Positive $Q$-matrices of graphs
Volume 179 / 2007
Studia Mathematica 179 (2007), 81-97
MSC: Primary 05C50; Secondary 43A35.
DOI: 10.4064/sm179-1-7
Abstract
The $Q$-matrix of a connected graph $\mathcal{G}=(V,E)$ is $Q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial(x,y)$ is the graph distance. Let $q(\mathcal{G})$ be the range of $q\in(-1,1)$ for which the $Q$-matrix is strictly positive. We obtain a sufficient condition for the equality $q(\widetilde{\mathcal{G}})=q(\mathcal{G})$ where $\widetilde{\mathcal{G}}$ is an extension of a finite graph $\mathcal{G}$ by joining a square. Some concrete examples are discussed.
