1,790 research outputs found
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
Quantum Computing and Quantum Algorithms
The field of quantum computing and quantum algorithms is studied from the ground up. Qubits and their quantum-mechanical properties are discussed, followed by how they are transformed by quantum gates. From there, quantum algorithms are explored as well as the use of high-level quantum programming languages to implement them. One quantum algorithm is selected to be implemented in the Qiskit quantum programming language. The validity and success of the resulting computation is proven with matrix multiplication of the qubits and quantum gates involved
COMPLEX HADAMARD MATRICES AND APPLICATIONS
A complex Hadamard matrix is a square matrix H ∈ M N (C) whose entries are on the unit circle, |H ij | = 1, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, F N = (w ij) with w = e 2πi/N. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. CONTENT
Matrix positivity preservers in fixed dimension. I
A classical theorem proved in 1942 by I.J. Schoenberg describes all
real-valued functions that preserve positivity when applied entrywise to
positive semidefinite matrices of arbitrary size; such functions are
necessarily analytic with non-negative Taylor coefficients. Despite the great
deal of interest generated by this theorem, a characterization of functions
preserving positivity for matrices of fixed dimension is not known.
In this paper, we provide a complete description of polynomials of degree
that preserve positivity when applied entrywise to matrices of dimension .
This is the key step for us then to obtain negative lower bounds on the
coefficients of analytic functions so that these functions preserve positivity
in a prescribed dimension. The proof of the main technical inequality is
representation theoretic, and employs the theory of Schur polynomials.
Interpreted in the context of linear pencils of matrices, our main results
provide a closed-form expression for the lowest critical value, revealing at
the same time an unexpected spectral discontinuity phenomenon.
Tight linear matrix inequalities for Hadamard powers of matrices and a sharp
asymptotic bound for the matrix-cube problem involving Hadamard powers are
obtained as applications. Positivity preservers are also naturally interpreted
as solutions of a variational inequality involving generalized Rayleigh
quotients. This optimization approach leads to a novel description of the
simultaneous kernels of Hadamard powers, and a family of stratifications of the
cone of positive semidefinite matrices.Comment: Changed notation for extreme critical value from to
. Addressed referee remarks to improve exposition, including
Remarks 1.2 and 3.3. Final version, 39 pages, to appear in Advances in
Mathematic
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