The path polynomial of a complete graph

Electroic Joural of Liear Algebra Volume 10 Article 12 2003 The path polyomial of a complete graph C M da Foseca cmf@matucpt Follow this ad additioal wors at: http://repositoryuwyoedu/ela Recommeded Citatio da Foseca, C M 2003, "The path polyomial of a complete graph", Electroic Joural of Liear Algebra, Volume 10 DOI: https://doiorg/1013001/1081-38101103 This Article is brought to you for free ad ope access by Wyomig Scholars Repository It has bee accepted for iclusio i Electroic Joural of Liear Algebra by a authorized editor of Wyomig Scholars Repository For more iformatio, please cotact scholcom@uwyoedu

Volume 10, pp 155-162, Jue 2003 THE PATH POLYNOMIAL OF A COMPLETE GRAPH C M DA FONSECA Abstract Let P x deote the polyomial of the path o vertices A complete descriptio of the matrix that is the obtaied by evaluatig P x at the adjacecy matrix of the complete graph, alog with computig the effect of evaluatig P x with Laplacia matrices of a path ad of a circuit Key words Graph, Adjacecy matrix, Laplacia matrix, Characteristic polyomial AMS subject classificatios 05C38, 05C50 1 Itroductio ad prelimiaries For a fiite ad udirected graph G without loops or multiple edges, with vertices, let us defie the polyomial of G, P G,as the characteristic polyomial of its adjacecy matrix, AG, ie, P G x =detxi AG Whe the graph is a path with vertices, we simply call P G the path polyomial ad deote it by P Defie A as the adjacecy matrix of a path o vertices For several iterestig classes of graphs, AG i is a polyomial i AG, where G i is the ith distace graph of G [5]Actually, for distace-regular graphs, AG i isa polyomial i AG ofdegreei, ad this property characterizes these id of graphs [14] I [4], Beezer has ased whe a polyomial of a adjacecy matrix will be the adjacecy matrix of aother graphbeezer gave a solutio i the case that the origial graph is a path Theorem 11 [4] Suppose that px is a polyomial of degree less tha The pa is the adjacecy matrix of graph if ad oly if px =P 2i+1 x, forsome i, with0 i 2 1 I the same paper, Beezer gave a elegat formula for P A with =1,,, ad Bapat ad Lal, i [1], completely described the structure of P A, for all itegers This result was also reached by Foseca ad Petroilho [10] i a oiductive way Theorem 12 [1],[4],[10] For 0 1, beig a positive iteger, { 1 if i + j = +2r,with1 r mi {i, j, } P A ij = 0 otherwise I [12], Shi Roghua obtaied some geeralizatios of the oes achieved by Bapat ad LalLater, i [10], Foseca ad Petroilho determied the matrix P C, where C is the adjacecy matrix of a circuit o vertices Received by the editors o 15 March 2003 Accepted for publicatio o 02 May 2003 Hadlig Editor: Ravidar B Bapat Departameto de Matemática, Uiversidade de Coimbra, 3001-454 COIMBRA, PORTUGAL cmf@matucpt Supported by CMUC - Cetro de Matemática da Uiversidade de Coimbra 155

Volume 10, pp 155-162, Jue 2003 156 C M da Foseca Cosider the permutatio σ =12 Theorem 13 [10] For ay oegative iteger, P C = δ 2r,+2+j ṅ P σ j, j=0 r=0 where δ is the Kroecer fuctio, σ is the permutatio 12, P σ j is the permutatio matrix of σ j ad ṅ rus over the multiples of Accordig to Bapat ad Lal cf[1], a graph G is called path-positive of order m if P G 0, for =1, 2,,m,adG is simply called path-positive if it is pathpositive of ay orderi [3], Bapat ad Lal have characterized all graphs that are path-positivethe followig corollary is immediate from the theorem above Corollary 14 The circuit C is path-positive We defie the complete graph K, to be the graph with vertices i which each pair of vertices is adjacetthe adjacecy matrix of a complete graph, which we idetify also by K,isthe matrix 11 K = 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 I this ote, we evaluate P K 2 The polyomial P Let us cosider the tridiagoal matrix A whose etries are give by { 1 if i j =1 A ij = 0 otherwise The expasio of the determiat det xi A =P x alog the first row or colum gives us the recurrece relatio 21 P x =xp 1 x P 2 x, for ay positive iteger, with the covetio P 1 x =0adP 0 x =1 It is well ow that x 22 P x =U, x C, =0, 1,, 2 where U x are the Chebyshev polyomials of the secod id

Volume 10, pp 155-162, Jue 2003 The Path Polyomial of a Complete Graph 157 From 22, it is straightforward to prove that 23 P x P y x y 1 = P l x P 1 l y The, from 21 ad 23, we may coclude the followig lemma Lemma 21 For ay positive iteger ad square matrices A ad B, 1 P A P B = P l AA B P 1 l B As i Bapat ad Lal [1], ote that a coected graph is path-positive if it has a spaig subgraph which is path-positivethus we have this immediate corollary from Corollary 14 Corollary 22 The complete graph K is path-positive 3 Evaluatig P of a complete graph If a matrix A =a ij satisfiesthe relatio a ij = a 1σ 1 i j we say that A is a circulat matrixtherefore, to defie a circulat matrix A is equivalet to presetig a tuple, say a 1,,a The ad its eigevalues are give by A = a i P σ i, i=0 31 λ h = ζ hl a l, where ζ =exp i 2π Give a polyomial px, the image of A is pa =p The, a i P σ i = 1 ζ hj p i=0 j=0 h=0 ζ hl a l P σ j P a i P σ i = 1 ζ hj P λ h P σ j, where λ h is defied as i 31 i=0 j=0 h=0

Volume 10, pp 155-162, Jue 2003 158 C M da Foseca The matrix K, defied i 11, is a circulat matrix ad it ca be writte K = P σ i i=1 By 31, K has the eigevalues λ 0 = 1adλ l = 1, for l =1,, 1 Therefore, P K =P P σ i i=1 = 1 ζ hj P λ h P σ j j=0 h=0 = 1 P 1 + P 1 ζ hj P σ j j=0 h=1 = P 1 P σ 0 + 1 P 1 P 1 P σ j Note that P σ 0 is the idetity matrix We have thus proved the mai result of this sectio: Theorem 31 For ay oegative iteger, the diagoal etries of P K are the weighted average 1 P 1 + P 1 ad the off-diagoal etries are 1 P 1 1 P 1 We ca easily evaluate the differet values of each term of the sum P K Accordigto22, 1 if 1mod3 P 1 = 0 if 2mod3 1 if 0mod3 Aother relatio already ow [11, p72] for P x is j=0 P x = /2 1 l l l x 2l, where z deotes the greatest iteger less or equal to ztherefore we have also P 1 P 1 = = /2 /2 1 l l l 2l 1 j+l j=1 1 2l 1 2l l! l!j! 2l j! j 1

Volume 10, pp 155-162, Jue 2003 The Path Polyomial of a Complete Graph 159 4 Evaluatig P of some Laplacia matrices Let G be a graphdeote DG the diagoal matrix of its vertex degrees ad by AG its adjacecy matrix The LG =DG AG is the Laplacia matrix of G I this sectio, expressios for P LA ad P LC, the path polyomials of the Laplacia matrices of a path ad a circuit, respectively, with vertices, are determied Let us cosider the followig recurrece relatio: P 0 x =1, P1 x =x +1, ad P x =x +2 P 1 x P 2 x, for 2 1, P x =x +1 P x P 2 x Therefore x x P x =U 2 +1 U 1 2 +1, for 2 1, ad P x =xu x 2 +1 where U x are the Chebyshev polyomials of the secod id The the zeroes of P x are λ j =2cos jπ 2, j =0,, 1 The recurrece relatio above ca be writte i the followig matricial way: P 0 x 1 1 0 P 0 x 0 P 1 x 1 2 1 P 1 x x = + P 0 x P 2 x 1 2 1 P 2 x 0 P x 0 1 1 P x 1 Thus, for j =0,, 1, the vector P 0 λ j P 1 λ j 41 = cos jπ 2 P 2 λ j P λ j 1 cos jπ 2 cos 3 jπ 2 cos2 3 jπ 2 cos2 1 jπ 2

Volume 10, pp 155-162, Jue 2003 160 C M da Foseca is a eigevector associated to the eigevalue λ j of LA Therefore the matrix LA is diagoalizable ad, for 0, thei, jth etry of P LA is give by P LA ij = 1 P i 1 λ l P λ l P j 1 λ l 2 s=1 Ps 1 λ l which is equal to 1 cos π 2 + 1 2 cos2i 1 lπ 2 U cos lπ 1 cos2j 1 lπ 2 If we defie l=1 α p m = cos m lπ lπ cosp, l=1 the α p m = 1 2 α p 1 m 1 + αp 1 m+1 ad 42 with α p m = 1 2 p p p l α 0 m+2l p, α 0 m = δ m,2ṅ 1 2 1 + 1m, where ṅ represets a multiple of Usig the trigoometric trasformatio formula ad the Taylor formula U cos lπ 1 = p=0 U p 1 cos p lπ p!, we ca state the followig propositio Theorem 41 For 0, beig a positive iteger, P LA ij = 1 cos π 2 + 1 2 where α p m is defied as i 42 p=0 U p 1 α p i j p! + i+j 1 αp,

Volume 10, pp 155-162, Jue 2003 The Path Polyomial of a Complete Graph 161 Note that U p 1 ca be easily evaluated, sice ad the U p U x = /2 1 l l l 2x 2l, /2 1 = 1 l p 2 2l l! l! 2l p! Now, we ca fid the matrix P LC usig the same techiques of the last sectio LC is the circulat matrix 2 1 1 1 2 1 0 0 1 2 1 1 1 2 Hece The eigevalues of LC are for l =0,, 1 ad thus LC =2P σ 0 P σ P σ 2 2cos 2lπ, P LC =P 2P σ 0 P σ P σ = 1 j=0 = 1 j=0 p=0 2ljπ i e U 1 cos 2lπ p P σ j U p 1 l!p l!2 p δ j+2l p,ṅp σ j REFERENCES [1] RB Bapat ad AK Lal Path-positive Graphs Liear Algebra ad Its Applicatios, 149:125 149, 1991 [2] RB Bapat ad VS Suder O hypergroups of matrices Liear ad Multiliear Algebra, 29:125 140, 1991 [3] RB Bapat ad AK Lal Path positivity ad ifiite Coxeter groups Liear Algebra ad Its Applicatios, 196:19 35, 1994 [4] Robert A Beezer O the polyomial of a path Liear Algebra ad Its Applicatios, 63:221 225, 1984

Volume 10, pp 155-162, Jue 2003 162 C M da Foseca [5] NL Biggs Algebraic Graph Theory Cambridge Uiversity Press, Cambridge, 1974 [6] TS Chihara A itroductio to orthogoal polyomials Gordo ad Breach, New Yor, 1978 [7] DMCvetović, M Doobad H Sachs Spectra of Graphs, Theory ad Applicatios Academic Press, New Yor, 1979 [8] PJ Davis Circulat Matrices Joh Wiley & Sos, New Yor, 1979 [9] A Erdélyi, W Magus, F Oberhettiger ad FG Tricomi Higher Trascedetal Fuctios Vol II Robert E Krieger Publishig Co, Melboure, FL, 1981 [10] CM da Foseca ad J Petroilho Path polyomials of a circuit: a costructive approach Liear ad Multiliear Algebra, 44:313 325, 1998 [11] László Lovász Combiatorial Problems ad Exercises North-Hollad, Amsterdam, 1979 [12] Shi Roghua Path polyomials of a graph Liear Algebra ad Its Applicatios, 31:181 187, 1996 [13] Ala C Wilde Differetial equatios ivolvig circulat matrices Rocy Moutai Joural of Mathematics, 13:1 13, 1983 [14] PM Weichsel O distace-regularity i graphs Joural of Combiatorial Theory Series B, 32:156 161, 1982