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Peer reviewed
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Sun, Gaoxing; Liu, Feng; Lan, Kaiyang
Linear algebra and its applications, 11/2022, Volume: 652Journal Article
Suppose that Σ is a signed graph with n vertices and m edges. Let λ1≥λ2≥⋯≥λn 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 ωb be the balanced clique number of Σ, which is the maximum order of a balanced complete subgraph of Σ. In this paper, we prove thatλ1≤2ωb−1ωbm. 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≤λ2≤0.
