On the Spectrum of Locally Linear Graphs

TL;DR

This paper explores the spectral properties and structural relationships between locally linear graphs and their associated graphs constructed from triangles, revealing unique spectra and reconstructibility.

Contribution

It introduces a method to construct a related graph from locally linear graphs and proves that the spectrum of this new graph uniquely determines the original.

Findings

The spectrum of the constructed graph is uniquely determined by the original graph.

Both graphs share the same number of quadrilaterals and pentagons.

The constructed graph does not contain certain subgraphs like K4-e and K_{1,4}.

Abstract

For a locally linear graph $G$, which is a graph built out of triangles, it is possible to construct another graph $G^*$ that would consist of triangles of $G$ as vertices, while sharing (or not sharing) a common vertex between a pair of triangles would define a binary relation for edges of $G^*$. In this paper we show that the spectrum of $G^*$ is uniquely defined by $G$. We will also show some structural similarities of these graphs; in particular, that the number of quadrilaterals and pentagons in both graphs are the same; that $G^*$ does not contain $K_4-e$ and $K_{1,4}$; and that $G$ can be reconstructed from $G^*$.