Some signed graphs whose eigenvalues are main

Abstract

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. 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.