Universal Adjacency Matrices with Two Eigenvalues
نویسندگان
چکیده
منابع مشابه
On the Main Eigenvalues of Universal Adjacency Matrices and U-Controllable Graphs
A universal adjacency matrix U of a graph G is a linear combination of the 0–1 adjacency matrix A, the diagonal matrix of vertex degrees D, the identity matrix I and the matrix J each of whose entries is 1. A main eigenvalue of U is an eigenvalue having an eigenvector that is not orthogonal to the all–ones vector. It is shown that the number of distinct main eigenvalues of U associated with a s...
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The universal connected correlations proposed recently between eigenvalues of unitary random matrices is examined numerically. We perform an ensemble average by the Monte Carlo sampling. Although density of eigenvalues and a bare correlation of the eigenvalues are not universal, the connected correlation shows a universal behavior after smoothing. Typeset using REVTEX 1 Brézin and Zee [1–3] hav...
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Proof. We first recall that every non-singular matrix B can be written B = QR, where Q is an orthonormal matrix Q and R is upper-triangular matrix R with positive diagonals1 We will use a slight variation of this fact, writing B = RQ. Now, since QT = Q−1, QAQT has exactly the same eigenvalues as A. Let Rt be the matrix t ∗R+ (1− t)I, and consider the family of matrices Mt = RtQAQR t , as t goes...
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ژورنال
عنوان ژورنال: SSRN Electronic Journal
سال: 2010
ISSN: 1556-5068
DOI: 10.2139/ssrn.1717756