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

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.

💡 Research Summary

The paper provides a complete characterization of connected graphs G whose second largest distance eigenvalue λ₂(G) is less than –½. The distance matrix D(G) records the pairwise shortest‑path lengths in G, and its eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λₙ are real because D(G) is symmetric. The authors combine structural graph theory with spectral analysis to determine exactly which graphs satisfy λ₂(G) < –½.

First, they recall the known result of Guo and Zhou (2019) that any graph with λ₂ < –½ must be chordal, i.e., every cycle of length at least four has a chord. In chordal graphs, minimal vertex separators (mvs) are cliques, and each separator may appear multiple times in the clique‑tree representation; the multiplicity µ(S) counts how often a separator S occurs. The paper shows that if an mvs has size three, then λ₂ > –½ (Proposition 1). Likewise, if an mvs has size two but multiplicity at least two, λ₂ > –½ (Proposition 2). Consequently, any graph meeting the eigenvalue condition can only have mvs of size one or two, and any size‑two separator must have multiplicity one.

These restrictions force the graph to be Ptolemaic (gem‑free chordal, equivalently distance‑hereditary chordal). The authors then split the analysis according to the graph’s diameter.

Diameter 2. If G has diameter two and possesses at least one size‑two separator, the authors prove that G must be an induced subgraph of a “relaxed block star”. A relaxed block star consists of a universal vertex together with blocks that are cliques, diamonds, or a “full house” (a house graph with two diagonal chords). This family generalizes the previously studied block stars (block graphs where all blocks share a common vertex). Theorem 2 establishes that every connected induced subgraph of a relaxed block star satisfies λ₂ < –½, and conversely any diameter‑2 graph with λ₂ < –½ belongs to this family.

Diameter 3. For graphs of diameter three, the paper identifies two new families, denoted Pt₁ and Pt₂(p,q) (with p,q ≥ 2). Both are Ptolemaic and have exactly two minimal separators, each of size two and multiplicity one. Pt₁ is a specific split graph configuration, while Pt₂(p,q) consists of two cliques of sizes p and q that intersect in a single vertex. Theorem 3 proves that any connected graph of diameter three with λ₂ < –½ is an induced subgraph of either Pt₁ or Pt₂(p,q), and that these graphs indeed have λ₂ < –½.

The spectral verification is carried out in Section 5. Using Descartes’ Rule of Signs and Sturm’s Theorem, the authors analyze the characteristic polynomials of the distance matrices of relaxed block stars, Pt₁, and Pt₂(p,q). They show that the second largest root is always less than –½, confirming that the structural families identified are not only necessary but also sufficient.

Finally, Corollaries 1 and 2 summarize the main outcome: the only connected chordal graphs whose distance matrix has λ₂ < –½ are (i) induced subgraphs of relaxed block stars (diameter 2) and (ii) induced subgraphs of Pt₁ or Pt₂(p,q) (diameter 3). The paper thus resolves the problem completely by linking the eigenvalue bound to precise combinatorial constraints on minimal separators, diameter, and block structure. This work advances the understanding of distance spectra and provides a clear template for tackling similar eigenvalue‑based graph classification problems.