Starlike tree

Spectral properties

Two finite starlike trees are isospectral, i.e. their 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 (the largest eigenvalue of its adjacency matrix) can be bounded in terms of its maximum degree ${\displaystyle \Delta }$. For starlike trees ${\displaystyle S(n_{1},\ldots ,n_{k})}$ with ${\displaystyle k\geq 4}$ and ${\displaystyle n_{1},\ldots ,n_{k}\geq 2}$, the spectral radius ${\displaystyle \lambda _{1}}$ satisfies:

${\displaystyle {\frac {k-1}{\sqrt {k-2}}}<\lambda _{1}(S(n_{1},\ldots ,n_{k}))<{\frac {k}{\sqrt {k-1}}}}$

or equivalently, in terms of the maximum degree ${\displaystyle \Delta =k}$:

${\displaystyle {\frac {\Delta -1}{\sqrt {\Delta -2}}}<\lambda _{1}<{\frac {\Delta }{\sqrt {\Delta -1}}}}$

These bounds show that the spectral radius of such starlike trees is asymptotically ${\displaystyle {\sqrt {\Delta }}}$ as the maximum degree grows large.

For specific cases:

• If ${\displaystyle k=3}$ and all branches have length 1, then ${\displaystyle \lambda _{1}={\sqrt {3}}}$
• If ${\displaystyle k=3}$ and all branches have length 2, then ${\displaystyle \lambda _{1}=2}$
• If ${\displaystyle k\geq 4}$ and all branches have length 1 (i.e., the tree is a star ${\displaystyle S_{k+1}}$), then ${\displaystyle \lambda _{1}={\sqrt {k}}}$

Eigenvalues in the interval (−2, 2)

The eigenvalues of starlike trees have been characterized with respect to the interval ${\displaystyle (-2,2)}$. A starlike tree ${\displaystyle S(n_{1},n_{2},n_{3})}$ with three branches has all of its eigenvalues in the open interval ${\displaystyle (-2,2)}$ if and only if it is isomorphic to one of the following:

• ${\displaystyle S(1,1,m)}$ for any positive integer ${\displaystyle m}$
• ${\displaystyle S(1,2,2)}$, ${\displaystyle S(1,2,3)}$, or ${\displaystyle S(1,2,4)}$

For starlike trees with four or more branches ${\displaystyle (k\geq 4)}$, at least one eigenvalue lies outside the interval ${\displaystyle (-2,2)}$.

Topological indices

Vertex-degree-based topological indices are molecular descriptors defined as ${\displaystyle TI(G)=\sum _{1\leq i\leq j\leq n-1}m_{ij}(G)\varphi _{ij}}$, where ${\displaystyle m_{ij}(G)}$ is the number of edges between vertices of degree ${\displaystyle i}$ and degree ${\displaystyle j}$, and the values ${\displaystyle \varphi _{ij}}$ determine the specific index. Examples include the Randić index, first Zagreb index, harmonic index, and atom-bond connectivity index.

For a starlike tree ${\displaystyle X}$ with ${\displaystyle n}$ vertices and central degree ${\displaystyle k}$, any such index satisfies ${\displaystyle TI(X)=k_{1}P_{k}+kQ_{k}+(n-1)\varphi _{22}}$, where ${\displaystyle k_{1}}$ is the number of branches of length 1, ${\displaystyle P_{k}=\varphi _{1k}+\varphi _{22}-\varphi _{12}-\varphi _{2k}}$, and ${\displaystyle Q_{k}=\varphi _{12}+\varphi _{2k}-2\varphi _{22}}$. This shows that the index value depends primarily on the number of unit-length branches.

Among all starlike trees on ${\displaystyle n}$ vertices, the extremal values are typically attained by the star graph ${\displaystyle S(1,1,\ldots ,1)}$ with ${\displaystyle n-1}$ branches and the tree ${\displaystyle S(2,2,n-5)}$. For indices where ${\displaystyle P_{k}\leq 0}$ for all ${\displaystyle k\geq 3}$ (including the Randić, harmonic, sum-connectivity, geometric-arithmetic, and augmented Zagreb indices), the star graph attains the minimum and ${\displaystyle S(2,2,n-5)}$ attains the maximum. The reverse holds for indices where ${\displaystyle P_{k}\geq 0}$ (including the first Zagreb, Albertson, and atom-bond connectivity indices).