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

TL;DR

This paper investigates how to maximize the largest eigenvalue of signed complete graphs with a given spanning tree structure and a specified number of pendant vertices, identifying extremal configurations.

Contribution

It characterizes the extremal signed graphs with maximum eigenvalue among those with a fixed spanning tree and pendant vertices.

Findings

Identifies the extremal signed graph configurations for maximum eigenvalue.

Provides a characterization of the structure of such extremal graphs.

Enhances understanding of spectral properties of signed graphs with spanning trees.

Abstract

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