arXiv subjects

Kenneth Dadedzi

Publications and source records attributed to Kenneth Dadedzi.

2 recordsLinked to original sources

arXiv2022

On the distribution of eigenvalues of increasing trees

We prove that the multiplicity of a fixed eigenvalue $α$ in a random recursive tree on $n$ vertices satisfies a central limit theorem with mean and variance asymptotically equal to $μ_α n$ and $σ^2_α n$ respectively. It is also shown that $μ_α$ and $σ^2_α$ are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue $0$, the constants $μ_0$ and $σ^2_0$ can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.

math.CO↗

arXiv2018

The ancestral matrix of a rooted tree

Given a rooted tree $T$ with leaves $v_1,v_2,\ldots,v_n$, we define the ancestral matrix $C(T)$ of $T$ to be the $n \times n$ matrix for which the entry in the $i$-th row, $j$-th column is the level (distance from the root) of the first common ancestor of $v_i$ and $v_j$. We study properties of this matrix, in particular regarding its spectrum: we obtain several upper and lower bounds for the eigenvalues in terms of other tree parameters. We also find a combinatorial interpretation for the coefficients of the characteristic polynomial of $C(T)$, and show that for $d$-ary trees, a specific value of the characteristic polynomial is independent of the precise shape of the tree.

math.CO↗