On main eigenvalues of certain graphs
Nair Abreu, Domingos M. Cardoso, Francisca A. M. França, Cybele T. M. Vinagre
aa r X i v : . [ m a t h . C O ] M a y On main eigenvalues of certain graphs
Nair Abreu , Domingos M. Cardoso , Francisca A.M. Fran¸ca , andCybele T.M. Vinagre PEP/COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil.e-mail: [email protected] Center for Research and Development in Mathematics and Applications Department of Mathematics, Universidade de Aveiro, 3810-193, Aveiro,Portugal. e-mail: [email protected] Instituto de Ciˆencias Exatas, Universidade Federal Fluminense, Volta Redonda,Brasil. e-mail: [email protected]ff.br Instituto de Matem´atica e Estat´ıstica, Universidade Federal Fluminense, Niter´oi,Brasil. e-mail: [email protected]ff.br
October 15, 2018
Abstract
An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is not orthogonal to the associated eigenspace.In this work, we approach the main eigenvalues of some graphs. Thegraphs with exactly two main eigenvalues are considered and a relationbetween those main eigenvalues is presented. The particular case ofharmonic graphs is analyzed and they are characterized in terms oftheir main eigenvalues without any restriction on its combinatorialstructure. We give a necessary and sufficient condition for a graph G to have − − λ min as an eigenvalue of its complement, where λ min denotes the least eigenvalue of G . Also, we prove that among connectedbipartite graphs, K r,r is the unique graph for which the index of thecomplement is equal to − − λ min . Finally, we characterize all pathsand all double stars (trees with diameter three) for which the smallesteigenvalue is non-main. Main eigenvalues of paths and double starsare identified. Keywords:
Main eigenvalue, harmonic graph, path, double star.
Classification MSC (2010):
Introduction and preliminaries
In 1970, Cvetkovi´c [3] introduced the concept of main eigenvalue ofa graph, that is, the ones for which the associated eigenspaces are non-orthogonal to the vector whose entries equal 1. One year later, the sameauthor [8] related the main eigenvalues directly to the number of walks in agraph. It is well known that a graph is regular if and only if it has only onemain eigenvalue [20] but the characterization of graphs with exactly s > G whose complement G has a main eigen-value between its first eigenvalue ( index ) and − − λ min ( G ), where λ min ( G )denotes the least eigenvalue of G . The second raises the possibility of char-acterizing graphs G whose spectrum of G contains − − λ min ( G ) as a maineigenvalue. The third question approaches the characterization of connectedgraphs for which the least eigenvalue is non-main. To answer these questionswas the first motivation for the results of Sections 3 and 4.In Section 3 we answer, in the negative form, to the first question posedin [1]. The largest and the second largest eigenvalues of the complement ofa graph are also analyzed and we conclude that − − λ min ( G ) belongs toits spectrum if and only if it coincides with its second largest eigenvalue.We show that, among all connected bipartite graphs, the balanced completebipartite graphs K r,r are those whose respective complements contain − − min ( G ) as a main eigenvalue.In Section 4 we determine the main spectrum (the set of distinct maineigenvalues) of a path with n vertices and conclude that the least eigenvalueof such graph is non-main if and only if n is even. Finally, we conclude thatamong the trees of diameter three (double stars) only the balanced ones havethe least eigenvalue non-main. On the other hand, the main eigenvalues ofan arbitrary double star are determined and it is shown that their mainspectra has cardinality four when they are not balanced.Throughout this paper, unless otherwise stated, G denotes a simplegraph of order n with edge set E ( G ) and vertex set V ( G ) = { , · · · , n } .The edges with end-vertices i and j are simply denoted ij and the comple-ment of the graph G is denoted G . The adjacency matrix of G , A = [ a ij ],is the n × n matrix for which the entries are a ij = 1 if ij ∈ E ( G ), and 0otherwise. The eigenvalues of A are also called the eigenvalues of G . Wewrite Spec ( G ) for the multi-set of eigenvalues of G . The characteristic poly-nomial of A is called the characteristic polynomial of G . Unless otherwisestated, the eigenvalues of G are considered in non-increasing order, that is, λ max = λ ≥ λ ≥ · · · ≥ λ n = λ min . When necessary, we write A ( G )instead of A and λ i ( G ) instead of λ i , for i ∈ { , , . . . , n } . The eigenvec-tors associated to the eigenvalues of G are also called the eigenvectors of G and the eigenspace associated to the eigenvalue λ of G is denoted ε G ( λ ).An eigenvector associated to the largest eigenvalue of G is usually called principal eigenvector of G . The all one n × n matrix is denoted J and j denotes a column of the matrix J . An eigenvalue λ of G is said to be a maineigenvalue if there is an associated eigenvector v which is not orthogonal to j . Otherwise, we say that λ is a non-main eigenvalue of G . Notice that forevery graph G , its largest eigenvalue λ is a main eigenvalue. In particular,when G is r -regular with spectrum λ = r, λ , . . . , λ n , all eigenvalues but λ are non-main [5].For the basic notions and notation from spectral graph theory not hereindefined the reader is referred to [7]. For further mention, we also recall thefollowing consequence of the theorem of Perron-Frobenius (see [6]). Theorem 1
A graph G is connected if and only if its largest eigenvalue issimple and there exists an associated eigenvector for which all the coordinatesare positive. Graphs with exactly two main eigenvalues
In this section, a relation between the main eigenvalues of a graph withexactly two main eigenvalues is presented and special attention is given tothe harmonic non regular graphs which are characterized using just theirmain spectral properties. Furthermore, the bipartite graphs with just twomain eigenvalues are also analyzed. From now on, d G = [ d , . . . , d n ] ⊤ de-notes the degrees vector of G , where d i is the degree of the vertex i ∈ V ( G ). Proposition 2 If G is a graph with m edges and exactly two main (distinct)eigenvalues λ and λ i then λ i = P i ∈ V ( G ) d i − mλ m − nλ . (1) Proof.
It is immediate that there are scalars α and β and orthonormaleigenvectors u and v of A associated to λ and λ i , respectively, such that j = α u + β v and then Aj = d G = αλ u + βλ i v . Then, it follows j ⊤ j = α + β = n (i) j ⊤ d G = λ α + λ i β = 2 m (ii) d ⊤ G d G = λ α + λ i β = P i ∈ V ( G ) d i (iii).Let us consider two cases (a) λ i = − λ and (b) λ i > − λ .(a) Replacing λ i by − λ in (iii), it follows that λ (cid:0) α + β (cid:1) = P i ∈ V ( G ) d i .Therefore, applying (i), we obtain λ = P i ∈ V ( G ) d i n which in this caseis equivalent to (1).(b) Considering the pairs of equations (i)-(ii) and (i)-(iii), we obtain( i ) − ( ii ) ( α = m − nλ i λ − λ i β = nλ − mλ − λ i ( i ) − ( iii ) α = P j ∈ V ( G ) d j − nλ i λ − λ i β = nλ − P j ∈ V ( G ) d j λ − λ i . Therefore, from 2 m − nλ i = P i ∈ V ( G ) d i − nλ i λ + λ i , the equality (1) follows. (cid:3) co-main-spectral graphs. Despite the relation (1) between the index of G and the othermain eigenvalue, in [2] infinite families of non isomorphic co-main-spectralgraphs with exactly two main eigenvalues were presented. For instance, thebidegreed graphs (that is, graphs where all vertices have one of two possibledegrees) H qk obtained from a connected k -regular graph H k of order p , afterattaching q ≥ H k (then the order of H qk is n = ( q + 1) p ) were considered. All of these graphs (independently of p )have exactly the two main eigenvalues λ ,i ( H qk ) = k ± √ k +4 q . If k = 2, thatis, H is the cycle C p , then ∀ p ≥ λ ( H q ) = 1 + p q and λ i ( H q ) = 1 − p q, are its two main eigenvalues. Notice that for p = 3 , , . . . we obtain an infi-nite sequence of co-main-spectral graphs of increasing order equal to ( q + 1) p (see [2, Fig. 2]). In spite of this, taking into account that H q has m = n edges and P i ∈ V ( H s ) d i = p ( q + 2) + pq it is easy to check that the equality(1) holds.The next immediate corollary of Proposition 2 it will be useful for thestudy of harmonic graphs. Corollary 3 If λ i = 0 , then λ = P i ∈ V ( G ) d i m . A graph H is said to be harmonic when d H is an eigenvector associated toa (necessarily) integer eigenvalue, that is, if there is a positive integer ℓ suchthat Ad H = ℓ d H . It is immediate that every regular graph is harmonic.The harmonic graphs were introduced in [11], [10]. In [19], such a graphwithout isolated vertices is called pseudo-regular graph and it is defined asbeing a graph H such that P j ∈ N H ( i ) d H ( j ) d H ( i ) is constant for every i ∈ V ( H ).The particular case of harmonic trees was studied in [11], where theauthor consider the trees T ℓ , with ℓ ≥ , such that one of its vertices v hasdegree ℓ − ℓ +1, while every neighbor of v has degree ℓ and all the remainingvertices have degree 1. He proved that these are the unique harmonic trees.These trees are among the trees with two main eigenvalues which have beencharacterized in [15] (see also [14]). Theorem 4 ([15])
The stars, the balanced double stars and the harmonictrees T ℓ , for ℓ ≥ , are the unique trees with exactly two main eigenvalues. emma 5 If H is a bipartite harmonic graph with at least one edge andlargest eigenvalue λ , then − λ is a non-main eigenvalue of H . Proof.
Let us consider that the bipartite harmonic graph H has q ≥ H , . . . , H q and H q +1 , . . . , H q + p trivial compo-nents, with p ≥ H k of order n k is a connectedbipartite harmonic subgraph with the same largest eigenvalue λ and thesame simple least eigenvalue − λ . Assuming that V ( H k ) admits the bipar-tition S k and T k such that each edge of H k has one end-vertex in S k andthe other in T k , then the vectors (cid:18) d S k d T k (cid:19) and (cid:18) − d S k d T k (cid:19) , where d S k and d T k denote the subvectors of degrees of the vertices in S k and T k , are theprincipal eigenvector and the eigenvector associated to − λ , respectively, of H k , for k = 1 , . . . , q . The vectorsˆ u Tk = ( 0 , . . . , , d TS k , d TT k , , . . . , , , . . . , v Tk = ( 0 , . . . , , − d TS k , d TT k , , . . . , , , . . . , p zero coordinates correspond to the p trivial components,the k − d TS k and the q − k zeros on the right of d T k correspond to the vertices in the components H , . . . , H k − , H k +1 , . . . , H q ,respectively, are also a principal eigenvector and the eigenvector associatedto − λ , respectively, of H . Therefore, since for each component H k the sumof the degrees of the vertices in S k is equal to the sum of the degrees of thevertices in T k , it follows that the vectors in (3) are all orthogonal to the allone vector. Since every vector of the eigenspace associated to − λ is a linearcombination of those vectors in (3), − λ is non-main. (cid:3) Nikiforov in [19, Th. 8] proved that every main eigenvalue of an harmonicgraph H belongs to the set {− λ , , λ } . It is also stated in [19, Th. 8] thatif H is a graph without a bipartite component such that all main eigenvaluesare in {− λ , , λ } , then it is harmonic. A similar result for connected graphsis obtained in [20, Pr. 3.3], using a different approach. The next propositiongives a spectral characterization of harmonic graphs without any restrictionregarding theirs combinatorial structure. Proposition 6
A graph H is harmonic if and only if every main eigenvalueof H belongs to the set { , λ } . Proof. If H is harmonic, as direct consequence of Lemma 5 and Theorem 8in [19], it follows that every main of its eigenvalues are in { , λ } . Converse-ly, let us consider that the main eigenvalues of H are in { , λ } . If H has6nly one main eigenvalue, then H is regular and the result follows. Other-wise, assuming that { v , . . . , v p } is a basis for ε H (0) and { u , . . . , u q } isa basis for ε H ( λ ), it follows that j = P pi =1 α i u j + P qj =1 β j v j for somescalars α , . . . , α p , β , . . . , β q and d H = A H j = λ P pi =1 α i u j . Therefore, d H ∈ ε H ( λ ). (cid:3) The following proposition gives an alternative characterization of har-monic graphs.
Proposition 7
A graph G with m edges is harmonic if and only if λ = P i ∈ V ( G ) d i m and it has no more than two main eigenvalues. Proof.
Suppose that G is harmonic. By Proposition 6, all its main eigen-values are in { , λ } and we have two cases: (i) G is regular, with degree say k , and then λ = k is the unique main eigenvalue or (ii) G is non regularand then it has two main eigenvalues.(i) λ = k = nk nk = P i ∈ V ( G ) d i m .(ii) By Corollary 3, it follows that λ = P i ∈ V ( G ) d i m .Conversely, assume that λ = P i ∈ V ( G ) d i m and G has no more than two maineigenvalues. If G is regular then the conclusion is immediate. Else, byProposition 2, the main eigenvalues of G , λ i ( G ) and λ , are related by theequality (1). Replacing λ in (1) by P i ∈ V ( G ) d i m it follows that the maineigenvalues of G are in { , λ } . Therefore, by Proposition 6, the resultfollows. (cid:3) From now on, we consider the all distinct eigenvalues µ , . . . , µ s , 1 ≤ s ≤ n , of the graph G having the respective associated eigenspace not or-thogonal to the vector j as the main eigenvalues of G and the remainingdistinct eigenvalues µ s +1 , · · · , µ p , s + 1 ≤ p ≤ n , as the non-main eigenval-ues. The set of distinct main eigenvalues of G is herein called the mainspectrum of G and it is denoted M ainSpec ( G ). Therefore, Spec ( G ) = { µ [ q ]1 , . . . , µ [ q s ] s , µ [ q s +1 ] s +1 , . . . , µ [ q p ] s } , where µ [ q j ] j means that the eigenvalue µ j has multiplicity q j .Before to proceed, it is worth to recall the following theorem.7 heorem 8 ([8]) M ainSpec ( G ) and M ainSpec ( G ) have the same numberof elements. Furthermore, if λ ∈ M ainSpec ( G ) and λ ∈ M ainSpec ( G ) ,then λ + λ = − . Taking into account this theorem and the definition of main/non-maineigenvalue it is immediate to obtain the basic results stated in the nextproposition partially proved in [12].
Proposition 9
Consider a graph G and λ ∈ Spec ( G ) . Then the followingassertions are equivalent:1. the eigenvalue λ is non-main or it is main with multiplicity greaterthan ;2. there is some eigenvector v of G associated to λ such that j ⊤ v = 0 ;3. the scalar − − λ belongs to Spec ( G ) . As direct consequence of this proposition, we may note that a necessaryand sufficient condition for a simple eigenvalue λ of a graph G to be non-main is − − λ to be an eigenvalue of G (see [12]).Furthermore, we also may conclude the following corollary of Proposi-tion 9. Corollary 10 If − − λ ( G ) is a simple eigenvalue of G then it is non-main. Proof.
It follows from Proposition 9, in view of the known relation A ( G ) = J − I n − A ( G ). (cid:3) Now, it is worth to recall the following consequence of Weyl’s inequalitieswhich proof can be found in [6]: λ ( G ) ≤ − − λ n ( G ) ≤ λ ( G ) . (4)The relations (4) furnish a (negative) answer to the question raised in[1] about the existence of a graph G for which the complement G has aneigenvalue less than its index and greater than − − λ n ( G ). Proposition 11 If G is a graph of order n , then G has no eigenvalue be-longing to the open interval ( − − λ n ( G ) , λ ( G )) . G for which − − λ n ( G ) is an eigenvalue of G . We have two cases: (a) λ ( G ) = − − λ n ( G )and (b) λ ( G ) = − − λ n ( G ).In the case (a), we have that − − λ n ( G ) = λ ( G ) is a main eigenvalue of G . Therefore, Theorem 8 guarantees that λ n ( G ) is a non-main eigenvalue,since − λ n ( G ) + ( − − λ n ( G )). In fact, regarding the equality (a), wemay establish the following proposition. Proposition 12
Let G be a graph of order n . Then λ ( G ) = − − λ n ( G ) ifand only if λ n ( G ) is non-main and the multiplicity of λ ( G ) is greater thanone. Proof. If λ ( G ) = − − λ n ( G ), λ n ( G ) is non-main and from Proposition 9, − − λ n has an eigenvector v such that v ∈ ε G ( λ n ) and j ⊤ v = 0. Onthe other hand (by Perron-Frobenius theorem), there is an eigenvector v ,associated to λ ( G ), with nonnegative entries and then j ⊤ v = 0. Therefore, v and v are linearly independent. This implies that the multiplicity of λ ( G ) is greater than 1. Conversely, if λ n ( G ) is non-main then − − λ n ( G ) ∈ Spec ( G ) by Proposition 9. Since the multiplicity of λ ( G ) is greater thanone, from (4) the result follows. (cid:3) According to Theorem 1 and Proposition 12, when λ ( G ) = − − λ n ( G )it follows that λ n ( G ) is non-main and G is disconnected. On the other hand,for the case (b) we have: Proposition 13
Let G be a graph of order n . Then λ ( G ) = − − λ n ( G ) <λ ( G ) if and only if λ n ( G ) is main with multiplicity greater than one or itis non-main and λ ( G ) is simple. The inequalities in (4) and Propositions 12 and 13 allow us to concludethat − − λ n ( G ) ∈ Spec ( G ) if and only if λ ( G ) = − − λ n ( G ).From Corollary 10, for an arbitrary graph G of order n such that λ n ( G ) isa simple eigenvalue, we have that λ n ( G ) is non-main if and only if − − λ n ( G )is an eigenvalue of G . Since the least eigenvalue of a connected bipartitegraph is simple, for these graphs we may conclude the following:(a) For a connected bipartite graph G , λ ( G ) = − − λ n ( G ) if and only if λ n ( G ) is non-main and λ ( G ) has multiplicity greater than one.(b) If G is connected and bipartite then λ ( G ) = − − λ n ( G ) < λ ( G ) ifand only if λ n ( G ) is non-main and λ ( G ) is simple.9he next result gives a combinatorial characterization of bipartite graphs G of order n for which λ ( G ) = − − λ n ( G ). Theorem 14
Let G be a bipartite graph of order n . Then λ ( G ) = − − λ n ( G ) if and only if G is complete (bipartite) and balanced. Proof.
Let us consider a bipartite graph G with vertex set V = V ˙ ∪ V ,where | V | = r and | V | = s . If λ ( G ) = − − λ n ( G ) then (a) above implies G is disconnected, and thus G = K r ˙ ∪ K s . Since λ ( G ) is a multiple eigenvaluethen r = s . Conversely, if G = K s,s , for some positive integer s , then G = K s,s is a disconnected graph with two components which are completegraphs with s vertices. It follows that λ n ( G ) = − s and λ ( G ) = s − λ ( G ) = − λ n ( G ) − (cid:3) Concerning the third question of [1], we may note that among the con-nected graphs for which the least eigenvalue is non-main we can count theharmonic graphs (see the Proposition 6) which includes the regular graphs.In this section, the graphs with non-main least eigenvalue of two families oftrees are characterized. We start by determining the paths with non-mainleast eigenvalue. For sake of completeness, we determine the main spectrumof an arbitrary path.It is worth to recall the following lemma which can be found in [5] (theeigenvectors are described in [17]).
Lemma 15 ([5],[17])
Let P n be the path on n vertices. Then its eigen-values are simple and given by λ j ( P n ) = 2 cos (cid:18) jπn + 1 (cid:19) , ≤ j ≤ n . Eachof these eigenvalues λ j has an associated eigenvector with entries v ( j ) i =sin (cid:18) i jπn + 1 (cid:19) , for i ∈ { , , . . . , n } . Theorem 16
For n ≥ and ≤ j ≤ n , λ j is a non-main eigenvalue of thepath P n if and only if j is even. In particular, the least eigenvalue of P n isnon-main if and only if n is even. Proof.
Let us fix j , 1 ≤ j ≤ n . For the λ j -eigenvector v ( j ) = ( v ( j )1 , . . . , v ( j ) n ) ⊤ we have λ v ( j ) i = P t ∼ i v ( j ) t , whence λ P i v ( j ) i = P i d i v ( j ) i = 2 P i v ( j ) i − ( j )1 − v ( j ) n . From Lemma 15, λ j = 2 and then P i v ( j ) i = 0 if and onlyif v ( j )1 + v ( j ) n = 0. Since v ( j )1 + v ( j ) n = 2 sin (cid:16) jπ (cid:17) cos (cid:16) ( n − jπ n +1) (cid:17) , we mayverify that λ j is a non-main eigenvalue if and only if j is even. In fact,cos (cid:16) ( n − jπ n +1) (cid:17) = 0 if and only if ( n − jπ n +1) = π + kπ , for k ∈ N . From this, wehave that 1 + 2 k < j ≤ n . Also, it holds that n = j +(1+2 k ) j − (1+2 k ) , which implies n < nj − (1+2 k ) and then, j < k + 3. Thus 1 + 2 k < j < k + 3, that is, j iseven. The another case is straightforward. (cid:3) Corollary 17
The path P n on n vertices has ⌈ n ⌉ main eigenvalues, where ⌈ x ⌉ denotes the least integer no less than x . The following result characterizes the semi-regular bipartite graphs interms of theirs main eigenvalues.
Theorem 18 ([20])
A non-trivial connected graph G is semi-regular bipar-tite if and only if its main eigenvalues are only λ ( G ) and − λ ( G ) . Combining Theorem 18 with Proposition 2, it follows that when G isa connected semi-regular bipartite graph of order n , λ ( G ) = q P i ∈ V ( G ) d i n .This is a known result obtained in [13] where it was stated that if a graph H has order n , then P i ∈ V ( H ) d i ≤ λ ( H ) n and the equality holds if and onlyif H is a semi-regular bipartite graph.Since diameter-2 trees (the stars) are connected semi-regular bipartitegraphs, there exists no diameter-2 tree with non-main least eigenvalue. Re-garding diameter-3 trees, it should be noted that these trees (the doublestars) are not semi-regular bipartite graphs and then, combining Theorems18 and 4, we may conclude that the least eigenvalue of a balanced double staris non-main. On the other hand, we claim that there are no non-balanceddouble stars with least eigenvalue non-main.In order to prove our assertion, we first remember that a walk in G isa sequence i , i , · · · i r of vertices in G such that i t is adjacent to i t +1 , for0 ≤ t ≤ r −
1. The length of a walk is its number of edges. For a squarematrix B , the walk-matrix of B is given by W ( B ) = [ j B j B j · · · B n − j ].In the particular case B = A , the adjacency matrix of G , W ( A ) = [ w ij ] issuch that w ij gives the number of walks in G of length j starting at vertex i , 1 ≤ i ≤ n and 1 ≤ j ≤ n −
1, and then it is called the walk-matrix of G .We also recall that a partition π of the vertex set V ( G ) of the graph G is equitable when, given two cells V i and V j of π , there is a constant m ij such11ach vertex v ∈ V j has exactly m ij neighbors in V j . The matrix M = [ m ij ]is called the divisor of G with respect to π . It is known (Theorem 3 of [4]),that the main eigenvalues of G are eigenvalues of M . A fundamental resulton the number of eigenvalues of a graph is the following Theorem 19 ([12])
The rank of the walk-matrix of G is equal to the num-ber of its main eigenvalues. It was recently proved ([23], Lemma 2.4) that the number of main eigen-values of G is equal to the rank of the walk-matrix W ( M ) of M . Theorem 20
Let T be a double star with n vertices. Then its least eigen-value is non-main if and only if T is balanced. Proof.
Let T = T ( k, s ) be a double star of order n = k + s + 2 whosevertices are labeled as in Figura 1. Let us consider V = { } , V = { } , · · · · · · k + 2 k + 3 k + s + 2 Figure 1: Double star T ( k, s ). V = { , . . . , k + 2 } and V = { k + 3 , . . . , k + s + 2 } . Then V ˙ ∪ V ˙ ∪ V ˙ ∪ V isan equitable partition of V ( T ) with associated divisor M = [ m ij ] = k
01 0 0 s , for which the walk-matrix is W ( M ) = k + 1 k + s + 1 s + k + 2 k + 11 s + 1 k + s + 1 s + 2 s + k + 11 1 k + 1 k + s + 11 1 s + 1 k + s + 1 . It can be verified that det W ( M ) = − ks ( s − k ) , which is equal to zero ifand only if s = k . Since M has characteristic polynomial q ( x ) = x − ( k +12 + 1) x + ks and, according to [9], the characteristic polynomial of T is p ( x ) = x s − x k − ( x − x ( k + s + 1) + ks ), we conclude that in case k = s thefour non-zero eigenvalues of the graph T = T ( k, s ) are main. In particular,it follows that λ n is a main eigenvalue (clearly, the others are λ , λ and λ n − ). Considering that the case k = r is already known, the assertion isproved. (cid:3) By combining Theorem 8, Proposition 13 and Theorems 16 and 20 wemay conclude immediately the next corollary.
Corollary 21
If the graph G is a path (respectively, a balanced double star)on n vertices then its complement G has ⌈ n ⌉ (resp., two) main eigenvaluesand the second largest eigenvalue of G is equal to − − λ n ( G ) . Acknowledgements
The research of Domingos M. Cardoso is partially supported by thePortuguese Foundation for Science and Technology (“FCT-Funda¸c˜ao paraa Ciˆencia e a Tecnologia ”), through the CIDMA - Center for Research andDevelopment in Mathematics and Applications, within project UID/MAT/04106/2013. This author also thanks the support of Project Universal CNPq442241/2014 e Bolsa PQ 1A CNPq, 304177/2013-0 and the hospitality ofPEP/COPPE/UFRJ where this paper was started.
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