A tree is a graph without cycles, but what does the additional word "homogeneous" mean?
2026-04-02 15:21:41.1775143301
What is a homogeneous tree?
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Firstly tree is a connected graph without cycles. As it is mentioned in comments finite regular tree is either $K_2$ or $K_1$. However term homogeneous tree concerns not finite, but locally finite trees. A tree $T$ is locally finite if set $N(v)$ of vertices adjacent to $v$ is finite for all $v \in V(T)$. Locally finite tree is homogeneous of degree $d$ if each vertex is incident to $d$ edges. See more in book "Harmonic Analysis for Anisotropic Random Walks on Homogeneous Trees" by Alessandro Figà-Talamanca and Tim Steger.