Characterizing graphs with the second largest distance eigenvalue less than -1/2

TL;DR

The paper resolves the problem of characterizing connected graphs $G$ with $\lambda_2(G)<-1/2$ for their distance matrix by showing such graphs are exactly the chordal graphs that are not block graphs and fall into two structural regimes: diameter $2$ subgraphs of relaxed block stars and diameter $3$ subgraphs of Pt1 or Pt2(p,q) with $p,q\ge2$. It then proves that the identified graph families—relaxed block stars, Pt1, and Pt2(p,q)—indeed satisfy $\lambda_2<-1/2$ using distance-matrix spectral tools, including equitable partitions and rigorous root-bounds via Descartes’ Rule of Signs and Sturm’s Theorem. The results hinge on a tight interplay between minimal vertex separators and graph classes (Ptolemaic, split, block, and distance-hereditary structures) to achieve a complete, finite description. This work advances the understanding of distance matrices in graphs, providing a precise taxonomy that can guide rapid identification of graphs with strong distance-spectral constraints in chordal graph theory.

Abstract

Let $G$ be a connected graph with vertex set $V$. The distance, $d_G(u, v)$, between vertices $u$ and $v$ of $G$ is defined as the length of a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the matrix $\mathbf{D}(G) =[d_G(u, v)]_{u,v\in V}$. The second largest distance eigenvalue $λ_2(G)$ of $G$ is the second largest one in the spectrum of $\mathbf{D}(G)$. In this work, we completely characterize the connected graphs $G$ for which $λ_2(G)<-1/2$ through approaches both spectral and structural.

Figures (14)

• Figure 1: Gem and diamond graphs.
• Figure 2: Forbidden subgraphs for graphs satisfying $\lambda_2 <-1/2$guo2024graphs.
• Figure 3: Graphs $BG(p,q,3,2,2)$ and $BGA$ of Lemma \ref{['lem:block-graph']}.
• Figure 4: Split graphs of Theorem \ref{['theo:split']}.
• Figure 5: (a) Smallest graph with a $mvs$$S$ such that $|S|=3$; (b) smallest graph with a mvs$S$ for which $|S|=2$ and $\mu(S)=2$.

Theorems & Definitions (44)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4: Cauchy's Interlace Theorem
• Remark 1
• Lemma 5: guo2024graphs, Theorem 3.1
• Lemma 6
• Theorem 1
• proof
• Proposition 1