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

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