On connected graphs with finite spectral redundancy index and Pythagorean triplets

TL;DR

This paper explores the spectral redundancy in connected graphs, analyzing how multiple subgraphs can share the same spectral radius, and investigates the relationship between this redundancy and Pythagorean triplets.

Contribution

It introduces the spectral redundancy index for families of graphs and examines its connection to Pythagorean triplets, providing new insights into spectral graph theory.

Findings

Determined the spectral redundancy index for a specific family of graphs.

Established a link between spectral redundancy and Pythagorean triplets.

Identified conditions under which spectral redundancy is maximized.

Abstract

This article investigates spectral redundancy, a concept initially introduced by Alberto Seeger. Spectral redundancy arises when different connected induced subgraphs of a graph share the same spectral radius in their adjacency spectrum. Let \(b(G)\) denote the total number of non-isomorphic induced subgraphs of \(G\), and \(c(G)\) represents the cardinality of the set of spectral radius of all connected induced subgraphs of \(G\). The spectral redundancy of a graph \( G \) is defined as the ratio \( \frac{b(G)}{c(G)} \). The supremum of this ratio across all graphs in a family is called the spectral redundancy index of that family. We focus on a family of graphs that exhibit spectral redundancy and we find out the spectral redundancy index of this family. Furthermore, we investigate the connection between the spectral redundancy of these graphs and the presence of Pythagorean triplets.