More on signed graphs with at most three eigenvalues

Abstract

We consider signed graphs with just 2 or 3 distinct eigenvalues, in particular (i) those with at least one simple eigenvalue, and (ii) those with vertexdeleted subgraphs which themselves have at most 3 distinct eigenvalues. We also construct new examples using weighing matrices and symmetric 3-class association schemes.

Star complements in signed graphs with two symmetric eigenvalues

“…In the domain of signed graphs the problem of characterizing such signed graphs is more intriguing. Some recent results can be found in (And¯elić et al, 2020;Belardo et al, 2018;Ghorbani et al, 2020;Hou et al, 2019;Ramezani et al, 2022;Stanić I, 2019;Stanić II, 2019;Stanić, 2020).…”

“…In (Ramezani et al, 2022) constructions of non-regular signed graphs having just 3 eigenvalues are considered. Here we proceed with a construction of a family of such signed graphs which may be interesting because, apart from non-regularity, it has another relevant property: the eigenvalues are equal in multiplicity.…”

“…It is worth mentioning that signed graphs with 2 eigenvalues play a crucial role in Huang's resolution of the Sensitivity Conjecture (Huang, 2019); see also (Stanić I, 2019). Some particular constructions can be found in (Ramezani et al, 2022;Stanić I, 2019;Stanić II, 2019).…”