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arXiv · 2409.15001

On the Spectrum of Locally Linear Graphs

Reimbay Reimbayev
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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 $G^*$.

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Reimbay Reimbayev. 2024-09-23. On the Spectrum of Locally Linear Graphs. https://arxiv.org/abs/2409.15001

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