Note on “hook-length” as a graph invariant of trees

The hook-length of a vertex $v$ in a rooted tree $T$, analogous to that defined in the Ferrers diagrams of integer partitions, is the number of descendants of $v$ (including $v$ itself) in $T$. In this note we consider two different types of “average” hook-length of vertices in a rooted tree, yielding a graph invariant that has interesting correlation to the distance functions in trees. This correlation is observed and used throughout the study of related properties and extremal questions.

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Published 2 December 2016