On some classes of bivalent and trivalent planar graphs

TL;DR

This paper characterizes the structure of bivalent and trivalent graphs within various planar graph families, highlighting their importance in engineering applications like vibrating systems.

Contribution

It provides a detailed structural analysis of bivalent and trivalent graphs in trees, unicyclic, bicyclic, and cactus families, expanding understanding of their properties.

Findings

Characterized bivalent and trivalent graphs in trees, unicyclic, bicyclic, and cactus families.

Identified structural properties relevant for engineering applications.

Enhanced understanding of eigenvector configurations in planar graphs.

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.