The matrix calculus is widely applied in various branches of mathematics and control system engineering. In this paper properties of real matrices with nonnegative elements are studied. The classical Collatz theorem is unique and immediately applicable to estimating the spectral radius of nonnegative irreducible matrices. The coherence property is identified. Then the Perron–Frobenius theorem a...

The matrix calculus is widely applied in various branches of mathematics and control system engineering. In this paper properties of real matrices with nonnegative elements are studied. The classical Collatz theorem is unique and immediately applicable to estimating the spectral radius of nonnegative irreducible matrices. The coherence property is identified. Then the Perron–Frobenius theorem a...

In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the ex...

The largest eigenvalue of the adjacency matrix of a network (referred to as the spectral radius) is an important metric in its own right. Further, for several models of epidemic spread on networks (e.g., the ‘flu-like’ SIS model), it has been shown that an epidemic dies out quickly if the spectral radius of the graph is below a certain threshold that depends on the model parameters. This motiva...

We prove a number of relations between the number of cliques of a graph G and the largest eigenvalue (G) of its adjacency matrix. In particular, writing ks (G) for the number of s-cliques of G, we show that, for all r 2; r+1 (G) (r + 1) kr+1 (G) + r X s=2 (s 1) ks (G) r+1 s (G) ; and, if G is of order n; then kr+1 (G) (G) n 1 + 1 r r (r 1) r + 1 n r r+1 :

We consider spectral radius algebras associated to operators of the form h(S), where h ∈ H∞ and S is the unilateral shift. We show that, for a large class of H∞ functions, Bh(S) is weakly dense in LH. In this paper we continue the study of the spectral radius algebras (SRA) initiated in [2]. These algebras represent a very interesting class of non-selfadjoint, non-closed operator algebras. Furt...

The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm always exists. This approach lends itself easily to the analysis of further properties of the generali...

We prove an a priori lower bound for the pressure, or p-norm joint spectral radius, of a measure on the set of d × d real matrices which parallels a result of J. Bochi for the joint spectral radius. We apply this lower bound to give new proofs of the continuity of the affinity dimension of a selfaffine set and of the continuity of the singular-value pressure for invertible matrices, both of whi...

Let G be a simple connected graph with vertex set V = {1, 2, . . . , n}. Let d(i, j) denote the distance between vertices i and j. For i ∈ V , the degree of i and the average of the degree of the vertices adjacent to i are denoted by di and mi, respectively. The 2-degree of vertex i is denoted by ti, which is the sum of degrees of the vertices adjacent to i, that is ti = midi. Let Ni be the sum...