The spectrum of an I-graph

TL;DR

This paper derives the complete eigenvalue spectrum of I-graphs, enabling spectral analysis of their properties, including connectedness, bipartiteness, and nullity, thus advancing understanding of their structural characteristics.

Contribution

The paper provides a full spectral characterization of I-graphs, linking eigenvalues to graph properties and calculating nullity for specific subfamilies, which was previously unexplored.

Findings

Eigenvalues of I-graphs are explicitly determined.

Connectedness and bipartiteness are characterized spectrally.

Nullity of certain I-graph subfamilies is computed.

Abstract

We completely determine the spectrum of an $I$-graph, that is, the eigenvalues of its adjacency matrix. We apply our result to prove known characterizations of connectedness and bipartiteness in $I$-graphs by using an spectral approach. With our result, we also determine the nullity of a certain subfamily of $I$-graphs.