Some signed graphs whose eigenvalues are main

Let G be a graph. For a subset X of V(G), the switching sigma of G is the signed graph G(sigma) obtained from G by reversing the signs of all edges between X and V(G) \ X. 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. Let S-n,S-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 sigma such that all eigenvalues of G(sigma) are main when G is a complete multipartite graph, or G is a harmonic tree, or G is a S-n,S-k. These results partly confirm a conjecture of Akbari et al. (c) 2022 Elsevier Inc. All rights reserved.