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arxiv: 1907.04979 · v1 · pith:EQJ2XPPSnew · submitted 2019-07-11 · 💻 cs.DM · math.CO

Integer Laplacian Eigenvalues of Chordal Graphs

Nair Maria Maia de Abreu , Claudia Marcela Justel , Lilian Markenzon This is my paper

Pith reviewed 2026-05-24 23:03 UTC · model grok-4.3

classification 💻 cs.DM math.CO
keywords chordal graphsLaplacian eigenvaluesalgebraic connectivityvertex connectivityminimal vertex separatorsmaximal cliquesinteger eigenvalues
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The pith

Chordal graphs with equal vertex and algebraic connectivities are characterized by the vertices in their minimal vertex separators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines structural features of chordal graphs to connect those features to the integer values that appear in the Laplacian spectrum. It gives a characterization of the chordal graphs in which vertex connectivity equals algebraic connectivity, expressed directly in terms of the vertices that make up the minimal vertex separators. It also supplies a sufficient condition under which the cardinality of a maximal clique becomes an integer Laplacian eigenvalue. Two subclasses of chordal graphs are then reviewed to derive additional properties of this kind. A reader cares because the results turn spectral questions about these graphs into questions about their separators and cliques.

Core claim

In chordal graphs the equality of vertex connectivity and algebraic connectivity is characterized by the vertices that compose the minimal vertex separators; a sufficient condition is also given for the cardinality of a maximal clique to appear as an integer Laplacian eigenvalue.

What carries the argument

Minimal vertex separators (which are cliques in chordal graphs) used to characterize connectivity equality and to identify integer Laplacian eigenvalues.

If this is right

  • The equality between the two connectivities can be verified by inspecting the vertices of the minimal separators instead of computing the algebraic connectivity directly.
  • The size of any maximal clique that satisfies the given condition is guaranteed to be an eigenvalue of the Laplacian matrix.
  • New structural properties appear in the two reviewed subclasses that relate their separators or cliques to the Laplacian spectrum.
  • Integer eigenvalues of this form are tied to the clique structure rather than to arbitrary subgraphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The characterization may yield a polynomial-time test for connectivity equality restricted to chordal graphs.
  • Similar separator-based arguments could be tested on other classes where minimal separators are cliques, such as interval graphs.
  • The sufficient condition might be checked computationally by enumerating maximal cliques and verifying the eigenvalue equation.

Load-bearing premise

The graphs must be chordal so that every minimal vertex separator is itself a clique.

What would settle it

A single chordal graph in which vertex connectivity equals algebraic connectivity yet the vertices of at least one minimal separator fail the stated condition, or in which a maximal-clique size fails to be an eigenvalue even though the sufficient condition holds.

Figures

Figures reproduced from arXiv: 1907.04979 by Claudia Marcela Justel, Lilian Markenzon, Nair Maria Maia de Abreu.

Figure 1
Figure 1. Figure 1: Graphs with a(G) = κ(G) P i=1,r |Si | ≤ m this step has also O(m) complexity time. It is immediate that G has κ(G) = a(G) if and only if this intersection is for itself a minimal vertex separator composed by universal vertices. 4. Quasi-threshold graphs The quasi-threshold graphs are the object of our study in this section. An important subclass of chordal graphs, they were defined in 1962 by Wolk [15] as … view at source ↗
Figure 2
Figure 2. Figure 2: (a) presents W d(4, 3). • the split-complete graph, which is the join of a Kk and a set of n − k independent vertices [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: displays a (2, 3)-split graph. In this case, r = 3. If G is a (k, t)-split graph, the following statements are valid: 1. G is a biregular graph. 2. n = (k + t)r. 3. For r > 1, G has a unique split partition. So the split partition equals the degree partition. 4. The splitness H of G can be obtained taking off all edges of the clique determined by its non-simplicial vertices. 5. The splitness H of G is isom… view at source ↗
Figure 4
Figure 4. Figure 4: A chordal graph with ℓ = 6 Proof. Let G be a non-complete chordal graph and Qi ∈ Q, |Qi | = ni . Let u, v be two simplicial vertices belonging Qi . Since G is a chordal graph and u and v are simplicial vertices in the same clique, they are true twins. So N[u] = N[v] and, as |Qi | = ni , then dG(u) = dG(v) = ni − 1. Considering the entries of L(G), we observe that: L(G)u,u = L(G)v,v = dG(u) = dG(v) and L(G)… view at source ↗
Figure 5
Figure 5. Figure 5: A hierarchy of chordal graphs In all these subclasses we could notice the importance of the graph structure, its minimal vertex separators, its simplicial vertices and the maximal cliques. Other subclasses can be explored in the future. References [1] N.M.M. de Abreu, C.M.Justel, L. Markenzon, C.S.Oliveira, C.F.E.M.Waga, Block-indifference graphs: Characterization, struc￾tural and spectral properties, Disc… view at source ↗
read the original abstract

In this paper, structural properties of chordal graphs are analysed, in order to establish a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a sufficient condition for the cardinality of a maximal clique to appear as an integer Laplacian eigenvalue. Finally, we review two subclasses of chordal graphs, showing for them some new properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript analyzes structural properties of chordal graphs to relate them to integer Laplacian eigenvalues. It characterizes chordal graphs with equal vertex and algebraic connectivities in terms of the vertices comprising minimal vertex separators; it establishes a sufficient condition for the cardinality of a maximal clique to be an integer Laplacian eigenvalue; and it derives additional properties for two subclasses of chordal graphs.

Significance. If the characterizations hold, the work supplies concrete structural criteria linking minimal separators and maximal cliques to Laplacian spectral features in chordal graphs. Such links are potentially useful for algorithmic and combinatorial applications involving connectivity and spectra on perfect graphs.

minor comments (2)
  1. Abstract: 'stablish' should be 'establish'.
  2. The abstract states the main claims but the full text should be checked for explicit statements of the characterizations (e.g., the precise condition on separator vertices) and for concrete examples verifying the sufficient condition on clique cardinality.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. The report provides a positive summary of the manuscript's contributions but lists no specific major comments requiring point-by-point response. We remain available to incorporate any minor suggestions during revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives characterizations of chordal graphs with equal vertex and algebraic connectivities via minimal vertex separators, plus a sufficient condition for maximal clique cardinality to be an integer Laplacian eigenvalue. These rest on the standard fact that chordal graphs have clique minimal separators, which is an external structural property, not defined or fitted within the paper. No equations reduce results to inputs by construction, no parameters are fitted then relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems appear in the provided claims. The derivation is self-contained against external graph-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities appear. The work rests on standard domain assumptions from graph theory.

axioms (1)
  • domain assumption Chordal graphs have the property that every minimal vertex separator is a clique.
    This standard property of chordal graphs is invoked to link separators to connectivity and eigenvalues.

pith-pipeline@v0.9.0 · 5602 in / 1098 out tokens · 27197 ms · 2026-05-24T23:03:36.801838+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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