Maximizing the index of signed complete graphs with spanning trees on k pendant vertices

A signed graph Σ=(G,σ) consists of an underlying graph $G=(V,E)$ with a sign function σ:E→\-1,1\. Let A(Σ) be the adjacency matrix of Σ and λ₁(Σ) denote the largest eigenvalue (index) of Σ.Define (Kₙ,H⁻) as a signed complete graph whose negative edges induce a subgraph H. In this paper, we focus on the following problem: which spanning tree T with a given number of pendant vertices makes the λ₁(A(Σ)) of the unbalanced (Kₙ,T⁻) as large as possible? To answer the problem, we characterize the extremal signed graph with maximum λ₁(A(Σ)) among graphs of type (Kₙ,T⁻).