Emergence of a spectral gap in a class of random matrices associated with split graphs

TL;DR

This paper investigates the spectral properties of random split graph adjacency matrices, revealing a spectral gap between -1 and 0 with a single eigenvalue, supported by analytical and numerical analysis.

Contribution

It introduces an analytic expression for the bulk spectrum of split graph adjacency matrices and explores their relation to chiral ensembles, advancing understanding of their spectral behavior.

Findings

A spectral gap between -1 and 0 is observed in the spectrum.

An explicit analytic form for the bulk distribution is derived.

The spectral properties are related to those of chiral ensembles.

Abstract

Motivated by the intriguing behavior displayed in a dynamic network that models a population of extreme introverts and extroverts (XIE), we consider the spectral properties of ensembles of random split graph adjacency matrices. We discover that, in general, a gap emerges in the bulk spectrum between -1 and 0 that contains a single eigenvalue. An analytic expression for the bulk distribution is derived and verified with numerical analysis. We also examine their relation to chiral ensembles, which are associated with bipartite graphs.