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

A signed graph $\mathrm{\Sigma }=(G,\sigma )$ consists of an underlying graph $G=(V,E)$ with a sign function $\sigma :E\to \{-1,1\}$. Let $A(\mathrm{\Sigma })$ be the adjacency matrix of $\mathrm{\Sigma }$ and ${\lambda }_1(\mathrm{\Sigma })$ denote the largest eigenvalue (index) of $\mathrm{\Sigma }$.Define $(K_n,H^{-})$ 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 ${\lambda }_1(A(\mathrm{\Sigma }))$ of the unbalanced $(K_n,T^{-})$ as large as possible? To answer the problem, we characterize the extremal signed graph with maximum ${\lambda }_1(A(\mathrm{\Sigma }))$ among graphs of type $(K_n,T^{-})$.