dries Che Holst Zero Forcig Parameters ad Miimum Rak Problems Liear Algebra ad Its Alicatios Vol 4 No doi:006/laa [5] L DeLoss J rout L Hogbe T McKay J Smith ad Tims Techiques for Determiig the Miimum Rak of a Small rah Liear Algebra ad Its Alicatios Vol 4 No doi:006/laa [6] F Barioli S M Fallat ad L Hogbe A Variat o the rah Parameters of Coli de Verdiere: Imlicatios to the Miimum Rak of rahs Electroic Joural of Liear Algebra Vol [7] L Hogbe ad H va der Holst Forbidde Miors for the Class of rahs with Liear Algebra Alicatios Vol 4 No doi:006/laa [8] A Berma S Friedlad L Hogbe U Rothblum ad B Shader A Uer Boud for the Miimum Rak of a rah Liear Algebra ad Its Alicatios Vol 49 No doi:006/laa [9] L DeLoss J rout L Hogbe T McKay J Smith ad Tims Table of Miimum Raks of rahs of Order at Most 7 ad Selected Otimal Matrices htt://arxivorg/abs/ [0] America Istitute of Mathematics Worksho Sectra of Families of Matrices Described by rahs Digrahs ad Sig Patters -7 October 006 Palo Alto Coyright 0 SciRes