On convexity properties of the spectral radius of nonnegative matrices

Elementary matrix-theoretic proofs are given for the following well-known results: r(D) = max{Re λ : λ an eigenvalue of A + D} and s(D) = lnρ(e^DA) are convex. Here D is diagonal, A a nonnegative n × n matrix, and ρ the spectral radius.     

On the preservation of log convexity and log concavity under some classical Bernstein-type operators

This paper analyzes the preservation of both the log convexity and the log concavity under certain Bernstein-type operators. Some results are provided for the Bernstein, Szász, Baskakov, the gamma-type and the Weierstrass operators. Probabilistic methods support the proofs of these results.     

Sharp upper and lower bounds for the Laplacian spectral radius and the spectral radius of graphs

In this paper, sharp upper bounds for the Laplacian spectral radius and the spectral radius of graphs are given, respectively. We show that some known bounds can be obtained from our bounds. For a bipartite graph G, we also present sharp lower bounds for the Laplacian spectral radius and the spectral radius, respectively.     

On an inequality for spectral radius

In this note, we first extend and then give a related result to an inequality involing the spectral radius of nonnegative matrices that recently appcared in the literature.     

Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy

It was recently shown by Nikodem that a function defined on an open convex subset of R ⁿ is convex if and only if it is midpoint convex and quasiconvex. It is shown that quasiconvexity can be replaced by strict quasiconvexity and that the openness condition can be removed altogether. The domain can then be taken from a general real linear space. There will also be given some related results of a “local” nature     

Convexity theories III. Classification of certain real convexity theories

In this paper we classify all real convexity theories that contain the standard convexity theory _c. For this purpose we consider three subcases: finitary; infinitary and (_sc\_c)Ø; infinitary and _sc=_c. In each of these subcases one encounters a phenomenon resembling bifurcation.This research was supported by the Deutsche Forschungsgemeinschaft.     

Sharp bounds for the spectral radius of digraphs

Let $G=(V\text{,}E)$ be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius $\rho (G)$ of G is the largest eigenvalue of its adjacency matrix. In this paper, the following sharp bounds on $\rho (G)$ have been obtained.$\min \{\sqrt{t_i^{+}{t}_j^{+}}:(v_i\text{,}{v}_j)\in E\}?\rho (G)?\max \{\sqrt{t_i^{+}{t}_j^{+}}:(v_i\text{,}{v}_j)\in E\}$where G is strongly connected and $t_i^{+}$ is the average 2-outdegree of vertex $v_i$. Moreover, each equality holds if and only if G is average 2-outdegree regular or average 2-outdegree semiregular.     

On the joint spectral radius of matrices of order 2 with equal spectral radius

We provide explicit formulas for the joint spectral radius of certain classes of pairs of real matrices of order 2 with equal spectral radius.     

Walks and the spectral radius of graphs

Given a graph G, write μ(G) for the largest eigenvalue of its adjacency matrix, ω(G) for its clique number, and w_k(G) for the number of its k-walks. We prove that the inequalities

     

Functions of operators and the spectral radius

This paper contains a connected account of results concerning the maximum problem raised by the first-named author in [21] and of its generalizations. For a number of results simplified proofs are given, new estimates are obtained, and important connections with stability theory and with classical function theory are pointed out.     

Duality results for the joint spectral radius and transient behavior

For linear inclusions in discrete or continuous time several quantities characterizing the growth behavior of the corresponding semigroup are analyzed. These quantities are the joint spectral radius, the initial growth rate and (for bounded semigroups) the transient bound. It is discussed how these constants relate to one another and how they are characterized by various norms. A complete duality theory is developed in this framework, relating semigroups and dual semigroups and extremal or transient norms with their respective dual norms.