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Eigenvalues of signed graphs

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, $ρ(M)=\mbox{max}\{λ_1(M),-λ_n(M)\}$. 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 $Σ\in (K_n,T^-)$. 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].