Maximizing the Index of Signed Complete Graphs Containing a Spanning Tree with k Pendant Vertices

A signed graph Sigma=(G,sigma) consists of an underlying graph G=(V,E) with a sign function sigma:E ->{-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 question: 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 question, we characterize the extremal signed graph with maximum lambda(1)(A(Sigma)) among graphs of type (K-n,T-).