On the spectrum of threshold graphs

Abstract

The antiregular connected graph on 𝑟 vertices is defined as the connected graph whose vertex degrees take the values of 𝑟−1 distinct positive integers. We explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal number 𝑟 of parts. Structural and combinatorial properties can be deduced for related classes of graphs and in particular for the minimal configurations in the class of singular graphs.