Starlike tree - Reference.org

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Starlike tree

Tree graph with exactly one vertex of degree >2

In the area of mathematics known as graph theory, a tree is said to be starlike if it has exactly one vertex of degree greater than 2. This high-degree vertex is the root and a starlike tree is obtained by attaching at least three linear graphs to this central vertex.

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Properties

Two finite starlike trees are isospectral, i.e. their graph Laplacians have the same spectra, if and only if they are isomorphic.¹ The graph Laplacian has always only one eigenvalue equal or greater than 4.²

External links

Weisstein, Eric W. "Spider Graph". MathWorld.
(sequence A004250 in the OEIS)

References

M. Lepovic, I. Gutman (2001). No starlike trees are cospectral. https://www.sciencedirect.com/science/article/pii/S0012365X01001698 ↩
Nakatsukasa, Yuji; Saito, Naoki; Woei, Ernest (April 2013). "Mysteries around the Graph Laplacian Eigenvalue 4". Linear Algebra and Its Applications. 438 (8): 3231–46. arXiv:1112.4526. doi:10.1016/j.laa.2012.12.012. /wiki/ArXiv_(identifier) ↩