Original version
Linear Algebra and its Applications. 2019, 584, 197-220, DOI: https://doi.org/10.1016/j.laa.2019.09.015
Abstract
This work introduces a general concept of center for graphs, built on the model of the characteristic set ([12], [14]) of trees. We define it as the set of cycles in a specific directed graph associated with the original graph G, and we let it depend on a function μ. In the case of trees we consider particular instances of μ given as weights of rooted subtrees, thus retrieving the characteristic set and, interestingly, the eccentricity-center. We investigate when the center of a graph G is simple – i.e., consisting of a unique cycle – and quasi-simple – i.e., inducing a connected subgraph of G. In particular, we prove that the center of a caterpillar tree associated with the so-called combinatorial Perron parameter ρc (studied in [2] and [3]) is always simple. We also make use of a discrete version of concavity to generate examples of simple and quasi-simple centers for graphs.