Sustainable trees: A deep dive into the spectral radii and centers of trees

Abstract

In this thesis, we introduce and study a new property of trees, termed sustainability. We say a tree is sustainable if any edge can be replaced by an edge between non-adjacent vertices such that the resulting graph remains a tree and its spectral radius does not decrease. The introduction of this property naturally motivates the use of tools from spectral graph theory and the study of various notions of tree centers. We define a new notion of centrality, which we use to show that certain trees are non-sustainable. The central techniques used in this thesis are the edge rotation lemma and the generalised tree shift. Using these, we identify several families of trees that are sustainable and others that are not. The main result of this thesis establishes that, for sufficiently large vertices, any tree with a high maximum degree is non-sustainable. This thesis contributes to the study of several known areas within graph theory, focusing in particular on the behaviour of the spectral radius under graph perturbations and on properties such as the centers of trees.