Maximizing the Index of Signed Complete Graphs Containing a Spanning Tree with 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 question: 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 question, we characterize the extremal signed graph with maximum λ1 (A (Σ) ) among graphs of type (Kn, T−).