In the area of mathematics known as
graph theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, a
tree
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...
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.
Properties
Two finite starlike trees are
isospectral
In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity.
The theory of isospectr ...
, i.e. their
graph Laplacians have the same spectra, if and only if they are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
. The graph Laplacian has always only one eigenvalue equal or greater than 4.
References
External links
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Trees (graph theory)
Spectral theory
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