A note on the main eigenvalues of signed graphs

A signed graph $G^{\sigma}$ consists of the underlying graph $G$ and a function $\sigma$ defined on its edge set $E(G)$, $\sigma:E(G)\to\{+1,-1\}$. Let $A(G^{\sigma})$ be the adjacency matrix of $G^{\sigma}$. An eigenvalue of $A(G^{\sigma})$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Akbari et al. [Linear Algebra Appl. 614(2021)270-278] proposed the conjecture that there exists a switching $\sigma$ such that all eigenvalues of $A(G^{\sigma})$ are main where $G\neq K_{2},K_{4}\backslash\{e\}$. Let $S_{n,r}$ be the graph obtained from the complete graph $K_{n-r}$ by attaching $r$ pendent edges at some vertex of $K_{n-r}$. We show this conjecture holds for the graph $S_{n,r}$ and for the complete multipartite graph