A generalization for the clique and independence numbers

This paper is concerned with some rank and inertia optimization problems of the Hermitian matrix-valued functions A + BXB * subject to restrictions. We first establish several groups of explicit formula for calculating the maximum and minimum ranks and inertias of matrix sum A + X subject to a Hermitian matrix X that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximum and minimum ranks and inertias of the matrix-valued function A + BXB * subject to a Hermitian matrix X that satisfies a fixed-rank and semi-definiteness restrictions, and show various properties A + BXB * from these ranks and inertias formulas. In particular, we give necessary and sufficient conditions for the equality A + BXB * = 0 and the inequality A + BXB * 0 (0, ≺ 0, 0) to hold respectively for these specified Hermitian matrices X.

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New bounds for the spread of the signless Laplacian spectrum

Güngör¹,

Çevik²,

Habibi³

2014

Mathematical Inequalities &Amp; Applications

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Abstract. The spread of the singless Laplacian of a simple graph G is defined as SQ(G) = μ 1 (G) − μ n (G) , where μ 1 (G) and μ n (G) are the maximum and minimum eigenvalues of the signless Laplacian matrix of G , respectively. In this paper, we will present some new lower and upper bounds for SQ(G) in terms of clique and independence numbers. In the final section, as an application of the theory obtained in here, we will also show some new upper bounds for the spread of the singless Laplacian of tensor products of any two simple graphs.Mathematics subject classification (2010): 05C50, 15A36, 15A42.

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A generalization for the clique and independence numbers

Abstract: Abstract. In this paper, lower and upper bounds for the clique and independence numbers are established in terms of the eigenvalues of the signless Laplacian matrix of a given graph G.

Cited by 3 publications

References 12 publications

On the dot product of graphs over monogenic semigroups

On the dot product of graphs over monogenic semigroups

A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$

New bounds for the spread of the signless Laplacian spectrum

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