The spectral radius of a matrix is a widely used concept in linear algebra. It expresses the asymptotic growth rate of successive powers of the matrix. This concept can be extended to sets of matrices, leading to the notion of "joint spectral radius". The joint spectral radius of a set of matrices was defined in the 1960's, but has only been used extensively since the 1990's. This concept is useful to study the behavior of multi-agent systems, to determine the continuity of wavelet basis functions or for expressing the capacity of binary codes. Although the joint spectral radius shares some properties with the usual spectral radius, it is much harder to compute, and the problem of approximating it is NP-hard. In this thesis, we first review theoretical results that lead to basic approximations of the joint spectral radius. Then, we list various specific cases where it is effectively computable, before presenting a specific type of sets of matrices, for which we solve the problem of computing it with a polynomial computational cost.

abstract: Persistence theory provides a mathematically rigorous answer to the question of population survival by establishing an initial-condition- independent positive lower bound for the long-term value of the population size. This study focuses on the persistence of discrete semiflows in infinite-dimensional state spaces that model the year-to-year dynamics of structured populations. The map which encapsulates the population development from one year to the next is approximated at the origin (the extinction state) by a linear or homogeneous map. The (cone) spectral radius of this approximating map is the threshold between extinction and persistence. General persistence results are applied to three particular models: a size-structured plant population model, a diffusion model (with both Neumann and Dirichlet boundary conditions) for a dispersing population of males and females that only mate and reproduce once during a very short season, and a rank-structured model for a population of males and females. / Dissertation/Thesis / Ph.D. Mathematics 2014

This thesis is devoted to the analysis of problems that arise when long products of matrices taken in a given set are constructed. A typical application is the stability of switched linear systems. The stability of a discrete-time linear system is a classical engineering problem that has been well understood for long: the dynamics can be expressed in terms of the eigenvalues of the matrix ruling the system. A more complicated problem arises when the dynamical system can switch, that is, if the matrix changes over time. If this matrix is taken from a given set but can be chosen arbitrarily in this set at every time, the stability problem turns to the computation of a quantity, the joint spectral radius of the set of matrices, introduced in the early sixties. While this quantity appears to be hard to compute, it has acquired more and more importance during the last decades, and new applications of the joint spectral radius in engineering or mathematics are frequently discovered. It has for instance been proved useful for the analysis of regularity of fractals, for the continuity of wavelets, or for autonomous agents detection in sensor networks. In the first part of this thesis, we present a theoretical survey of the joint spectral radius, including old and new results. The joint spectral subradius, which is its stabilizability counterpart, is also considered. In a second part, we study some applications related to long products of matrices. We first analyse in detail a problem in coding theory, that has been recently shown to involve a joint spectral radius computation. We then propose a new application of the joint spectral radius (and related quantities) to a classical problem in number theory, namely the counting of overlap-free words. We then turn to problems related with autonomous agents detection: we analyse the trackability of sensor networks, and introduce and analyse a new notion, namely the observability of sensor networks.

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We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of the vertices induced by breadth-first search. For trees the resulting structure is uniquely determined up to isomorphism. We also show that the largest spectral radius in such classes of trees is strictly monotone with respect to majorization. This paper is the revised final version of the preprint no. 35 of this research report series. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency / Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence. / Series: Research Report Series / Department of Statistics and Mathematics

A semiregular tree is a tree where all non-pendant vertices have the same degree. Belardo et al. (MATCH Commun. Math. Chem. 61(2), pp. 503-515, 2009) have shown that among all semiregular trees with a fixed order and degree, a graph with index is caterpillar. In this technical report we provide a different proof for this theorem. Furthermore, we give counter examples that show that this result cannot be generalized to the class of trees with a given (non-constant) degree sequence. / Series: Research Report Series / Department of Statistics and Mathematics

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We show that amongst all trees with a given degree sequence it is a ball where the vertex degrees decrease with increasing distance from its center that maximizes the spectral radius of the graph (i.e., its adjacency matrix). The resulting Perron vector is decreasing on every path starting from the center of this ball. This result it also connected to Faber-Krahn like theorems for Dirichlet matrices on trees. The above result is extended to connected graphs with given degree sequence. Here we give a necessary condition for a graph that has greatest maximum eigenvalue. Moreover, we show that the greatest maximum eigenvalue is monotone on degree sequences with respect to majorization. (author's abstract). Note: There is a more recent version of this paper available: "Graphs with Given Degree Sequence and Maximal Spectral Radius", Research Report Series / Department of Statistics and Mathematics, no. 72. / Series: Research Report Series / Department of Statistics and Mathematics