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{λ 1 (M), −λ n (M)}. Let A(Σ) be the adjacency matrix of Σ and (K n , 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 (K n , T − ) where T is a spanning tree? To answer the problem, we characterize the extremal signed graph with maximum λ 1 (A(Σ)) and minimum λ n (A(Σ)) among (K n , 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 ∈ (K n , 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].