Spectral bounds for the k-independence number of a graph

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1 Spectral bouds for the k-idepedece umber of a graph Aida Abiad a, Sebastia M. Cioabă b ad Michael Tait c a Dept. of Quatitative Ecoomics, Operatios Research Maastricht Uiversity, Maastricht, The Netherlads A.AbiadMoge@maastrichtuiversity.l b Dept. of Math. Scieces Uiversity of Delaware, Newark, DE 19707, USA cioaba@udel.edu c Departmet of Mathematics Uiversity of Califoria Sa Diego, La Jolla, CA 92037, USA mtait@math.ucsd.edu October 23, 2015 Abstract I this paper, we obtai two spectral upper bouds for the k-idepedece umber of a graph which is is the maximum size of a set of vertices at pairwise distace greater tha k. We costruct graphs that attai equality for our first boud ad show that our secod boud compares favorably to previous bouds o the k-idepedece umber. Keywords: k-idepedece umber; graph powers; eigevalues; Expader-Mixig lemma. 1 Itroductio The idepedece umber of a graph G, deoted by α(g), is the size of the largest idepedet set of vertices i G. A atural geeralizatio of the idepedece umber is the k-idepedece umber of G, deoted by α k (G) with k 0, which is the maximum umber of vertices that are mutually at distace greater tha k. Note that α 0 (G) equals the umber of vertices of G ad α 1 (G) is the idepedece umber of G. The k-idepedece umber of a graph is related to its ijective chromatic umber [16], packig chromatic umber [13], strog chromatic idex [21] ad has also coectios to 1

2 codig theory, where codes ad aticodes are k-idepedet sets i appropriate associated graphs. This parameter has bee studied i various other cotexts by may researchers [1, 6, 9, 10, 11, 12, 22, 18]. It is kow that determiig α k is NP-Hard i geeral [19]. I this article, we prove two spectral upper bouds for α k that geeralize two well-kow bouds for the idepedece umber: Cvetković s iertia boud [3] ad the Hoffma ratio boud (see [2, Theorem 3.5.2] for example). Note that α k is the idepedece umber of G k, the k-th power of G. The graph G k has the same vertex set as G ad two distict vertices are adjacet i G k if their distace i G is k or less. I geeral, eve the simplest spectral or combiatorial parameters of G k caot be deduced easily from the similar parameters of G (see [4, 5, 17] for example). Our bouds deped oly o the spectrum of the adjacecy matrix of G ad do ot require the spectrum of G k. We prove our mai results i Sectio 3 ad Sectio 4. We ed with a compariso of our bouds to previous work ad some directios for future work. 2 Prelimiaries Throughout this paper G = (V, E) will be a graph (udirected, simple ad loopless) o vertex set V with vertices, edge set E ad adjacecy matrix A with eigevalues λ 1 λ. The followig result was proved by Haemers i his Ph.D. Thesis (see [15] for example). Lemma 2.1 (Eigevalue Iterlacig, [15]). Let A be a symmetric matrix with eigevalues λ 1 λ 2 λ. For some iteger m <, let S be a real m matrix such that S S = I (its colums are orthoormal), ad cosider the m m matrix B = S AS, with eigevalues µ 1 µ 2 µ m. The, the eigevalues of B iterlace the eigevalues of A, that is, λ i µ i λ m+i, for 1 i m. If we take S = [ I O ], the B is just a pricipal submatrix of A ad we have: Corollary 2.2. If B is a pricipal submatrix of a symmetric matrix A, the the eigevalues of B iterlace the eigevalues of A. 3 Geeralized iertia boud Cvetković [3] (see also [2, p.39] or [14, p.205]) obtaied the followig upper boud for the idepedece umber. Theorem 3.1 (Cvetković s iertia boud, [3]). If G is a graph, the α(g) mi{ i : λ i 0, i : λ i 0 }. (1) Let w k (G) = mi i (A k ) ii be the miimum umber of closed walks of legth k where the miimum is take over all the vertices of G. Similarly, let W k (G) = max i (A k ) ii be the 2

3 maximum umber of closed walks of legth k where the maximum is take over all the vertices of G. Our first mai theorem geeralizes Cvetković s iertia boud which ca be recovered whe k = 1. Theorem 3.2. Let G be a graph o vertices. The, α k (G) {i : λ k i w k (G)} ad α k (G) {i : λ k i W k (G)}. (2) P roof. Because G has a k-idepedet set U of size α k, the matrix A k has a pricipal submatrix (with rows ad colums correspodig to the vertices of U) whose off-diagoal etries are 0 ad whose diagoal etries equal the umber of closed walks of legth k startig at vertices of U. Corollary 2.2 leads to α k (G) {i : λ k i w k (G)} ad α k (G) {i : λ k i W k (G)}. 3.1 Costructio attaiig equality I this sectio, we describe a set of graphs for which Theorem 3.2 is tight. For k, m 1 we will costruct a graph G with α 2k+2 (G) = α 2k+3 (G) = m. Le H be the graph obtaied from the complete graph K by removig oe edge. The eigevalues of H are 3± (+1) 2 8, 0 (each with multiplicity 1), ad 1 with multiplicity 2 3. This implies λ i (H) < 2 for i > 1. Let H 1,..., H m be vertex disjoit copies of H with u i, v i V (H i ) ad u i v i for 1 i m. Let x be a ew vertex. For each 1 i m, create a path of legth k with x as oe edpoit ad u i as the other. Let G be the resultig graph which has m + (k 2)m + 1 vertices with m (( 2) 1 ) + mk edges. Because the distace betwee ay distict v i s is 2k + 4, we get that α 2k+2 (G) α 2k+3 (G) m. (3) We will use Theorem 3.2 to show that equality occurs i (3) for sufficietly large. Startig from ay vertex of G, oe ca fid a closed walk of legth 2k + 2 or 2k + 3 that cotais a edge of some H i. Therefore, w 2k+2 (G) 2 ad w 2k+3 (G) 2. Choose so that 2 > ( m + 4) 2k+3. If we ca show that λ i (G) m + 4 (4) for all i > m, the Theorem 3.2 will imply that α 2k+3 (G) α 2k+2 (G) m ad we are doe. To show (4), ote that the edge-set of G is the uio of m edge disjoit copies of H, the star K 1,m, ad m vertex disjoit copies of P k 1. Sice the star K 1,m has spectral radius m ad a disjoit uio of paths has spectral radius less tha 2, applyig the Courat- Weyl iequalities agai alog with the triagle iequality yields that λ i (G) < m + 4 for all i > m ad fiishes our proof. 3

4 4 Geeralized Hoffma boud The followig boud o the idepedece umber is a upublished result of Hoffma kow as the Hoffma s ratio boud (see [2, p.39] or [14, p.204]). Theorem 4.1 (Hoffma boud). If G is regular the α(g) λ λ 1 λ ad if a coclique C meets this boud the every vertex ot i C is adjacet to precisely λ vertices of C. Let G be a d-regular graph o vertices (udirected, simple, ad loopless) havig a adjacecy matrix A with eigevalues d = λ 1 λ 2 λ d. Let λ = max{ λ 2, λ }. We use Alo s otatio ad say G is a (, d, λ)-graph (see also [20, p.19]). Let W k = max i k j=1 (Aj ) ii be the maximum over all vertices of the umber of closed walks of legth at most k. Our secod theorem is a extesio of the Hoffma boud to k-idepedet sets. Theorem 4.2. Let G be a (, d, λ)-graph ad k a atural umber. The α k (G) W k + k j=1 λj k j=1 dj +. (5) k j=1 λj The proof of Theorem 4.2 will be give as a corollary to a type of Expader-Mixig Lemma. For k a atural umber, deote λ (k) = λ + λ λ k, ad d (k) = d + d d k. Theorem 4.3 (k-expader Mixig Lemma). Let G be a (, d, λ)-graph. For S, T G let W k (S, T ) be the umber of walks of legth at most k with oe edpoit i S ad oe edpoit i T. The for ay S, T V, we have ( W k(s, T ) d(k) S T λ(k) S T 1 S ) ( 1 T ) < λ (k) S T. P roof. Let S, T V (G) ad let 1 S ad 1 T be the characteristic vectors for S ad T respectively. The ( k ) W k (S, T ) = 1 t S A j 1 T. Let x 1,..., x be a orthoormal basis of eigevectors for A. The 1 S = i=1 α ix i ad 1 T = i=1 β ix i, where α i = 1 S, x i ad β i = 1 T, x i. Note that αi 2 = 1 S, 1 S = S ad similarly, βi 2 = T. Because G is d-regular, we get that x 1 = 1 1 ad so α 1 = S ad β 1 = T. Now, sice i j implies x i, x j = 0, we have j=1 4

5 ( ) t ( k ) ( ) W k (S, T ) = α i x i A j β i x i i=1 j=1 i=1 = (α i x i )((β j (λ j + λ 2 j + + λ k j )x j ) i,j = (λ i + λ 2 i + + λ k i )α i β i i=1 = d k S T + (λ i + λ 2 i + + λ k i )α i β i Therefore, we have W k(s, T ) d k S T = (λ i + λ 2 i + + λ k i )α i β i λ (k) α i β i λ (k) ( where the last iequality is by Cauchy-Schwarz. Now sice ad we have the result. α 2 i α 2 i = S S 2 2 βi 2 = T T 2 2, Now we are ready to prove the boud from Theorem 4.2. ) 1/2 ( ) 1/2 βi 2, P roof. [Proof of Theorem 4.2] Let S be a k-idepedet set i G with S = α k (G), ad let W k (S, S) be equal to the umber of closed walks of legth at most k startig i S. Theorem 4.3 gives ( d (k) S 2 W k (S, S) λ (k) S 1 S ). Recallig that W k = max i k j=1 (Aj ) ii, we have W k (S, S) S W k. This yields d (k) S ( W k λ (k) 1 S ). 5

6 Solvig for S ad substitutig S = α k gives α k W k + λ (k ) d (k) + λ (k). Note that the boud from Theorem 4.2 behaves icely if W k ad λ k are small with respect of d k. It is easy to see that W k dk 1 (we expad d i each step but the last step we d 1 do ot have ay freedom sice we assume that we are coutig closed walks). Sice G is d-regular ad we kow that W k d k 1, the above boud performs well for graphs with a good spectral gap. 5 Cocludig Remarks I this sectio, we ote how our theorems compare with previous upper bouds o α k. Our geeralized Hoffma boud for α k is best compared with Firby ad Havilad [12], who proved that if G is a coected graph of order 2 the 2( ɛ) α k (G) (6) k + 2 ɛ where ɛ k (mod 2). If d is large compared to k ad λ = o(d), the Theorem 4.2 is much better tha (6). We ote that almost all d-regular graphs have λ = o(d) as d. I [8], Fiol (improvig work from [9]) obtaied the boud α k (G) 2 P k (λ 1 ), (7) whe G is a regular graph (later geeralized to oregular graphs i [7]), ad P k is the k-alteratig polyomial of G. The polyomial P k is defied by the solutio of a liear programmig problem which depeds o the spectrum of the graph G. It is otrivial to compute P k, ad it is uclear how (7) compares with our theorems. However, we ote that (7) caot be strictly better i geeral tha our Theorem 3.2, as there are cases where Theorem 3.2 is sharp, show i subsectio 3.1. Furthermore, our theorems require less iformatio to apply tha (7). If p is a polyomial of degree at most k, ad U is a k-idepedet set i G, the p(a) has a pricipal submatrix defied by U that is diagoal, with diagoal etries defied by a liear combiatio of various closed walk. Theorems 3.2 ad 4.2 are obtaied by takig p(a) = A k, but hold also for i geeral for other polyomials of degree at most k. It is ot clear how to choose a polyomial p(a) to optimize our bouds ad we leave as a ope problem. Fially, we were able to costruct graphs attaiig equality i Theorem3.2 but ot i Theorem 4.2. We leave ope whether the boud i Theorem 4.2 is attaied for some graphs or ca be improved i geeral. Ackowledgmets The authors would like to thak Rady Elziga for helpful discussios. Some of this work was doe whe the first ad third authors were at the SP Codig ad Iformatio 6

7 School i Campias, Brazil. We gratefully ackowledge support from UNICAMP ad the school orgaizers. Refereces [1] G. Atkiso ad A. Frieze, O the b-idepedece Number of Sparse Radom Graphs, Comb., Prob., ad Comp. 13 (2004), [2] A.E. Brouwer ad W.H. Haemers, Spectra of Graphs, Spriger, New York (2012). [3] D.M. Cvetković, Chromatic umber ad the spectrum of a graph, Publ. Ist. Math. (Beograd) 14(28) (1972), [4] K. Ch. Das ad J.-M. Guo, Laplacia eigevalues of the secod power of a graph, Discrete Math. 313 (2013), [5] M. DeVos, J. McDoald ad D. Scheide, Average degree i graph powers, J. Graph Theory 72 (2013), [6] W. Duckworth ad M. Zito, Large 2-idepedet sets of regular graphs, Electro. Notes i Theo. Comp. Sci. 78 (2003), [7] M.A. Fiol, Eigevalue iterlacig ad weight parameters of graphs, Liear Algebra ad its Applicatios, 290, (1999), [8] M.A. Fiol, A eigevalue characterizatio of atipodal distace-regular graphs, Electro. J. Combi. 4 (1997), R30. [9] M.A. Fiol ad E. Garriga, The alteratig ad adjacecy polyomials, ad their relatio with the spectra ad diameters of graphs, Discrete Applied Mathematics 87, Issues 13 (1998), [10] M.A. Fiol, E. Garriga, ad J.L.A. Yebra, Locally pseudo-distace-regular graphs, J. Combi. Theory Ser. B 68 (1996), [11] M.A. Fiol, E. Garriga, ad J.L.A. Yebra, The alteratig polyomials ad their relatio with the spectra ad coditioal diameters of graphs, Discrete Mathematics (1997), [12] P. Firby ad J. Havilad, Idepedece ad average distace i graphs, Discrete Applied Mathematics, 75 (1997), [13] W. Goddard, S.M. Hedetiemi, S.T. Hedetiemi, J.M. Harris, ad D.F. Rall, Broadcast chromatic umbers of graphs, Ars. Combi. 86 (2008), [14] C. Godsil ad G. Royle, Algebraic Graph Theory, Spriger-Verlag, New York (2001). [15] W.H. Haemers, Iterlacig eigevalues ad graphs, Liear Algebra Appl (1995),

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