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

2024/05/18 by Li, Dan, Yan, Minghui, Teng, Zhaolin
#05C35 #05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.11214

Abstract

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 (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 λ₁(A(Σ)) of the unbalanced (K_n,T^-) as large as possible? To answer the problem, we characterize the extremal signed graph with maximum λ₁(A(Σ)) among graphs of type (K_n,T^-).

Related

• On the Signed Complete Graphs with Maximum Index
• Signed graphs with maximal index
• The maximum index of signed complete graphs whose negative edges induce a bicyclic graph
• Eigenvalues of signed graphs
• On the index of unbalanced signed bicyclic graphs