Eigenvalues of signed graphs

Abstract: Signed graphs have their edges labeled either as positive or negative. ρ(M) denote the M-spectral radius of Σ, where M = M(Σ) is a real symmetric graph matrix of Σ. Obviously, ρ(M) = max{λ1(M),−λn(M)}. Let A(Σ) be the adjacency matrix of Σ and (Kn,H−) be a signed complete graph whose negative edges induce a subgraph H . In this paper, we first focus on a central problem in spectral extremal graph theory as follows: Which signed graph with maximum ρ(A(Σ)) among (Kn,T −) where T is a spanning tree? To answer the problem, we characterize the extremal signed graph with maximum λ1(A(Σ)) and minimum λn(A(Σ)) among (Kn,T −), respectively. Another interesting graph matrix of a signed graph is distance matrix, i.e. D(Σ)which was defined by Hameed, Shijin, Soorya, Germina and Zaslavsky [8]. Note that A(Σ) = D(Σ) when Σ ∈ (Kn,T−). In this paper, we give upper bounds on the least distance eigenvalue of a signed graph Σ with diameter at least 2. This result implies a result proved by Lin [11] was originally conjectured by Aouchiche and Hansen [1].