Some regular signed graphs with only two distinct eigenvalues

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 $k$.