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

Abstract

A signed graph $\Sigma =(G,\sigma )$ consists of an underlying graph $G=(V,E)$ with a sign function $\sigma :E\to \{-1,1\}$. 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 $(K_n,T^{-})$.