A note on eigenvalues of signed graphs.

Suppose that Σ is a signed graph with n vertices and m edges. Let λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n be the eigenvalues of Σ. A signed graph is called balanced if each of its cycles contains an even number of negative edges, and unbalanced otherwise. Let ω b be the balanced clique number of Σ, which is the maximum order of a balanced complete subgraph of Σ. In this paper, we prove that λ 1 ≤ 2 ω b − 1 ω b m. 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 ≤ λ 2 ≤ 0. [ABSTRACT FROM AUTHOR]