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

A signed graph $\Sigma=(G,\sigma)$ consists of an underlying graph $G=(V,E)$ with a sign function $\sigma:E\rightarrow\{-1,1\}$. Let $A(\Sigma)$ be the adjacency matrix of $\Sigma$ and $\lambda_1(\Sigma)$ denote the largest eigenvalue (index) of $\Sigma$.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 $\lambda_1(A(\Sigma))$ of the unbalanced $(K_n,T^-)$ as large as possible? To answer the problem, we characterize the extremal signed graph with maximum $\lambda_1(A(\Sigma))$ among graphs of type $(K_n,T^-)$.