UPPER AND LOWER BOUNDS FOR THE POWER OF EIGENVALUES IN SEIDEL MATRIX

Abstract

In this paper, we generalize the concept of the energy of Seidel matrix S(G) which denoted by S^α(G) and obtain some results related to this matrix. Also, we obtain an upper and lower bound for S^α(G) related to all of graphs with |detS(G)| ≥ (n - 1); n ≥ 3.

1. Introduction

All of graphs considered in this paper are finite, undirected and simple. Let G be a graph with vertex set V(G) and edge set E(G). Let A(G) be the (0,1)-adjacency matrix of G. In 1966, Van Lint and Seidel in [13] introduced a symmetric(0,-1,1)-adjacency matrix for a graph G, called the Seidel matrix of G as S(G) = J − 2A(G) − I, where J is a square matrix which all of entries are equal to 1. Thus S(G) has 0 on the diagonal and ±1 off diagonal, where -1 indicates adjacency, unless is equal to 1. It is obvious that -S(G) is the Seidel matrix of the complement of G. Haemers in [10], similar to the normal energy, defined the Seidel energy Es(G) of G which is the sum of the absolute values of the eigenvalues of the Seidel matrix. For example consider the complete graph Kn, its Seidel matrix is I − J. Hence the eigenvalues of S(Kn) are (1-n) and 1 with multiplicity (n-1). So Es(Kn) = 2n − 2. The Seidel matrix of a graph can be interpreted as the incidence matrix of a design, or as the generator matrix of an alternative binary code. We refer the reader to [5, 6, 9] for more information related to eigenvalue and adjacency matrix and their properties.

In section 2, we proceed with the study of generalization of Seidel matrix and define Sα(G) and we obtain a lower bound for Sα(G), α ≥ 2.

Section 3 contains a brief summary of KKT method and establishes the relation between nonlinear programming and upper bound.

In [7], Ghorbani obtained a lower bound for Sα(G), 0 ≤ α ≤ 2. In this section we obtain an upper bound for Sα(G), α ≥ 2. As for prerequisites, the reader is expected to be familiar with nonlinear programming. Undefined notations and terminology from nonlinear programming, can be found in [2, 12].

 

2. The generalization of Seidel matrix

At first, we define the concept of generalized Seidel matrix and then we obtain some results related to this concept.

Definition 2.1. Let λ1, λ2, …, λn be the eigenvalues of the Seidel matrix S(G). The power of eigenvalues of Seidel matrix S(G) is denoted by Sα(G) and define as follows:

Remark 2.1. If α = 1, then Sα(G) = Es(G); i.e., Sα(G) is a generalization of Seidel energy of G.

For the proof of the next theorem, we need the concept of conference graph.

Definition 2.2 (Conference matrix [10]). A conference matrix is a square matrix C of order n with zero diagonal and ±1 off diagonal, such that CCT = (n − 1)I. If C is symmetric, then C is the Seidel matrix of a graph and this graph is called a conference graph.

Conference matrices are a class of Hadamard matrices and its have the Hadamard properties. For more details we refer the reader to [3, 10]. The remainder of this section will be devoted to the proof of a lower bound for Sα(G) for all α ≥ 2.

Definition 2.3 (Hölder’s Inequalities [8]). Let with p, q > 1. Then Hölder’s inequality for the n-dimensional Euclidean space, when the set S is {1, …, n} with the counting measure, we have

for all X = (x1, x2, …, xn), Y = (y1, y2, …, yn) ∈ ℝn with equality when |bk| = c|ak|p−1. If p = q = 2, this inequality becomes Cauchy’s inequality.

Theorem 2.4. Let G be a graph with n vertices, then for all α ≥ 2, n ≥ 3, and equality holds if and only if G is a conference graph.

Proof. Let λ1, λ2, …, λn be the eigenvalues of the Seidel matrix S(G), then the trace of S2(G) is equal . Let and Y = (1, 1, …, 1). By the Hölder’s Inequalities, we have |XTY | ≤ ║X║p║Y║q, . But since = n(n - 1), we have n(n - 1) ≤ , and hence , therefore ≥ n(n - 1)p. We assume that 2p = α, then we get Sα(G) = with equality if and only if |λi| = for i = 1, …, n. Moreover, if each eigenvalue equal to , then S(G)ST(G) = S2(G) = (n − 1)I, which means that the Seidel matrix S(G) is a symmetric and hence the graph G is a conference graph. □

 

3. Computation of upper bound of Sα(G) by using KKT method

In nonlinear programming, the Karush-Kuhn-Tucker (KKT) conditions are necessary for a local solution to a maximization problem provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. This construction is adapted from [1, 2, 12]. We consider the following nonlinear optimization problem:

where I and J are finite sets of indices. Suppose that the objective function f : ℝn → ℝ and the constraint functions gi : ℝn → ℝ, i ∈ I and hj : ℝn → ℝ, j ∈ J are continuously differentiable at a point X∗.

Definition 3.1. Let X∗ be a point satisfying the constraints:

and let I∗ be the set of indices i for which gi(X∗) = 0. Then X∗ is said to be a regular point of the constraints (3.2), if the gradient vectors ∇hj , j ∈ J , ∇gi(X∗), i ∈ I∗ are linearly independent.

Definition 3.2 (Karush-Kuhn-Tucker Conditions(KKT) [1, 12]). Let X∗ be a relative maximum point for the problem (3.1) and suppose X∗ is a regular point for the constraints, then there exist constants μj (j ∈ J), λi (i ∈ I), which these constants are called KKT multipliers, such that the following conditions are hold:

Stationarity

Primal feasibility

Dual feasibility

Complementary slackness

In the particular case, set I is empty, i.e., when there are no inequality constraints, the KKT conditions turn into the Lagrange conditions, and the KKT multipliers are called Lagrange multipliers.

Definition 3.3 (Linear Independence Constraint Qualification [2, 12]). Given the point X∗ is feasible and the active set I∗ = {i|gi(X∗) = 0, i ∈ I} defined in (3.2), we say that the linear independence constraint qualification (LICQ) holds if the set of active constraint gradients {∇gi(X∗), i ∈ I∗)} is linearly independent.

Theorem 3.4 (LICQ and Multipliers [12]). Given a point X∗, that satisfies the KKT conditions, along with an active set with multipliers , if LICQ holds at X∗, then the multipliers are unique.

Definition 3.5 (Mangasarian-Fromovitz Constraint Qualification [11]). Given X∗ is a local solution of (3.1), and active set is A(X∗). The Mangasarian-Fromovitz Constraint Qualification MFCQ is defined by linear independence of the equality constraint gradients and the existence of a search direction d such that ∇gi(X∗)T d < 0,∇hj(X∗)T d = 0, for all i, j in A(X∗).

The MFCQ is always satisfied if the LICQ is satisfied. Also, satisfaction of the MFCQ leads to bounded multipliers, μ∗, λ∗, although they are not necessarily unique.

Theorem 3.6 ([11]). If a local maximum X∗ of the function f(X) subject to the constraints gi(X∗) = 0, i ∈ I∗, hj(X∗) = 0, j ∈ J, satisfies MFCQ, then it satisfies the KKT conditions.

Now we continue by recalling the relevant upper bound of eigenvalue Seidel matrix. Let G = (V,E) be a simple, undirected graph on vertex set V = {v1, …, vn} and let λ1 ≥ λ2 ≥ … ≥ λn be the set of all eigenvalues of S(G). We formulate this as an optimization problem. For doing this case, we need to come up with appropriate constraints. The following assumption will be needed throughout the paper. The main constraint is made by the assumption |detS(G)| ≥ n − 1. The other ones are obtained by the following straightforward lemma.

Lemma 3.7. For any graph G with n vertices, we have:

Lemma 3.8 ([4]). Suppose α, β, ν, ω, a, b, c, d are positive numbers and that

Then the inequality αap + βbp ≤ νcp + ωdp holds for p ≥ 1.

The reminder of this section will be devoted to the proof of Theorem 3.9.

Theorem 3.9. Let G be a graph with n ≥ 3 vertices and let λ1, λ2, …, λn be the eigenvalues of S(G). If |detS(G)| ≥ n − 1, then the following condition is hold:

Proof. We prove this theorem by using of KKT method in nonlinear programming. Now, we can describe our problem as the maximization of the function f(X) with assume |λi|2 = xi. Hence, we have:

Since |detS(G)| ≥ n − 1, we have λi > 0, 1 ≤ i ≤ n and hence δ must be non zero and non-negative, as a fixed number. Also, for all i, we must have xi ≠ δ, because, if for some i, xi = δ, then < (n - 1)2 which is contradiction with (3.7). In continue, we need prove the following claim:

Claim 3.10. Let λ be a local maximum of f(X) according to the constraints (3.4)-(3.9). Then λ satisfies MFCQ.

Proof: Let λ = (λ1, …, λn). Without loss of generality, we can assume λ1 ≥ λ2 ≥ … ≥ λn. If λ1 = λn, then in view of (3.4), all of λi are equal to n−1. In this case, in the above formulas (3.5) till (3.9), we have the equality only in (3.6) and hence λ is not a local maximum where f(λ) = n(n − 1)p < (n − 1)2p + (n − 1), p > 1, n ≥ 3, and therefore λ is not satisfies MFCQ. If λ1 > λn, then MFCQ is fulfilled by setting d = (1, 0, …, 0,−1). □

Now, we continue the proof of Theorem 3.9. We show that the maximum value of f(X) according to conditions (3.4)-(3.9) is equal to (n − 1)2p + (n − 1). So assume that X = (x1, x2, …, xn) is a local maximum of f(X) subject to the constraints (3.4) - (3.9). With no loss of generality suppose that x1 ≥ x2 ≥ … ≥ xn. By Theorem (3.6) and Lemma (3.7), X satisfies KKT conditions, namely:

By the choice of δ, we have ni(X) < 0 for i = 1, 2, …, n and hence by (3.14), γ1 = γ2 = … = γn = 0. We assume that D = , then (3.10) can be written as

We consider the following cases:

Case 1:

Let x1 = (n − 1)2. Then by (3.11) and since X satisfies (3.7), we have

It turns out that x2 = x3 = … = xn = 1 and we have f(X) = (n − 1)2p + (n − 1).

Case 2:

Let x1 < (n − 1)2. So, by (3.13), ρ1 = ρ2 = … = ρn = 0. It turns out that x1, x2, …, xn must satisfy the following equation:

Assume that μ = μ2 − μ3, then we have:

The curves of y = pxp and Parabolic curve y = −μ4D −μ1x−2μx2 intersect in at most two points, i.e., the formula (3.17) at most two distinct positive roots. Now, we have two subcases:

Subcase(i): We have one positive root. Then by (3.11), x1 = x2 = … = xn = n − 1. Hence f(X) = n(n − 1)p which is smaller than (n − 1)2p + (n − 1), for n > 3, p > 1.

Subcase(ii): If (3.18) has two positive roots, say a and b, then by Lemma 3.8, we assume that c = (n − 1)2 and d = 1, we have f(X) ≤ (n − 1)2p + (n − 1), which is the desired conclusion and the proof is completed. □