Speaker
Description
The underlying graph $G$ of a symmetric matrix $M=(m_{ij})\in \mathbb{R}^{n\times n}$ is the graph with vertex set $\{v_1,\ldots,v_n\}$ such that a pair $\{v_i,v_j\}$ with $i\neq j$ is an edge if and only if $m_{ij}\neq 0$. For a graph $G$, let $q(G)$ denote the minimum number of distinct eigenvalues among all symmetric matrices whose underlying graph is $G$. A symmetric matrix $M$ is a realization of $q(G)$ if it has underlying graph $G$ and exactly $q(G)$ distinct eigenvalues. For any tree $T$ with diameter $d$, it is well known that $q(T)\ge d+1$. We call $T$ diminimal if equality holds, that is, if $q(T)=d+1$.
Johnson and Saiago (2013) introduced a decomposition of the set of trees of any fixed diameter $d$ based on a finite family of trees, called seeds, and on an operation known as combinatorial branch duplication (CBD, for short). A tree $T'$ obtained from a tree $T$ through a sequence of CBDs is called an unfolding of $T$. In particular, every tree with diameter $d$ can be obtained as the unfolding of a unique seed of diameter $d$.
Relating the concept of seed with the minimum number of distinct eigenvalues problem, a seed $S$ is called diminimal if every unfolding of $S$ is diminimal, and defective otherwise. In this talk, we present ongoing results showing that most seeds are defective. In particular, the proportion of defective seeds tends to 1 as the diameter grows to infinity.