Some regular signed graphs with only two distinct eigenvalues

TL;DR

This paper classifies and constructs infinite families of regular signed graphs with only two eigenvalues, including Ramanujan graphs, for various degrees, expanding understanding of their spectral properties.

Contribution

It determines parameters for regular signed graphs with two eigenvalues and constructs infinite families, including Ramanujan graphs, for degrees five and above.

Findings

Infinite families of signed graphs with two eigenvalues are constructed.

Existence of infinitely many connected signed k-regular graphs with maximum eigenvalue √k.

Construction of signed 8-regular graphs with specific spectra for all m ≥ 4.

Abstract

We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the $\{5,6,\ldots,10\}$-regular signed graphs which have only two distinct eigenvalues. For each obtained parameter we provide some examples of signed graphs having two distinct eigenvalues. It turns out to construction of infinitely many signed graphs of each mentioned valency with only two distinct eigenvalues. We prove that for any $k\geq 5$ there are infinitely many connected signed $k$-regular graphs having maximum eigenvalue $\sqrt{k}$. Moreover for each $m\geq 4$ we construct a signed $8$-regular graph with spectrum $[4^m,-2^{2m}]$. These yield infinite family of $k$-regular Ramanujan graphs, for each $k$.