A new generalization of Fielder's lemma with applications

TL;DR

This paper extends Fielder's lemma to a broader class of graph products, enabling the computation of various spectra for these products, including adjacency and universal adjacency spectra.

Contribution

It provides a generalized version of Fielder's lemma applicable to arbitrary graph H-products, expanding spectral analysis capabilities beyond previous results.

Findings

Generalized Fielder's lemma for H-products of graphs

Computed adjacency spectrum of H-product of commuting graphs

Derived universal adjacency spectrum of H-product of commuting regular graphs

Abstract

Very recently Ma and Wu \cite{wu2024generalization} obtained a generalization of Fielder's lemma and applied to find adjacency, Laplacian, and signless Laplacian spectra of $P_n-$ product of commuting graphs. In this paper, we give a generalization of Fielder's lemma applying which not only one gets generalized result in \cite{wu2024generalization} as a particular case, but also one can find several kind of spectra of $H$-product of graphs when $H$ is an arbitrary graph. Moreover, we compute adjacency spectrum of $H-$ product of commuting graphs and universal adjacency spectrum of $H-$ product of commuting regular graphs.