On the spectral redundancy of pineapple graphs

TL;DR

This paper investigates the spectral properties of pineapple graphs, focusing on their spectral redundancy, and identifies conditions under which their connected induced subgraphs have distinct spectral radii.

Contribution

It provides a detailed spectral analysis of pineapple graphs and characterizes their spectral redundancy, a novel exploration in the spectral graph theory domain.

Findings

Spectral radii of subgraphs are mostly distinct in pineapple graphs.

Identified specific eigenvalues of pineapple graphs, including 0 and -1.

Analyzed the implications of spectral redundancy for graph structure.

Abstract

In this article, we explore the concept of spectral redundancy within the class of pineapple graphs, denoted as $\mathcal{P}(\alpha,\beta)$. These graphs are constructed by attaching $\beta$ pendent edges to a single vertex of a complete graph $K_\alpha$. A connected graph $G$ earns the title of being spectrally non-redundant if the spectral radii of its connected induced subgraphs remain distinct. Spectral redundancy, on the other hand, arises when there is a repetition of spectral radii among the connected induced subgraphs within $G$. Specifically, we analyze the adjacency spectrum of $\mathcal{P}(\alpha,\beta)$, revealing distinct eigenvalues including $0$, $-1$, and additional eigenvalues, some negative and others positive. Our investigation focuses on determining the spectral redundancy within this class of graphs, shedding light on their unique structural properties and implications for graph theory.