ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

Article in press


J.-L. Shang


Branch-weight unique trees


Discussiones Mathematicae Graph Theory

Received: 2017-10-09, Revised: 2019-11-15, Accepted: 2019-11-15,


A branch at a vertex $x$ in a tree is a maximal subtree containing $x$ as an endvertex. The branch-weight of $x$ is the maximum number of edges in any branch at $x$. The branch-weight sequence of a tree is the multiset consisting of the branch-weights of all vertices arranged in nonincreasing order. Non-isomorphic trees may have the same branch-weight sequence. A tree $T$ is said to be branch-weight unique in a family of trees if $T$ is uniquely determined in the family by its branch-weight sequence. A spider is a tree in which exactly one vertex has degree exceeding two. It is known that spiders are branch-weight unique in the family of spiders but not in the family of all trees. In this study, a necessary and sufficient condition is obtained whereby a spider may be branch-weight unique in the family of all trees. Moreover, two types of trees are proposed to be branch-weight unique in the family of all trees.


branch-weight, branch-weight sequence, branch-weight unique, tree, spider