A signed adjacency matrix is a {−1, 0, 1}-matrix A obtained from the adjacency matrix A of a simple graph G by symmetrically replacing some of the 1’s of A by −1’s. Bilu and Linial have conjectured that if G is k-regular, then some A has spectral radius ρ(A) ≤ 2 √ k − 1. If their conjecture were true then, for each fixed k > 2, it would immediately guarantee the existence of infinite families o...

If A is a square matrix with spectral radius less than 1 then A k 0 as k c, but the powers computed in finite precision arithmetic may or may not converge. We derive a sufficient condition for fl(Ak) 0 as k x) and a bound on [[fl(Ak)[[, both expressed in terms of the Jordan canonical form of A. Examples show that the results can be sharp. We show that the sufficient condition can be rephrased i...

For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L−1 holds, where |H| = (|d| + |d|∗)2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G′ allows fo...

A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius prope...

The spectral radius (i.e., the largest eigenvalue) of the adjacency matrices of complex networks is an important quantity that governs the behavior of many dynamic processes on the networks, such as synchronization and epidemics. Studies in the literature focused on bounding this quantity. In this paper, we investigate how to maximize the spectral radius of interdependent networks by optimally ...

A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius prope...

In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from ‘strong stability’ forms of the corresponding (pure) extremal results. These results hold for the α-spectral radius defined using the α-norm for any α > 1; the usual spectral radius is the case α = 2. Our results imply that any hypergraph Turán problem ...

In this paper, we determine graphs with the largest spectral radius among all the unicyclic and all the bicyclic graphs with n vertices and k pendant vertices, respectively. © 2005 Elsevier Inc. All rights reserved. AMS classification: 05C50