Combinatorial Perron parameters for trees

Abstract

The notion of combinatorial Perron value was introduced in [2]. We continue the study of this parameter and also introduce a new parameter πe(M) which gives a new lower bound on the spectral radius of the bottleneck matrix M of a rooted tree. We prove a bound on the approximation error for πe(M). Several properties of these two parameters are shown. These ideas are motivated by the concept of algebraic connectivity. A certain extension property for the combinatorial Perron value is shown and it allows us to define a new center concept for caterpillars. We also compare computationally this new center to the so-called characteristic set, i.e., the center obtained from algebraic connectivity.