[Paper Review] The triangle-free graphs with rank 6

Long Wang, Yi-Zheng Fan+1 more|arXiv (Cornell University)|Jan 3, 2013

Graph theory and applications12 references3 citations

TL;DR

This paper characterizes all connected triangle-free graphs with adjacency matrix rank 6 by identifying their reduced forms—specifically, classifying them as induced subgraphs of one of seven unique graphs (G₁ to G₆ and G₅). The approach combines vertex multiplication/reduction operations with structural analysis of induced subgraphs like C₅, P₂, and P₄, proving that any such graph must be isomorphic to a subgraph of G₆ or one of the five base graphs, with rank preservation under reduction and multiplication.

ABSTRACT

The rank of a graph G is defined to be the rank of its adjacency matrix A(G). In this paper we characterize all connected triangle-free graphs with rank 6.

Motivation & Objective

• To classify all connected triangle-free graphs with adjacency matrix rank 6.
• To identify the reduced forms of such graphs, leveraging vertex multiplication and reduction operations.
• To determine which induced subgraphs (e.g., C₅, P₂, P₄) force the rank to exceed 6 or constrain the structure.
• To prove that any reduced triangle-free graph of rank 6 must be an induced subgraph of one of seven specific graphs, including G₆.
• To establish that non-bipartite triangle-free graphs of rank 6 must contain an induced C₅ and are subgraphs of G₆.

Proposed method

• Use of vertex multiplication and reduction operations to preserve adjacency matrix rank, enabling reduction to minimal forms.
• Application of structural lemmas (e.g., Lemma 2.6) to constrain distances from vertices outside induced subgraphs.
• Employment of rank preservation under reduction: r(G) = r(R(G)) for reduced graphs.
• Use of induced subgraph containment (G ⪯ H) to classify graphs based on known rank-6 base graphs.
• Analysis of neighborhood sets (N_H(v)) to enforce graph reduction and avoid isomorphic vertex neighborhoods.
• Proof by case analysis on the number and placement of vertices outside key induced subgraphs (e.g., 3P₂, C₅), using distance and adjacency constraints.

Experimental results

Research questions

• RQ1 Which reduced connected triangle-free graphs of rank 6 exist, and how can they be fully classified?
• RQ2 What induced subgraphs force the rank of a triangle-free graph to exceed 6, and how can such configurations be excluded?
• RQ3 How do vertex multiplication and reduction operations preserve rank in triangle-free graphs?
• RQ4 What structural constraints arise when a triangle-free graph of rank 6 contains an induced C₅?
• RQ5 Can all connected triangle-free graphs of rank 6 be embedded as induced subgraphs within a finite set of base graphs?

Key findings

• All connected triangle-free graphs of rank 6 are induced subgraphs of one of seven specific graphs: G₁ to G₅ and G₆.
• The unique reduced non-bipartite triangle-free graph of rank 6 is G₆, which contains an induced C₅ and is the maximal such graph.
• Any connected triangle-free graph of rank 6 containing an induced 3P₂ (three disjoint P₂s) must be an induced subgraph of G₁ to G₅.
• If a graph contains an induced C₅ and has rank 6, then all vertices outside the C₅ must have degree at most 2 to avoid creating triangles or increasing rank.
• Graphs with rank 6 and more than two vertices outside a C₅ cannot exist without violating the reduced form or rank condition, as shown by neighborhood and distance constraints.
• The graph G₆ is the unique maximal reduced triangle-free graph of rank 6, and all others are subgraphs of it.