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On non-bipartite graphs with integral signless Laplacian eigenvalues at most 6

Semin Oh, Jeong Rye Park, Jongyook Park, Yoshio Sano

TL;DR

This work solves the classification problem for connected non-bipartite $Q$-integral graphs with signless Laplacian eigenvalues all integral and $Q$-spectral radius at most $6$ by extending prior results to the $\rho(Q)=6$ case. It combines subgraph-based obstructions, interlacing and Perron–Frobenius theory with a computer-assisted, iterative subgraph-extension approach, starting from a distinguished adjacent pair of degrees $4$ and $3$. The authors derive rigorous constraints that exclude most configurations (two common neighbours, various $T_{3,2}$- and $S_{3,2}$-containing subgraphs) and identify the striped fish as the unique $\rho=6$ maximum-edge-degree-5 example. The culmination is a complete list of eight connected non-bipartite $Q$-integral graphs with $\rho(Q)\le 6$, presented as eight explicit graphs, with the methodology enabling potential extension to higher radii via the provided pseudocode in the appendix.

Abstract

In this paper, we completely classify the connected non-bipartite graphs with integral signless Laplacian eigenvalues at most 6.

On non-bipartite graphs with integral signless Laplacian eigenvalues at most 6

TL;DR

This work solves the classification problem for connected non-bipartite -integral graphs with signless Laplacian eigenvalues all integral and -spectral radius at most by extending prior results to the case. It combines subgraph-based obstructions, interlacing and Perron–Frobenius theory with a computer-assisted, iterative subgraph-extension approach, starting from a distinguished adjacent pair of degrees and . The authors derive rigorous constraints that exclude most configurations (two common neighbours, various - and -containing subgraphs) and identify the striped fish as the unique maximum-edge-degree-5 example. The culmination is a complete list of eight connected non-bipartite -integral graphs with , presented as eight explicit graphs, with the methodology enabling potential extension to higher radii via the provided pseudocode in the appendix.

Abstract

In this paper, we completely classify the connected non-bipartite graphs with integral signless Laplacian eigenvalues at most 6.

Paper Structure

This paper contains 7 sections, 15 theorems, 19 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:fish']}
  4. The case where $x$ and $y$ have two common neighbors
  5. The case where $x$ and $y$ have one common neighbor
  6. The case where $x$ and $y$ have no common neighbor
  7. Main Results

Key Result

Theorem 1.1

Let $G$ be a connected $Q$-integral graph with $Q$-spectral radius $6$. Suppose that $G$ has the maximum edge-degree $5$. If $G$ is non-bipartite, then $G$ is the graph with the vertex set $V(G) = \{v_1, v_2, v_3, v_4, v_5, v_6 \}$ and the edge set $E(G) = \{v_1v_2, v_1v_3, v_2v_3, v_3v_4, v_3v_5, v

Figures (8)

  • Figure 1: The striped fish graph.
  • Figure 2: Connected non-bipartite $Q$-integral graphs with $Q$-spectral radius at most $6$
  • Figure 3: $x$ and $y$ have two common neighbors $y_0$ and $y_1$.
  • Figure 4: $T_{3,2}$
  • Figure 5: $T_{3,2}'$
  • ...and 3 more figures

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: OPPS, PS19 and PS23
  • Proposition 2.2: PS19
  • Theorem 3.1: SS08PS19OPPS
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 15 more