A note on eigenvalues of signed graphs

Suppose that sigma is a signed graph with n vertices and m edges. Let lambda 1 > lambda 2 > . . . > lambda n be the eigenvalues of sigma. A signed graph is called balanced if each of its cycles contains an even number of negative edges, and unbalanced otherwise. Let wb be the balanced clique number of sigma, which is the maximum order of a balanced complete subgraph of sigma. In this paper, we prove that lambda 1 <= root 2 omega b - 1/omega bm. This inequality extends a conjecture of ordinary graphs, which was confirmed by Nikiforov (2002) [8], to the signed case. In addition, we completely characterize the signed graphs with -1 < lambda 2 < 0.(c) 2022 Elsevier Inc. All rights reserved.