Signed graphs cospectral with the path

Abstract

A signed graph $\Gamma$ is said to be determined by its spectrum if every signed graph with the same spectrum as $\Gamma$ is switching isomorphic with $\Gamma$. Here it is proved that the path $P_n$, interpreted as a signed graph, is determined by its spectrum if and only if $n\equiv 0, 1$, or 2 (mod 4), unless $n\in\{8, 13, 14, 17, 29\}$, or $n=3$.