Starlike tree


In the mathematical subdiscipline of 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 can be seen as resulting from attaching to this central vertex at least three linear graphs. Starlike trees are also referred to as spider graphs.

Definition

More formally, let and be positive integers. The starlike tree is a tree with a central vertex of degree such that, where denotes the path graph on vertices, and every neighbor of in has degree one or two. The total number of vertices in is. The simplest starlike tree is the star graph with branches of length one.

Properties

Spectral properties

Two finite starlike trees are isospectral, i.e. their discrete [Laplace operator|graph Laplacians] have the same spectra, if and only if they are isomorphic. The graph Laplacian has always only one eigenvalue equal or greater than 4.

Spectral radius bounds

The spectral radius of a starlike tree can be bounded in terms of its maximum degree. For starlike trees with and, the spectral radius satisfies:
or equivalently, in terms of the maximum degree :
These bounds show that the spectral radius of such starlike trees is asymptotically as the maximum degree grows large.
For specific cases:
  • If and all branches have length 1, then
  • If and all branches have length 2, then
  • If and all branches have length 1, then

Eigenvalues in the interval (−2, 2)

The eigenvalues of starlike trees have been characterized with respect to the interval. A starlike tree with three branches has all of its eigenvalues in the open interval if and only if it is isomorphic to one of the following:
For starlike trees with four or more branches, at least one eigenvalue lies outside the interval.

Topological indices

Vertex-degree-based topological indices are molecular descriptors defined as, where is the number of edges between vertices of degree and degree, and the values determine the specific index. Examples include the Randić index, first Zagreb index, harmonic index, and atom-bond connectivity index.
For a starlike tree with vertices and central degree, any such index satisfies, where is the number of branches of length 1,, and. This shows that the index value depends primarily on the number of unit-length branches.
Among all starlike trees on vertices, the extremal values are typically attained by the star graph with branches and the tree. For indices where for all , the star graph attains the minimum and attains the maximum. The reverse holds for indices where .