• Citer

Partially ordering the class of invertible trees

par Guo, Krystal
Référence European journal of combinatorics, 83, 103019
Publication Publié, 2020-01

Article révisé par les pairs

• Détails
• Contenu

Résumé :

A tree T is invertible if and only if T has a perfect matching. In Godsil (1985), Godsil considers an invertible tree T and finds that the matrix A(T)−1 has entries in {0,±1} and is the signed adjacency matrix of a graph which contains T. In this paper, we give a new proof of this theorem, which gives rise to a partial ordering relation on the class of all invertible trees on 2n vertices. In particular, we show that given an invertible tree T whose inverse graph has strictly more edges, we can remove an edge from T and add another edge to obtain a non-isomorphic invertible tree T˜ whose median eigenvalue is strictly greater. This extends naturally to a partial ordering. We find the maximal and minimal elements of this poset and explore the implications about the median eigenvalues of invertible trees.