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On some classes of bivalent and trivalent planar graphs

Jorge Alencar, Jean-Guy Caputo, Leonardo de Lima, Arnaud Knippel

TL;DR

This paper addresses the problem of which connected graphs admit a Laplacian eigenvector with entries in {-1,1} (bivalent) or in {-1,0,1} (trivalent), focusing on planar families. It develops structural characterizations using Merris's edge principle, the notions of soft nodes and equal links, and builds up from trees and unicyclic graphs to bicyclic, multicyclic, and cactus graphs. The main contributions include complete classifications for bivalent and trivalent trees, unicyclic and bicyclic graphs, and a detailed description for cactus graphs, together with a formula for the cyclomatic number in regular bipartite multicyclic graphs and consequences for cactus structures. These results connect spectral constraints to precise graph constructions and have potential applications in vibrational modeling and energy concentration.

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.

On some classes of bivalent and trivalent planar graphs

TL;DR

This paper addresses the problem of which connected graphs admit a Laplacian eigenvector with entries in {-1,1} (bivalent) or in {-1,0,1} (trivalent), focusing on planar families. It develops structural characterizations using Merris's edge principle, the notions of soft nodes and equal links, and builds up from trees and unicyclic graphs to bicyclic, multicyclic, and cactus graphs. The main contributions include complete classifications for bivalent and trivalent trees, unicyclic and bicyclic graphs, and a detailed description for cactus graphs, together with a formula for the cyclomatic number in regular bipartite multicyclic graphs and consequences for cactus structures. These results connect spectral constraints to precise graph constructions and have potential applications in vibrational modeling and energy concentration.

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.
Paper Structure (8 sections, 19 theorems, 10 equations, 10 figures)

This paper contains 8 sections, 19 theorems, 10 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Trees and unicyclic graphs
  4. Bicyclic graphs
  5. Multicyclic graphs
  6. General results
  7. Cactus bivalent graphs
  8. Cactus trivalent graphs

Key Result

Theorem 2.3

Let $\mathbf{v}$ be an eigenvector of $L(G)$ affording an eigenvalue $\lambda$. If $v_i=v_j$, then $\mathbf{v}$ is an eigenvector of $L(G^{\prime})$ affording the eigenvalue $\lambda$, where $G^{\prime}$ is the graph obtained from $G$ by deleting or adding the edge $e_{ij}$ depending whether the equ

Figures (10)

  • Figure 1: Examples of bivalent and trivalent trees with eigenvalues $\lambda=2$ (left) and $\lambda=1$ (right).
  • Figure 2: A trivalent unicyclic graph with eigenvalue 1.
  • Figure 3: The trivalent (first two) and bivalent elementary cycles, with eigenvalues 2,3 and 4 from left to right.
  • Figure 4: Families $B_1(p,q)$, $B_2(p,q,r)$ and $B_3(p,q,r)$ of bicyclic graphs with no pendant vertices
  • Figure 5: Bivalent bicyclic graphs from left to right (i) chains $P_2$ connected by equal links, $\lambda=2$, (ii) even cycle with an additional equal link $\lambda=4$ and (iii) two even cycles connected by an equal link $\lambda=4$.
  • ...and 5 more figures

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2: Soft regular graph
  • Theorem 2.3: Edge principle merris98
  • Theorem 2.4: Extension of soft nodes, merris98
  • Theorem 2.5: Bivalent graphs
  • Definition 2.6: Hard degree
  • Theorem 2.7: Trivalent graphs
  • Theorem 2.8: Graph with a leaf
  • proof
  • ...and 25 more