Extremal spectral results related to spanning trees of signed complete graphs.

A signed graph Σ = (G , σ) consists of a underlying graph G = (V , E) with a signature function σ : E → { − 1 , 1 }. ρ (M) denotes 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: which spanning tree T makes the ρ (A (Σ)) of the unbalanced (K n , T −) as large as possible? To answer the problem, we characterize the extremal signed graph with maximum λ 1 (A (Σ)) and minimum λ n (A (Σ)) among graphs of type (K n , T −) , respectively. Another matrix associated to a signed graph is the distance matrix D (Σ) defined by Hameed, Shijin, Soorya, Germina and Zaslavsky. 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 originally conjectured by Aouchiche and Hansen. [ABSTRACT FROM AUTHOR]