Search

The spectral radius ρ (A) is the maximum absolute value of the eigenvalues of a matrix A. In this paper, we establish some relationship between the spectral radius of a symmetric matrix and its principal submatrices, i.e., if A is partitioned as a 2 × 2 block matrix A = ( 0 A 12 A 21 A 22 ) , then ρ (A) 2 ≤ ρ 2 2 + θ ⁎ , where θ ⁎ is the largest real root of the equation μ 2 = (x − ν) 2 (ρ 2 2 + x) and ρ 2 = ρ (A 22) , μ = ρ (A 12 A 22 A 21) , ν = ρ (A 12 A 21). Furthermore, the results are used to obtain several upper bounds of the spectral radius of graphs, which strengthen or improve some known results. [ABSTRACT FROM AUTHOR]

Let M n be the space of real matrices of order n. The sign-real spectral radius ξ : M n → R + , introduced in a 1997 paper by S.M. Rump, intervenes for instance in the problem of estimating the componentwise distance to singularity. The function ξ has also a bearing in the analysis of generalized absolute value equations and linear complementarity problems. Although ξ is not a norm, it is at least absolutely homogeneous and continuous. Furthermore, ξ is invariant under transposition, permutation similarity, and a few other linear isomorphisms on M n. A matrix A ∈ M n is called sign-real expansive if ξ (A) ≥ 1. Let Ω n be the set of such matrices. The purpose of this work is to discover new properties of the function ξ and to explore in detail the structure of the set Ω n. [ABSTRACT FROM AUTHOR]

The energy of a graph is defined as the sum of absolute values of all eigenvalues of its adjacency matrix. For a nonempty graph G , S. Akbari, A. Alazemi, M. Andelić and M.A. Hosseinzadeh proposed a conjecture: The energy of the line graph of G is at least | E (G) | + Δ (G) − 3 , where E (G) is the edge set of G and Δ (G) is the maximum degree of G. In this paper, we give a proof confirming the conjecture, and present a lower bound and an upper bound for the energy of line graphs of regular graphs. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the global and finite convergence of the semi-smooth Newton method for solving a piecewise linear system that arises in cone-constrained quadratic programming problems and absolute value equations. We first provide a negative answer via a counterexample to a conjecture on the global and finite convergence of the Newton iteration for symmetric and positive definite matrices. Additionally, we discuss some surprising features of the semi-smooth Newton iteration in low dimensions and its behavior in higher dimensions. Secondly, we present two iterative schemes inspired by the classical Jacobi and Gauss-Seidel methods for linear systems of equations for finding a solution to the problem. We study sufficient conditions for the convergence of both proposed procedures, which are also sufficient for the existence and uniqueness of solutions to the problem. Lastly, we perform some computational experiments designed to illustrate the behavior (in terms of CPU time) of the proposed iterations versus the semi-smooth Newton method for dense and sparse large-scale problems. Moreover, we included the numerical solution of a discretization of the Boussinesq PDE modeling a two-dimensional flow in a homogeneous phreatic aquifer. [ABSTRACT FROM AUTHOR]

It is well known that there are Q -linear discontinuous functions from R to R. Moreover, such functions have dense graph in the real plane. In this paper, we study the existence of linear spaces contained in the family of linear discontinuous functions in the generalized context of topological vector spaces and fields endowed with an absolute value. Furthermore, in the case of fields endowed with an absolute value, we study the algebraic genericity of the families of (1) linear discontinuous functions that are not everywhere surjective and (2) linear discontinuous functions that are strongly everywhere surjective. This extends a number of previous results of lineability theory. Generalizations of classical results of Q -linear discontinuous functions are also analyzed. [ABSTRACT FROM AUTHOR]

The energy of a graph G , E (G) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) [3] it was conjectured that for every graph G with maximum degree Δ (G) and minimum degree δ (G) whose adjacency matrix is non-singular, E (G) ≥ Δ (G) + δ (G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E (G). In this paper, first we show that E (L (G)) ≥ | E (G) | + Δ (G) − 5 , where L (G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with vertex set V and edge set E , and let d i be the degree of the vertex v i ∈ V. If the vertices v i and v j are adjacent, we denote the respective edge by v i v j ∈ E. A vertex-degree-based topological index φ is defined as φ (G) = ∑ v i v j ∈ E φ d i , d j , where φ i , j is a function with the property φ i , j = φ j , i. The general extended adjacency matrix A φ is defined as [ A φ ] i j = φ d i , d j if v i v j ∈ E , and 0 otherwise. The energy associated to φ of G is the sum of the absolute values of the eigenvalues of A φ. In this paper we show that ρ (G) E φ (G) ≥ 2 φ (G) for all connected graphs G , where ρ (G) is the spectral radius of G and φ a , b ≠ 0 for all a , b ∈ N. We also characterize the graphs where equality holds. As a consequence, for any tree T with n vertices, E φ (T) ≥ 2 n − 1 φ 1 , (n − 1) , with equality holding if and only if T ≅ S n. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and let T be an acyclic set of edges of G. A partial orientation σ of G with respect to T is an orientation of the edges of G except those edges of T , the resulting graph associated with which is denoted by G T σ. In this paper we prove that there exists a partial orientation σ of G with respect to T such that the largest eigenvalue of the Hermitian adjacency matrix of G T σ is at most the largest absolute value of the roots of the matching polynomial of G. [ABSTRACT FROM AUTHOR]