- Email: [email protected]
A new eigenvalue bound for independent sets
Discrete Mathematics (
)
–
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
A new eigenvalue bound for independent sets J. Harant a,∗ , S. Richter b a
Ilmenau University of Technology, Department of Mathematics, Germany
b
Chemnitz University of Technology, Department of Mathematics, Germany
article
info
Article history: Received 29 October 2013 Accepted 17 December 2014 Available online xxxx Keywords: Independence number Eigenvalues
abstract Let G be a simple, undirected, and connected graph on n vertices with eigenvalues λ1 ≤ · · · ≤ λn . Moreover, let m, δ , and α denote the size, the minimum degree, and the −λ λ independence number of G, respectively. W.H. Haemers proved α ≤ δ 2 −λ1 λn n and, if η
is the largest Laplacian eigenvalue of G, then α ≤ M.W. Newman. We prove α ≤
2σ −2
η−δ n η
1 n
is shown by C.D. Godsil and
m for the largest normalized eigenvalue σ of G, if σδ δ ≥ 1. For ε > 0, an infinite family Fε of graphs is constructed such that 2σσ−δ 2 m = η−δ 1 λn α < ( 32 + ε) min{ δ2−λ n, η n} for all G ∈ Fε . Moreover, a sequence of graphs is −λ λ 1 n
2 presented showing that the difference between 2σσ− m and D.M. Cvetković’s upper bound δ min{|{i ∈ {1, . . . , n}|λi ≤ 0}|, |{i ∈ {1, . . . , n}|λi ≥ 0}|} on α can be arbitrarily large. © 2014 Elsevier B.V. All rights reserved.
1. Introduction and result We use standard notation and terminology of graph theory and consider a finite, simple, and undirected graph G with vertex set V = {1, . . . , n} and edge set E, where m = |E |. Let di and δ denote the degree of i ∈ V in G and the minimum degree of G, respectively. Furthermore, we assume that G has no isolated vertices, i.e. δ ≥ 1. A set of vertices I ⊆ V in G is independent, if no two vertices in I are adjacent. The independence number α of G is the maximum cardinality of an independent set of G. The independence number is one of the most fundamental and well-studied graph parameters [13]. In view of its computational hardness [11] various bounds on the independence number have been proposed, for a survey see [12]. In this paper, we are interested in upper bounds on α involving eigenvalues of matrices assigned to G (lower bounds on α in terms of eigenvalues can be found in [16]). Let λ1 ≤ · · · ≤ λn denote the eigenvalues of the adjacency matrix A of G. Our starting point is the following Delsarte–Hoffman-bound [3,8,9,15]. If G is an r-regular graph, then
α≤
−λ1 n. r − λ1
Note that λn = r if G is r-regular [3]. In [10,9], W.H. Haemers proved the following extension of the Delsarte–Hoffman-bound for arbitrary graphs.
α≤
∗
−λ1 λn n. δ 2 − λ1 λn
Corresponding author. E-mail addresses: [email protected] (J. Harant), [email protected] (S. Richter).
http://dx.doi.org/10.1016/j.disc.2014.12.008 0012-365X/© 2014 Elsevier B.V. All rights reserved.
2
J. Harant, S. Richter / Discrete Mathematics (
)
–
If all eigenvalues of G are taken into consideration, then D.M. Cvetković [3,6,7] proved
α ≤ min{|{i ∈ {1, . . . , n}|λi ≤ 0}|, |{i ∈ {1, . . . , n}|λi ≥ 0}|}. Let D be the degree matrix of G, that is an (n × n) diagonal matrix, where di is the ith element of the main diagonal. Moreover, let 0 = η1 ≤ · · · ≤ ηn be the eigenvalues of the Laplacian matrix L = D − A of G [1]. In [8], C.D. Godsil and M.W. Newman established the following inequality, which is also a consequence of a result in [2] concerning the size of a cut in a graph.
α≤
ηn − δ n. ηn
For G without isolated vertices, the normalized Laplacian is the (n × n) matrix L = (lij ) with lij = 1 if i = j, lij = − √1
di dj
if
ij ∈ E, and lij = 0 otherwise. The eigenvalues σ1 ≤ · · · ≤ σn of L are the normalized eigenvalues of G [4,5]. It is known that σ1 = 0 and 1 < σn ≤ 2 [4,5]. Our result is the following inequality.
α≤
2σn − 2
σn δ
m.
For its proof, let {u1 , . . . , un } be an orthonormal basis of Rn consisting of eigenvectors of the symmetric matrix L such that ui is an eigenvector of σi for i = 1, . . . , n. Moreover, let y = (y1 , . . . , yn ) ∈ Rn and y = µ1 u1 + · · · + µn un for suitable µ1 , . . . , µn ∈ R. It follows yT Ly = σ2 µ22 + · · · + σn µ2n = −σn µ21 + (σ2 − σn )µ22 + · · · + (σn−1 − σn )µ2n−1 + σn (µ21 + · · · + µ2n ) ≤ −σn µ21 + σn (µ21 +
· · · + µ2n ) = −σn (yT u1 )2 + σn (yT y). Let M be an (n × n) diagonal matrix, where √1d is the ith element of the main diagonal, i and I be the (n × n) identity matrix. With M T = M and L = I − MAM, we obtain yT MAMy ≥ σn (yT u1 )2 + (1 − σn )yT y. We √ √ √ may choose uT1 = √1 ( d1 , . . . , dn ) and, substituting yi = xi di for i = 1, . . . , n, it follows that 2m σn
n
2 + 2m(1 − σn )
d i xi
i=1
n
di x2i ≤ 4m
xi xj
(1)
ij∈E
i =1
= 1 if i ∈ I for arbitrary real numbers x1 , . . . , xn . Let I be a maximum independent set of G and xn = (x1 , . . . , xn ) with xi n and xi = 0, otherwise. By inequality (1), σn ( i∈I di )2 + 2m(1 − σn ) i∈I di = σn ( i=1 di xi )2 + 2m(1 − σn ) i=1 di x2i ≤ 2 4m ij∈E xi xj , hence, with ij∈E xi xj = 0 and i∈I di ≥ δα , it follows α ≤ 2σσn − m. δ n
Let us remark that i=1 di xi = [5,14] and (1) is equivalent to
ij∈E
(xi + xj ) and
n
i =1
n
di x2i =
ij∈E
(x2i + x2j ). Hence, if G is bipartite, then σn = 2
2
(xi + xj )
≤m
(xi + xj )2 .
(2)
ij∈E
ij∈E
Note that inequality (2) is a consequence of the Cauchy–Schwarz inequality and, therefore, (2) is valid also for an arbitrary (not necessarily bipartite) graph n nG. Using (2), 2m i=1 di x2i −( i=1 di xi )2 = 2m ij∈E (x2i + x2j )−( ij∈E (xi + xj ))2 ≥ m(2 ij∈E (x2i + x2j )− ij∈E (xi + xj )2 ) =
m ij∈E (xi − xj )2 ≥ 0 and it follows that the coefficient c (σn ) of σn in inequality (1) is not positive. Hence, the left side of (1) is a non-increasing function in σn and, if σn < 2 and c (σn ) < 0, then (1) is stronger than (2). Next, for given ε > 0, we present the infinite family Fε mentioned in the abstract. For a positive integer k, consider the graph Gk on n = 3k + 1 vertices obtained from k pairwise disjoint paths each on 3 vertices, where the vertices of these paths are numbered arbitrarily from 1 up to 3k, an additional vertex n = 3k + 1, and additional 2k edges connecting n with the 2k endvertices of these paths. Obviously, Gk is bipartite, n has degree 2k, each other vertex has degree 2, m = 4k, δ = 2, and, since Gk is bipartite, σn = 2 and λ √1 = −λn [5,14]. Moreover, 2σn −2 2 n m = α = 2k = ( n − 1 ) . Let x ∈ R be defined by x = 1 if i is a neighbour of n, x = 2k, and xi = 0 otherwise. It i n σ δ 3
n
follows λn ≥
xT Ax xT x
√
=
λ2
−λ λ
n 2k by Rayleigh’s principle [3], hence, δ 2 −λ1 λn n = n > k+k 2 3k. If x is defined by xi = 1 if i = n 4+λ2n 1 n
and xi = 0 otherwise, then ηn ≥
xT Lx xT x
(xT Dx−xT Ax) xT x k−1 k
= 2k, hence,
ηn −δ n ηn
>
k−1 3k. k
Let the integer l = lε ≥ 2 be chosen large
η −δ 1 λn + ε) min{ δ2−λ n, nη n} −λ1 λn n
enough such that l+2 > 1 − It follows ≥ k+2 and ( > ( 32 + ε)(1 − 2+3ε3ε )3k = 2k for all k ≥ l, hence, we are done with Fε = {Gk | k ≥ lε }. 2 Eventually, we show that the difference between 2σσn − m and D.M. Cvetković’s bound min{|{i ∈ {1, . . . , n}|λi ≤ 0}|, |{i ∈ δ l
3ε . 2+3ε
=
k
2 3
n
{1, . . . , n}|λi ≥ 0}|} can be arbitrarily large. For two graphs G on V (G) and G′ on V (G′ ), the cartesian product GG′ is the
J. Harant, S. Richter / Discrete Mathematics (
)
–
3
graph on V (G)× V (G′ ) and two vertices (v, v ′ ) and (w, w ′ ) of GG′ are adjacent in GG′ if and only if v = w and v ′ w ′ ∈ E (G′ ) or v ′ = w ′ and vw ∈ E (G). For a positive integer k, consider the bipartite and 4-regular graph Hk = C4k C4k on n = 16k2 2 m = α = 2n for Hk . vertices, where C4k is the cycle on 4k vertices, and it follows 2σσn − δ n
If λ and λ′ are eigenvalues of G and G′ , respectively, then λ + λ′ is an eigenvalue of GG′ [3]. If the graph G is bipartite, then the set of its eigenvalues is symmetric w.r.t. 0 [3]. Since C4k has the eigenvalue 0 with multiplicity 2 [3], Hk has the eigenvalue 0 with multiplicity 4k + 2, and, consequently, min{|{i ∈ {1, . . . , n}|λi ≤ 0}|, |{i ∈ {1, . . . , n}|λi ≥ 0}|} = 2n + 2k + 1 for Hk .
Acknowledgements The authors would like to express their gratitude to Horst Sachs, Ilmenau University of Technology, and two anonymous referees for their valuable advice. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
W.N. Anderson Jr., T.D. Morley, Eigenvalues of the Laplacian of a graph, Linear Multilinear Algebra 18 (1985) 141–145. B. Bollobás, V. Nikiforov, Graphs and hermitian matrices: eigenvalue interlacing, Discrete Math. 289 (2004) 119–127. A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, 2011. S. Butler, Interlacing for weighted graphs using the normalized Laplacian, Electron. J. Linear Algebra 16 (2007) 90–98. F.R.K. Chung, Laplacian of graphs and Cheeger’s inequalities, in: Combinatorics, Paul Erdös is Eighty, Vol. 2, János Bolyai Math. Soc., Budapest, 1996, pp. 157–172. D.M. Cvetković, Graphs and their spectra, Univ. Beograd Publ. Elektrotechn. Fak. Ser. Mat. Fiz. 354–356 (1971) 1–50. D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Applications, Johann Abrosius Barth Verlag, Heidelberg-Leipzig, 1995, third revised and enlarged edition. C.D. Godsil, M.W. Newman, Eigenvalue bounds for independent sets, J. Combin. Theory Ser. B 98 (2008) 721–734. W.H. Haemers, Interlacing Eigenvalues and graphs, Linear Algebra Appl. 227–228 (1995) 593–616. W.H. Haemers, Eigenvalue Techniques in Design and Graph Theory, Reidel, Dordrecht, 1980, Thesis (T.H. Eindhoven, 1979) = Math. Centr. Tract 121 (Amsterdam, 1980) (pp. 28, 39, 41, 125). J. Håstad, Clique is hard to approximate within n1−ϵ , in: Proceedings of 37th Annual Symposium on Foundations of Computer Science (FOCS), 1996, pp. 627–636. C.E. Larson, The independence number project: α -bounds, http://independencenumber.wordpress.com/dynamic-surveys/. V. Lozin, D. de Werra, Foreword: Special issue on stability in graphs and related topics, Discrete Appl. Math. 132 (2003) 1–2. V.S. Shigehalli, V.M. Shettar, Spectral techniques using normalized adjacency matrices for graph matching, Int. J. Comput. Sci. Math. 2 (4) (2011) 371–378. J.H. van Lint, R.M. Wilson, A Course in Combinatorics, second ed., Cambridge University Press, 2001, p. 435. H.S. Wilf, Spectral bounds for the clique and independence numbers of graphs, J. Combin. Theory B 40 (1986) 113–117.
)
–
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
A new eigenvalue bound for independent sets J. Harant a,∗ , S. Richter b a
Ilmenau University of Technology, Department of Mathematics, Germany
b
Chemnitz University of Technology, Department of Mathematics, Germany
article
info
Article history: Received 29 October 2013 Accepted 17 December 2014 Available online xxxx Keywords: Independence number Eigenvalues
abstract Let G be a simple, undirected, and connected graph on n vertices with eigenvalues λ1 ≤ · · · ≤ λn . Moreover, let m, δ , and α denote the size, the minimum degree, and the −λ λ independence number of G, respectively. W.H. Haemers proved α ≤ δ 2 −λ1 λn n and, if η
is the largest Laplacian eigenvalue of G, then α ≤ M.W. Newman. We prove α ≤
2σ −2
η−δ n η
1 n
is shown by C.D. Godsil and
m for the largest normalized eigenvalue σ of G, if σδ δ ≥ 1. For ε > 0, an infinite family Fε of graphs is constructed such that 2σσ−δ 2 m = η−δ 1 λn α < ( 32 + ε) min{ δ2−λ n, η n} for all G ∈ Fε . Moreover, a sequence of graphs is −λ λ 1 n
2 presented showing that the difference between 2σσ− m and D.M. Cvetković’s upper bound δ min{|{i ∈ {1, . . . , n}|λi ≤ 0}|, |{i ∈ {1, . . . , n}|λi ≥ 0}|} on α can be arbitrarily large. © 2014 Elsevier B.V. All rights reserved.
1. Introduction and result We use standard notation and terminology of graph theory and consider a finite, simple, and undirected graph G with vertex set V = {1, . . . , n} and edge set E, where m = |E |. Let di and δ denote the degree of i ∈ V in G and the minimum degree of G, respectively. Furthermore, we assume that G has no isolated vertices, i.e. δ ≥ 1. A set of vertices I ⊆ V in G is independent, if no two vertices in I are adjacent. The independence number α of G is the maximum cardinality of an independent set of G. The independence number is one of the most fundamental and well-studied graph parameters [13]. In view of its computational hardness [11] various bounds on the independence number have been proposed, for a survey see [12]. In this paper, we are interested in upper bounds on α involving eigenvalues of matrices assigned to G (lower bounds on α in terms of eigenvalues can be found in [16]). Let λ1 ≤ · · · ≤ λn denote the eigenvalues of the adjacency matrix A of G. Our starting point is the following Delsarte–Hoffman-bound [3,8,9,15]. If G is an r-regular graph, then
α≤
−λ1 n. r − λ1
Note that λn = r if G is r-regular [3]. In [10,9], W.H. Haemers proved the following extension of the Delsarte–Hoffman-bound for arbitrary graphs.
α≤
∗
−λ1 λn n. δ 2 − λ1 λn
Corresponding author. E-mail addresses: [email protected] (J. Harant), [email protected] (S. Richter).
http://dx.doi.org/10.1016/j.disc.2014.12.008 0012-365X/© 2014 Elsevier B.V. All rights reserved.
2
J. Harant, S. Richter / Discrete Mathematics (
)
–
If all eigenvalues of G are taken into consideration, then D.M. Cvetković [3,6,7] proved
α ≤ min{|{i ∈ {1, . . . , n}|λi ≤ 0}|, |{i ∈ {1, . . . , n}|λi ≥ 0}|}. Let D be the degree matrix of G, that is an (n × n) diagonal matrix, where di is the ith element of the main diagonal. Moreover, let 0 = η1 ≤ · · · ≤ ηn be the eigenvalues of the Laplacian matrix L = D − A of G [1]. In [8], C.D. Godsil and M.W. Newman established the following inequality, which is also a consequence of a result in [2] concerning the size of a cut in a graph.
α≤
ηn − δ n. ηn
For G without isolated vertices, the normalized Laplacian is the (n × n) matrix L = (lij ) with lij = 1 if i = j, lij = − √1
di dj
if
ij ∈ E, and lij = 0 otherwise. The eigenvalues σ1 ≤ · · · ≤ σn of L are the normalized eigenvalues of G [4,5]. It is known that σ1 = 0 and 1 < σn ≤ 2 [4,5]. Our result is the following inequality.
α≤
2σn − 2
σn δ
m.
For its proof, let {u1 , . . . , un } be an orthonormal basis of Rn consisting of eigenvectors of the symmetric matrix L such that ui is an eigenvector of σi for i = 1, . . . , n. Moreover, let y = (y1 , . . . , yn ) ∈ Rn and y = µ1 u1 + · · · + µn un for suitable µ1 , . . . , µn ∈ R. It follows yT Ly = σ2 µ22 + · · · + σn µ2n = −σn µ21 + (σ2 − σn )µ22 + · · · + (σn−1 − σn )µ2n−1 + σn (µ21 + · · · + µ2n ) ≤ −σn µ21 + σn (µ21 +
· · · + µ2n ) = −σn (yT u1 )2 + σn (yT y). Let M be an (n × n) diagonal matrix, where √1d is the ith element of the main diagonal, i and I be the (n × n) identity matrix. With M T = M and L = I − MAM, we obtain yT MAMy ≥ σn (yT u1 )2 + (1 − σn )yT y. We √ √ √ may choose uT1 = √1 ( d1 , . . . , dn ) and, substituting yi = xi di for i = 1, . . . , n, it follows that 2m σn
n
2 + 2m(1 − σn )
d i xi
i=1
n
di x2i ≤ 4m
xi xj
(1)
ij∈E
i =1
= 1 if i ∈ I for arbitrary real numbers x1 , . . . , xn . Let I be a maximum independent set of G and xn = (x1 , . . . , xn ) with xi n and xi = 0, otherwise. By inequality (1), σn ( i∈I di )2 + 2m(1 − σn ) i∈I di = σn ( i=1 di xi )2 + 2m(1 − σn ) i=1 di x2i ≤ 2 4m ij∈E xi xj , hence, with ij∈E xi xj = 0 and i∈I di ≥ δα , it follows α ≤ 2σσn − m. δ n
Let us remark that i=1 di xi = [5,14] and (1) is equivalent to
ij∈E
(xi + xj ) and
n
i =1
n
di x2i =
ij∈E
(x2i + x2j ). Hence, if G is bipartite, then σn = 2
2
(xi + xj )
≤m
(xi + xj )2 .
(2)
ij∈E
ij∈E
Note that inequality (2) is a consequence of the Cauchy–Schwarz inequality and, therefore, (2) is valid also for an arbitrary (not necessarily bipartite) graph n nG. Using (2), 2m i=1 di x2i −( i=1 di xi )2 = 2m ij∈E (x2i + x2j )−( ij∈E (xi + xj ))2 ≥ m(2 ij∈E (x2i + x2j )− ij∈E (xi + xj )2 ) =
m ij∈E (xi − xj )2 ≥ 0 and it follows that the coefficient c (σn ) of σn in inequality (1) is not positive. Hence, the left side of (1) is a non-increasing function in σn and, if σn < 2 and c (σn ) < 0, then (1) is stronger than (2). Next, for given ε > 0, we present the infinite family Fε mentioned in the abstract. For a positive integer k, consider the graph Gk on n = 3k + 1 vertices obtained from k pairwise disjoint paths each on 3 vertices, where the vertices of these paths are numbered arbitrarily from 1 up to 3k, an additional vertex n = 3k + 1, and additional 2k edges connecting n with the 2k endvertices of these paths. Obviously, Gk is bipartite, n has degree 2k, each other vertex has degree 2, m = 4k, δ = 2, and, since Gk is bipartite, σn = 2 and λ √1 = −λn [5,14]. Moreover, 2σn −2 2 n m = α = 2k = ( n − 1 ) . Let x ∈ R be defined by x = 1 if i is a neighbour of n, x = 2k, and xi = 0 otherwise. It i n σ δ 3
n
follows λn ≥
xT Ax xT x
√
=
λ2
−λ λ
n 2k by Rayleigh’s principle [3], hence, δ 2 −λ1 λn n = n > k+k 2 3k. If x is defined by xi = 1 if i = n 4+λ2n 1 n
and xi = 0 otherwise, then ηn ≥
xT Lx xT x
(xT Dx−xT Ax) xT x k−1 k
= 2k, hence,
ηn −δ n ηn
>
k−1 3k. k
Let the integer l = lε ≥ 2 be chosen large
η −δ 1 λn + ε) min{ δ2−λ n, nη n} −λ1 λn n
enough such that l+2 > 1 − It follows ≥ k+2 and ( > ( 32 + ε)(1 − 2+3ε3ε )3k = 2k for all k ≥ l, hence, we are done with Fε = {Gk | k ≥ lε }. 2 Eventually, we show that the difference between 2σσn − m and D.M. Cvetković’s bound min{|{i ∈ {1, . . . , n}|λi ≤ 0}|, |{i ∈ δ l
3ε . 2+3ε
=
k
2 3
n
{1, . . . , n}|λi ≥ 0}|} can be arbitrarily large. For two graphs G on V (G) and G′ on V (G′ ), the cartesian product GG′ is the
J. Harant, S. Richter / Discrete Mathematics (
)
–
3
graph on V (G)× V (G′ ) and two vertices (v, v ′ ) and (w, w ′ ) of GG′ are adjacent in GG′ if and only if v = w and v ′ w ′ ∈ E (G′ ) or v ′ = w ′ and vw ∈ E (G). For a positive integer k, consider the bipartite and 4-regular graph Hk = C4k C4k on n = 16k2 2 m = α = 2n for Hk . vertices, where C4k is the cycle on 4k vertices, and it follows 2σσn − δ n
If λ and λ′ are eigenvalues of G and G′ , respectively, then λ + λ′ is an eigenvalue of GG′ [3]. If the graph G is bipartite, then the set of its eigenvalues is symmetric w.r.t. 0 [3]. Since C4k has the eigenvalue 0 with multiplicity 2 [3], Hk has the eigenvalue 0 with multiplicity 4k + 2, and, consequently, min{|{i ∈ {1, . . . , n}|λi ≤ 0}|, |{i ∈ {1, . . . , n}|λi ≥ 0}|} = 2n + 2k + 1 for Hk .
Acknowledgements The authors would like to express their gratitude to Horst Sachs, Ilmenau University of Technology, and two anonymous referees for their valuable advice. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
W.N. Anderson Jr., T.D. Morley, Eigenvalues of the Laplacian of a graph, Linear Multilinear Algebra 18 (1985) 141–145. B. Bollobás, V. Nikiforov, Graphs and hermitian matrices: eigenvalue interlacing, Discrete Math. 289 (2004) 119–127. A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, 2011. S. Butler, Interlacing for weighted graphs using the normalized Laplacian, Electron. J. Linear Algebra 16 (2007) 90–98. F.R.K. Chung, Laplacian of graphs and Cheeger’s inequalities, in: Combinatorics, Paul Erdös is Eighty, Vol. 2, János Bolyai Math. Soc., Budapest, 1996, pp. 157–172. D.M. Cvetković, Graphs and their spectra, Univ. Beograd Publ. Elektrotechn. Fak. Ser. Mat. Fiz. 354–356 (1971) 1–50. D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Applications, Johann Abrosius Barth Verlag, Heidelberg-Leipzig, 1995, third revised and enlarged edition. C.D. Godsil, M.W. Newman, Eigenvalue bounds for independent sets, J. Combin. Theory Ser. B 98 (2008) 721–734. W.H. Haemers, Interlacing Eigenvalues and graphs, Linear Algebra Appl. 227–228 (1995) 593–616. W.H. Haemers, Eigenvalue Techniques in Design and Graph Theory, Reidel, Dordrecht, 1980, Thesis (T.H. Eindhoven, 1979) = Math. Centr. Tract 121 (Amsterdam, 1980) (pp. 28, 39, 41, 125). J. Håstad, Clique is hard to approximate within n1−ϵ , in: Proceedings of 37th Annual Symposium on Foundations of Computer Science (FOCS), 1996, pp. 627–636. C.E. Larson, The independence number project: α -bounds, http://independencenumber.wordpress.com/dynamic-surveys/. V. Lozin, D. de Werra, Foreword: Special issue on stability in graphs and related topics, Discrete Appl. Math. 132 (2003) 1–2. V.S. Shigehalli, V.M. Shettar, Spectral techniques using normalized adjacency matrices for graph matching, Int. J. Comput. Sci. Math. 2 (4) (2011) 371–378. J.H. van Lint, R.M. Wilson, A Course in Combinatorics, second ed., Cambridge University Press, 2001, p. 435. H.S. Wilf, Spectral bounds for the clique and independence numbers of graphs, J. Combin. Theory B 40 (1986) 113–117.