Some signed graphs whose eigenvalues are main

Highlights：

• Let G be a graph. For a subset X of V (G), the switching σ of G is the signed graph Gσ obtained from G by reversing the signs of all edges between X and V (G) ∖ X. Let A(Gσ) be the adjacency matrix of Gσ. An eigenvalue of A(Gσ) is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Two switching equivalent graphs share the same spectrum, while they may have different main eigenvalues. Akbari et al. (2021) conjectured that let G≠K2,K4∖{e}. Then there exists a switching σsuch that all eigenvalues of Gσ are main. Let Sn,k be the graph obtained from the complete graph Kn−r by attaching r pendent edges at some vertex of Kn−r. In this paper we prove that there exists a switching σ such that all eigenvalues of Gσ are main when G is a complete multipartite graph, or G is a harmonic tree, or G is a Sn,k. These results partly confirm a conjecture of Akbari et al.