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arXiv · 2505.13199

On some classes of bivalent and trivalent planar graphs

Jorge Alencar·Jean-Guy Caputo·Leonardo de Lima·Arnaud Knippel
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Abstract

A graph is called bivalent or trivalent if there exists an eigenvector of the graph Laplacian composed from {-1,1} or {-1,0,1}, respectively. These bivalent and trivalent eigenvectors are important for engineering applications, in particular for vibrating systems. In this article, we determine the structure of bivalent and trivalent graphs in the following planar graph families: trees, unicyclic, bicyclic, and cactus.

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Jorge Alencar, Jean-Guy Caputo, Leonardo de Lima, Arnaud Knippel. 2025-05-19. On some classes of bivalent and trivalent planar graphs. https://arxiv.org/abs/2505.13199

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