On the minimum number of distinct eigenvalues of a threshold graph
Abstract
For a graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A\in S(G)$, the location of the nonzero offdiagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ be the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we give a characterization of all connected threshold graphs $G$ with $q(G)=2$. Moreover, we study the values of $q(G)$ for connected threshold graphs with trace $2$, $3$, $n2$, $n3$, where $n$ is the order of threshold graph. The values of $q(G)$ are determined for all connected threshold graphs with $7$ and $8$ vertices with two exceptions. Finally, a sharp upper bound for $q(G)$ over all connected threshold graph $G$ is given.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.10143
 Bibcode:
 2021arXiv211010143F
 Keywords:

 Mathematics  Combinatorics;
 05C50;
 15A29
 EPrint:
 25 pages, 1 figrue