Some parametric inequalities on strongly regular graphs spectra

Transcription

1 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 06 Some parametric inequalities on strongly regular graphs spectra Vasco Moço Mano and Luís de Almeida Vieira Faculty of Engineering, University of Porto, Portugal Abstract In this work we deal with the problem of finding suitable admissibility conditions for the parameter sets of strongly regular graphs To address this problem we analyze the regularity of these graphs through the introduction of a parameter and deduce some parametric admissibility conditions Applying an asymptotic analysis, the conditions obtained enabled us to extract some spectral conclusions over the class of strongly regular graphs Keywords-Strongly regular graph, Euclidean Jordan algebra, Matrix analysis I INTRODUCTION In this paper we obtain admissibility conditions on the parameters and spectra of strongly regular graphs The conditions are deduced by the analysis of a parameter such that the regularity of the graph is given as a function of its order Also, the relationship between this class of graphs and Euclidean Jordan algebras is explored We consider a generalized binomial series regarding an element of the Jordan frame of an Euclidean Jordan algebra associated to the strongly regular graph, in an analogous manner as it was done in [6] The paper is organized as follows A survey on the main concepts regarding Euclidean Jordan algebras and Strongly regular graphs is presented in Section II and in Section III, respectively Next, in Section IV, we present the algebraic environment of our work, that is, an Euclidean Jordan algebra spanned by the adjacency matrix of a strongly regular graph Our main results are presented in Section V, were the asymptotic parameters of the generalized binomial Hadamard series of a strongly regular graph are introduced, in a similar manner than in [6] Analyzing these parameters, we establish some asymptotic admissibility conditions over the parameters and the spectra of strongly regular graphs Finally, in Section VI, some conclusions and experimental results are presented II REAL EUCLIDEAN JORDAN ALGEBRAS In the following paragraphs we introduce the fundamental concepts on Euclidean Jordan algebras as well as some notation Such algebras were introduced in 934 by Pascual Jordan, John von Neumann and Eugene Wigner, [0] These algebras have applications in several branches of Mathematics such as statistics, [3], interior point methods, [6], [7] and combinatorics [3], [5], [6] Euclidean Jordan algebras are vector spaces, V, over the field R, where it is defined an inner product, <, >, and a bilinear form u, v) u v, from V V into V, such that: i) u v = v u; ii) u u v) = u u v), with u = u u; iii) e V: u V, e u = u [iv)]e = u; v) < u v, w >=< v, u w > The element e is usually called the unit element of V Along this paper we only deal with real finite dimensional Euclidean Jordan algebras with unit element An element u in V is an idempotent if u = u Two idempotents u and v of V are orthogonal if u v = 0 A complete system of orthogonal idempotents of V is a set {u, u,, u k } of orthogonal idempotents such that u u u k = e For every u in V, there are unique distinct real numbers λ, λ,, λ k, and an unique complete system of orthogonal idempotents {u, u,, u k } such that u = λ u λ u λ k u k, ) see [5], Theorem III) These λ j s are usually called the eigenvalues of u and ) is called the first spectral decomposition of u The rank of an element u in V is the least natural number l, such that the set {e, u,, u l } is linearly dependent where u k = u u k ) and we write ranku) = l This concept is expanded by defining the rank of the algebra V as the natural number rankv) = max{ranku) : u V} The elements of V with rank equal to the rank of V are called regular elements of V From now on, it is assumed that V has rank r An idempotent is called primitive if it is non-zero and it cannot be written as the sum of two other distinct nonzero orthogonal idempotents A Jordan frame is a complete system of orthogonal idempotents such that each idempotent is primitive For every u in V, where V is an Euclidean Jordan algebra with rank r, there exists a Jordan frame {u, u,, u r } and real numbers λ, λ,, λ r such that u = λ u λ u λ r u r ) The decomposition ) is called the second spectral decomposition of u ISSN:

2 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 06 Further information on real Euclidean Jordan algebras can be found in Koecher s lecture notes [] and in the monograph by Faraut and Korányi [5] III STRONGLY REGULAR GRAPHS In this section we present the family of strongly regular graphs The basic concepts of Graph Theory are assumed to be known by the reader and can be found in [8] Strongly regular graphs is a family of regular graphs in which each pair of vertices have a fixed number of neighbors This family was introduced in [] in the context of partial geometries and partially balanced designs One problem suggested when studying these graphs is to obtain feasibility conditions on their parameter set A simple, non-null, not complete and undirected graph X is strongly regular with parameter set n, k, a, c) if it is a k- regular graph with n vertices such that each pair of vertices has a common neighbors if they are adjacent and c common neighbors, otherwise The parameters of a strongly regular graph are related by the following equality: kk a ) = n k )c 3) If X is a strongly regular graph with parameters n, k, a, c), then its complement graph, X, that is, the graph with the same vertex set and such that two distinct vertices are adjacent in X if and only if they are not adjacent in X, is also strongly regular with parameters n, k, ā, c), where k = n k, 4) ā = n k c, 5) c = n k a 6) It is also well known see, for instance, [8]) that the eigenvalues of a strongly regular graph X with parameter set n, k, a, c) are k and θ = a c a c 4k c) and 7) τ = a c a c 4k c) 8) The eigenvalues θ and τ are usually called the restricted eigenvalues of X The multiplicities m θ of θ and m τ of τ are see the equalities for the multiplicities presented in [] and simplify): m θ = τn τ k, m τ = θn k τ Taking into account the information presented above, the following conditions are deduced: ) The nontrivial Krein conditions obtained in [4]: θ )k θ θτ) k θ)τ 9) τ )k τ θτ) k τ)θ 0) ) The Seidel s absolute bounds see [4]): n m θm θ 3) ) n m τ m τ 3) ) The conditions presented are used as feasibility conditions for parameter sets n, k, a, c), that is, if n, k, a, c) is a parameter set of a strongly regular graph, then inequalities 9)-) must hold With these conditions many parameter sets are discarded as possible parameter sets of strongly regular graphs However, there are still many parameter sets for which it is unknown if there exists a corresponding strongly regular graph The Krein conditions 9)-0) and the Seidel s absolute bounds )-) are special cases of general inequalities obtained for association schemes, which constitute more general combinatorial structures, since strongly regular graphs correspond to the particular case of symmetric association schemes with two classes see, for instance, []) IV AN EUCLIDEAN JORDAN ALGEBRA SPANNED BY THE ADJACENCY MATRIX OF A STRONGLY REGULAR GRAPH From now on, we consider the Euclidean Jordan algebra of real symmetric matrices of order n, V, with the Jordan product defined by AB BA A, B V, A B =, 3) where AB is the usual product of matrices Furthermore, the inner product of V is defined as < A, B >= trab), where tr ) is the classical trace of matrices, that is, the sum of its eigenvalues From now on, we will, we only consider n, k, a, c) strongly regular graphs such that 0 < c k < n Let X be a n, k, a, c)-strongly regular graph and let A be the adjacency matrix of X Then A has three distinct eigenvalues, namely the degree of regularity k of X, and the restricted eigenvalues θ and τ, given in 7) and 8) Recall that k and θ are the nonnegative eigenvalues and τ is the negative eigenvalue of A Now we consider the Euclidean Jordan subalgebra of V, V, spanned by the identity matrix of order n, I n, and the powers of A Since A has three distinct eigenvalues, then V is a three dimensional Euclidean Jordan algebra with rankv ) = 3 Let S = {E 0, E, E } be the unique complete system of orthogonal idempotents of V associated to A, with E 0 = A θ τ)a θτi n k θ)k τ) E = A k τ)a kτi n, )θ k) E = A k θ)a kθi n, τ θ)τ k) = J n n, where J n is the matrix whose entries are all equal to Since V is an Euclidean Jordan algebra that is closed for the Hadamard product of matrices, denoted by see the ISSN:

3 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 06 definition of Hadamard product, for instance, in [9]), and S is a basis of V, then there exist real numbers qα i and qαβ i, 0 i, α, β, α β, such that E α E α = qαe i i, 4) E α E β = qαβe i i 5) These numbers q p α and qp αβ, 0 α, β, α β, are called the Krein parameters of the graph X see []), since q 0 and q 0 yield the Krein admissibility conditions, 9) and 0), presented in [8, Theorem 3] In what follows, we introduce some important notation Firstly, considering S = {E 0, E, E } and rewriting the idempotents under the new basis {I n, A, J n A I n } of V we obtain E 0 = n) I n n) A n) J n A I n ), E = τ n τ k n) I n n τ k n) A τ k n) J n A I n ), E = θn k θ n) I n n k θ n) A k θ n) J n A I n ) 6) Now, we introduce another matrix product, the Kronecker product see [9]), denoted by C D, for matrices C = [C ij ] M m,n R) and D = [D ij ] M p,q R), with m, n, p, q N, defined by C D = C D C n D C m D C mn D Note that M m,n R) is the usual notation for the space of real m n matrices Finally, we introduce the following compact notation for the Hadamard and the Kronecker powers of the elements of S Let x, y, z, α and β be natural numbers such that 0 α, β, x and α < β Then, we define Eα x = E α ) x and Eα x = E α ) x, E yz αβ = E α) y E β ) z and E yz αβ = E α) y E β ) z, E x α β = E α E β ) x and E x α β = E α E β ) x V ASYMPTOTIC PARAMETRIC CONDITIONS FOR STRONGLY REGULAR GRAPHS Let X be a strongly regular graph with parameter set n, k, a, c) and adjacency matrix A Let S = {E 0, E, E } be the unique complete system of orthogonal idempotents of the Euclidean Jordan algebra V associated to A We will make use of the same notation introduced in the paper [6] Let x and ɛ be real positive numbers such that ɛ ]0, [ Since θn k θ)/n)) <, n k θ)/nθ τ)) < and k θ)/n)) <, then the series )i ) x ) i ɛ E i converges to a real number, Sxɛ, and we can write ) x ɛe S xɛ = ) i ) i i ) i x θn k θ = ) i ɛ i I n i n) ) i x n k θ ) i ɛ i A i n) ) i x k θ ) i ɛ i J n A I n ) i n) = θn n ) x I n ) x A ) x J n A I n ) 7) Note that the eigenvalues of S xɛ are real positive numbers Consider the first spectral decomposition of S xɛ, S xɛ = q 0 xɛe 0 q xɛe q xɛe We call the parameters q i xɛ, for i = 0,,, the asymptotic parameters of the strongly regular graph X Proceeding in an asymptotical analysis of the parameter q xɛ, we establish Theorem V which presents a feasibility condition for strongly regular graphs whose regularity is less than a half of its order Theorem V Let X be a strongly regular graph with parameter set n, k, a, c) and distinct restricted eigenvalues θ and τ If k = / β)n, with β ]0, /[, then τ < θθ β) 8) β Proof: Since qxɛ 0, for all x in R and for all ɛ in ]0, [, then, for any real numbers x and ɛ ]0, [, we have 0 θn ) x ) x τ ) x τ ) 9) ISSN:

4 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 06 After some algebraic manipulation of 9) we obtain x ) x θ n ) τ x ) x ) ) x ) x 0) θn Making x 0 in inequality 0) and noting that ) x θn lim x 0 ) x =, we obtain ln τ ln ) ) ) ) θn ) ) Applying the Mean value Theorem to the function f such that fu) [ = ln u), u ]0, [ on the intervals I = ɛ ) ], ɛ θn and I = [ ) ] ɛ, ɛ, by ) we deduce τ u u ɛ θ θn) ɛ θ τ ), ) with u I and u I After some straightforward simplifications of ), we obain τ u u θ θn) θ τ 3) θ τ) ) Since k = β)n, inequality 3) can be written as τ u u Finally, from 4) we have Therefore, θ θn β)n θ θ τ θ τ) β θ 4) θ τ τ u θθ β u β u θ β)θ τ lim ɛ 0 u β and 8) from Theorem V is attained Remark V Let X be a strongly regular graph with parameter set n, k, a, c) and eigenvalues k, θ and τ The inequality 8) presented in Theorem V provides us the following information The absolute value of τ is conditioned by the value of θ and we can say that this restriction is much more perceptible when the regularity k is much smaller than n/ Due to Remark V, we establish a restriction of inequality 8) when k < α, with α < n/ Accordingly, Corollary V presents an inequality for parameter sets with k < α = n/3, that can be easily deduced from Theorem V Corollary V Let X be a strongly regular graph with parameter set n, k, a, c) and distinct eigenvalues k, θ and τ If k < n/3, then τ < θ3θ ) 5) Applying Theorem V to the complement graph X, we establish inequality 6) in Corollary V for strongly regular graphs with parameter set n, k, a, c) such that k = n β), with β ]0, [ Corollary V Let X be a strongly regular graph with parameter set n, k, a, c) and distinct eigenvalues k, θ and τ If k = n/ β), with β ]0, /[, then θ < τ ) τ β) β 6) Remark V As we did for Theorem V with Remark V, we can conclude that the value of β in inequality 6) of Corollary V must not be near zero, in order for the inequality to be useful Therefore, k must be a natural number bigger, but not close, than n/ In the following Corollary, we present a restriction of the inequality 6) for β > /6, which is the same of considering k > n/3 Corollary V3 Let X be a strongly regular graph with parameter set n, k, a, c) and distinct eigenvalues k, θ and τ If k > n/3, then θ < τ )3 τ ) 7) From now on we will compose the idempotent E with the matrix S xɛ and analyze its eigenvalues Observe that all the eigenvalues of E S xɛ are positive since E S xɛ is a principal submatrix of E S xɛ and the eigenvalues of E and S xɛ are all positive here we apply the Eigenvalues Interlacing Theorem, see [9]) Considering the following spectral decomposition E S xɛ = q,xɛe 0 0 q,xɛe q,xɛe and analyzing the parameter q,xɛ 0 we deduce, in Theorem V, another admissibility condition for strongly regular graphs with parameter set n, k, a, c) Theorem V Let X be a strongly regular graph with parameter set n, k, a, c) and distinct eigenvalues k, θ and τ If k = / β)n, with β ]0, /[, then k < θ β)θθ β) 8) β)β ISSN:

5 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 06 Proof: Firstly, we note that the following inequality 9) is verified for any real numbers x and ɛ ]0, [ 0 θn k θ n) n k θ n) k θ n) θn ) x ) x k ) x n k ) 9) Since the elements of S are orthogonal, then E E 0 = 0 and we can conclude that k θ k θ n k ) = θn n k θ n) n) n) k and by some algebraic manipulation of 9) we obtain the inequality 30) ) x k θn k θ n k θ ) x θn x ) x ) ) x θn ) x 30) Making x 0 on the right hand side of 30) we obtain the following: k θn k θ n k θ ln ln ) ) ) ) θn ) 3) Applying the Mean-Value Theorem to the real function fu) = ln [ u), for all u in ]0, [, on the intervals I = ɛ ) ], ɛ θn and I = [ ) ] ɛ, ɛ, we obtain the inequality 3) k θn k θ n k θ u u ) θ θn) θ τ ) ), 3) with u I and u I After some algebraic manipulation of the inequality 3) we obtain k u )θn k θ)θθn k θ)) 33) u )n k θ)n k θ) Recalling that k = β)n) /, with β ]0, [, and after neglecting some terms in the multiplicative factors of both the numerator and denominator of the right hand side of 33) we obtain 34) k < u )θ β)θθ β) 34) u ) β)β Finally, making ɛ 0, then u, u 0 and inequality 8) follows Considering β > 6, which implies that k < n 3, we deduce the following consequence of Theorem V Corollary V4 Let X be a strongly regular graph with parameter set n, k, a, c) and distinct eigenvalues k, θ and τ If k < n/3, then k < 3θ )θ3θ ) 35) Applying Corollary V4 to the complement graph X, we establish Corollary V5 Corollary V5 Let X be a strongly regular graph with parameter set n, k, a, c) and distinct eigenvalues k, θ and τ If k > n/3, then n k < τ ) τ )3 τ ) 36) VI CONCLUSIONS AND EXPERIMENTAL RESULTS We conclude our paper with some experimental results which test the inequalities presented in Section V and deduce some spectral conclusions Firstly we analyze inequality 5) from Corollary V In Table I we consider the parameter sets X = 84, 48,, 6), X = 56, 66,, ) and X 3 = 75, 378, 57, 35) and we test them for the parameter Q = θ3θ) τ, obtained from the inequality 5) Note that each parameter set considered satisfies the condition of Corollary V, that is, k < n/3 X X X 3 θ τ Q TABLE I EXPERIMENTAL RESULTS FOR THE INEQUALITY 5) The results presented in Table I confirm the conclusion that one can deduce from the analysis of inequality 5), that is, when k < n/3, the value of τ cannot be too big relatively to the value of θ Next we analyze inequality 7) from Corollary V3 In Table II we consider the parameter sets from the complement graphs considered before, that is, X = 84, 35, 0, 90), X = 56, 89, 44, 6) and X 3 = 75, 896, 65, 576), and we test them for the parameter Q = τ )3 τ ) θ, obtained from the inequality 7) Note that each ISSN:

6 INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 06 X X X 3 θ τ Q TABLE II EXPERIMENTAL RESULTS FOR THE INEQUALITY 7) parameter set considered satisfies the condition of Corollary V3, that is, k > n/3 In this case we can draw a similar conclusion than before The results presented in Table II confirm that, when k > n/3, the value of θ cannot be too big relatively to the value of τ We proceed our analyzes with inequality 35) from Corollary V4 In Table III we consider the parameter sets X 4 = 300, 9, 0, 36), X 5 = 400, 4, 8, 4) and X 6 = 44, 8, 0, 48), which comply with the conditions of Corollary V4, and we also consider the parameter Q 3 = 3θ )θ3θ )/ k, obtained from inequality 35) X 4 X 5 X 6 θ τ Q TABLE III EXPERIMENTAL RESULTS FOR THE INEQUALITY 35) The results presented in Table III confirm the conclusions one can draw from the inequality 35) of Corollary V4, that is, if k < n/3, then k cannot be much bigger than the value of θ, since inequality 35) establishes that k is smaller than a polynomial in θ of degree 3 with positive coefficients Finally, in Table IV, we analyze the complement parameter sets of the graphs previously considered, that is, X 4 = 300, 07, 50, 6), X 5 = 400, 85,, 80) and X 6 = 44, 3, 3, 95), which comply with the conditions of Corollary V5, and we also consider the parameter Q 4 = τ ) τ )3 τ ) n k, obtained from inequality 36) In this paper we have established parametric admissibility conditions which enlightened some relations between the elements of the spectrum of a strongly regular graph These relations were highlighted in the previous paragraphs ACKNOWLEDGMENTS Luís Vieira in this work was partially supported by CMUP UID/MAT/0044/03), which is funded by FCT Portugal) with national MEC) and European structural funds FEDER), under the partnership agreement PT00 REFERENCES [] E Bannai and T Ito Algebraic Combinatorics I Association Schemes Benjamin-Cummings, Menlo Park, CA 984) [] R C Bose Strongly regular graphs, partial geometries and partially balanced designs Pacific J Math 3 963): [3] D M Cardoso and L A Vieira Euclidean Jordan Algebras with Strongly Regular Graphs Journal of Mathematical Sciences, 0:88 894, 004 [4] Ph Delsarte, J-M Goethals, JJ Seidel Bounds for systems of lines and Jacobi polynomials Philips Res Rep ): 9 05 [5] J Faraut and A Korányi Analysis on Symmetric Cones Oxford Science Publications, 994 [6] L Faybusovich Euclidean Jordan algebras and Interior-point algorithms J Positivity, 997): [7] L Faybusovich Linear systems in Jordan algebras and primal-dual interior-point algorithms Journal of Computational and Applied Mathematics ): [8] C Godsil and G Royle Algebraic Graph Theory Springer, 00 [9] R Horn and C R Johnson Matrix Analysis Cambridge University Press, 985 [0] P Jordan, J v Neuman and E Wigner On an algebraic generalization of the quantum mechanical formalism Annals of Mathematics ): 9 64 [] M Koecher The Minnesota Notes on Jordan Algebras and Their Applications, Edited and Annotated by Aloys Krieg and Sebastian Walcher Springer, Berlin, 999 [] J H van Lint and R M Wilson A Course in Combinatorics Cambridge University Press, Cambridge, 004 [3] H Massan and E Neher Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras Ann Statist 6 998): [4] L L Scott Jr A condition on Higman s parameters [5] VM Mano, E A Martins and LAA Vieira Feasibility Conditions on the Parameters of a Strongly Regular Graph Electronic Notes in Discrete Mathematics 38, 607-?63, 0) Notices Amer Math Soc 0 973), A-97 [6] VM Mano, L A Vieira and EA Martins On Generalized binomial series and strongly regular graphs Proyecciones 3 03), X 4 X 5 X 6 θ τ Q TABLE IV EXPERIMENTAL RESULTS FOR THE INEQUALITY 36) Our last experiments presented in Table IV are analogous to the ones from our previous experiments Here, one can say that the regularity of the complement graph, n k, cannot be much bigger than the absolute value of the negative restricted eigenvalue, since it is restricted by a polynomial in τ of degree 3 which is always positive ISSN: