A construction of cospectral signed line graphs

Abstract

For an ordinary graph $G$, we compute the eigenvalues and the eigenspaces of the signed line graph $\mathcal{L}(\ddot{G})$ , where $\ddots{G}$ is obtained from $G$ by inserting a negative parallel edge between every pair of adjacent vertices. As an application, we prove that if and share the same vertex degrees, then and share the same spectrum. To the best of our knowledge, this construction does not follow the line of any known construction developed for either graphs or signed graphs. Among the other consequences, we emphasize that is integral (i.e., its spectrum consists entirely of integers), which means that a construction of integral signed graphs has been established simultaneously.