On the Spectral Redundancy of Pineapple Graphs

In this article, we explore the concept of spectral redundancy within the class of pineapple graphs, denoted as P(α,β). These graphs are constructed by attaching β pendent edges to a single vertex of a complete graph Kα. A connected graph G earns the title of being spectrally non-redundant if the spectral radii of its connected induced subgraphs are all distinct. Spectral redundancy, on the other hand, arises when there is a repetition of spectral radii among the connected induced subgraphs within G. Our study analyzes the adjacency spectrum of P(α,β), identifying distinct eigenvalues such as 0, −1, along with other positive and negative eigenvalues. 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. Understanding spectral redundancy in these graphs is crucial for applications in network design, where distinct spectral radii can indicate different connectivity patterns and resilience features.