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arxiv: 2606.27328 · v1 · pith:37S5NETWnew · submitted 2026-06-25 · 🧮 math.CO

On the homology groups of clique complexes of strongly regular graphs

Sebastian M. Cioab\u{a} , Mutasim Mim This is my paper

Pith reviewed 2026-06-26 03:29 UTC · model grok-4.3

classification 🧮 math.CO
keywords strongly regular graphsclique complexeshomology groupseigenvaluesLatin square graphsNeumaier classification
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The pith

Non-vanishing first homology in clique complexes of strongly regular graphs occurs only for listed families or when smallest eigenvalues tend to negative infinity in infinite families.

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

The paper studies the first homology groups of the clique complexes of strongly regular graphs over arbitrary fields. It shows that these groups are trivial except in a short list of cases obtained from Neumaier's classification of graphs with smallest integral eigenvalue. The main theorem states that any infinite family of distinct strongly regular graphs whose clique complexes have non-vanishing first homology over some fields must consist of lattice graphs for infinitely many members or else have smallest eigenvalues diverging to negative infinity. The authors also compute the full homology for the graphs coming from Latin squares of order at least five.

Core claim

Using Neumaier's classification, non-vanishing H_1(Cl(G), F) can occur only for the Petersen graph, the Shrikhande graph, complete bipartite graphs, conference graphs on at most 255 vertices, lattice graphs, and the exceptional families E_m. For any infinite sequence of distinct strongly regular graphs with H_1(Cl(G_n), F_n) nonzero, either G_n is a lattice graph for infinitely many n or lim λ_min(G_n) = −∞. For a Latin square graph of order n ≥ 5 the homology satisfies H_i = 0 for i = 1 and i ≥ 3 while dim H_2 = (n−1)^3 − I(M).

What carries the argument

The first homology group H_1(Cl(G), F) of the clique complex of a strongly regular graph G, related to the graph parameters through Neumaier's classification of graphs with bounded smallest eigenvalue.

If this is right

  • The vast majority of strongly regular graphs have trivial first clique homology over every field.
  • For Latin square graphs of order n ≥ 5, H_1 and all H_i with i ≥ 3 vanish while the dimension of H_2 equals (n−1)^3 minus the number of intercalates.
  • Non-vanishing first homology is confined to the Petersen graph, Shrikhande graph, complete bipartite graphs, conference graphs up to 255 vertices, lattice graphs, and the E_m families.
  • Any infinite family of distinct strongly regular graphs with persistent non-trivial first homology must be lattice graphs infinitely often or have smallest eigenvalues unbounded below.

Where Pith is reading between the lines

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

  • Spectral bounds on the smallest eigenvalue can be used to force vanishing of low-dimensional homology in highly symmetric graphs.
  • The same classification technique might apply to higher homology groups or to other families of vertex-transitive graphs.
  • Explicit computation of homology for additional known strongly regular graphs could test the completeness of the listed exceptions.

Load-bearing premise

Neumaier's classification of strongly regular graphs with smallest integral eigenvalue is complete and accurate.

What would settle it

A strongly regular graph outside the listed families whose clique complex has nonzero first homology over some field, or an infinite family of non-lattice strongly regular graphs with bounded smallest eigenvalue yet nonvanishing H_1 over some sequence of fields.

Figures

Figures reproduced from arXiv: 2606.27328 by Mutasim Mim, Sebastian M. Cioab\u{a}.

Figure 1
Figure 1. Figure 1: From the left: the cases y1 = y4, y1 < y4 < y2, y4 < y1, y4 > y2, respectively. vertices on C. We consider several cases based on the relative position of y4 with respect to y1 and y2, see [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The case y1 < y4 < y2. Curved, dashed segments represent arbitrary length paths in corresponding cycles. From the left: cycles C, C1, C′ , and together, C2 and C3, respectively. Consider the case y4 < y1, see [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The case y4 < y1. Curved, dashed segments represent arbitrary length paths in correspond￾ing cycles. From the left: cycles C, C1, C′ , and together, C2 and C3, respectively. By Proposition 2.4, there is some nonzero c such that: β = T(C) − T(C1) = cT(C ′ ), or, T(C) = cT(C ′ ) + T(C1). Here, C ′ is a cycle in C of length s and no diagonal edges. Moreover, C ′ has three vertices with the same first coordina… view at source ↗
Figure 4
Figure 4. Figure 4: The case y3 < y4. Curved, dashed segments represent arbitrary length paths in correspond￾ing cycles. From the left: cycles C, C1, C′ , and together, C2 and C3, respectively. 3.2.2 Cycles with Diagonal Edges As before, assume that M is a Latin square of order n ≥ 4 and G is the strongly regular graph associated with M. Let F be an arbitrary but fixed field. In this section, we will show that if C is any cyc… view at source ↗
read the original abstract

In this paper, we study the first homology groups of clique complexes of strongly regular graphs over arbitrary fields and prove that most of these graphs have trivial first clique homology groups. Using Neumaier's classification of strongly regular graphs with smallest integral eigenvalue, we show that a non-vanishing first homology group may occur only in a short collection of cases: the Petersen graph, the Shrikhande graph, the complete bipartite graphs, the conference graphs on at most $255$ vertices, the lattice graphs, and the exceptional families $E_m$ in Neumaier's classification of strongly regular graphs with smallest adjacency eigenvalue $-m$, for some integer $m \geq 3$. Let $\text{Cl}(G)$ denote the clique complex of a graph $G$, $H_i(\text{Cl}(G),\mathbb{F})$ be the $i$-th homology group of $\text{Cl}(G)$ over the field $\mathbb{F}$, for some $i\geq 1$, and $\lambda_{min}(G)$ denote the smallest eigenvalue of the adjacency matrix of $G$. We prove that if $(G_n)_{n\geq 1}$ is an infinite family of pairwise distinct strongly regular graphs and $(\mathbb{F}_n)_{n\geq 1}$ is a sequence of fields such that $H_1(\text{Cl}(G_n), \mathbb{F}_n)\not=0$ for every $n$, then either $G_n$ is a lattice graph for infinitely many $n$, or $\lim_{n\rightarrow +\infty} \lambda_{\min}(G_n)=-\infty$. For Latin square graphs, we determine the clique homologies over arbitrary fields and show that if $G$ is the strongly regular graph associated with a Latin square $M$ of order $n \geq 5$ and $\mathbb{F}$ is any field, then $H_i(\text{Cl}(G),\mathbb{F})=0$ for $i=1$ or $i \geq 3$, and $\dim H_2(\text{Cl}(G),\mathbb{F})=(n-1)^3-I(M),$ where $I(M)$ is the number of $2 \times 2$ Latin subsquares or intercalates in $M$.

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

1 major / 1 minor

Summary. The paper claims that non-vanishing first homology H_1(Cl(G), F) for strongly regular graphs G occurs only in a short list of cases (Petersen graph, Shrikhande graph, complete bipartite graphs, conference graphs on ≤255 vertices, lattice graphs, and the E_m families from Neumaier's classification) by invoking that classification of SRGs with integral smallest eigenvalue. It then deduces that any infinite family of distinct SRGs with H_1(Cl(G_n), F_n) ≠ 0 for fields F_n must have either infinitely many G_n lattice graphs or λ_min(G_n) → −∞. Separately, for the strongly regular graph associated to a Latin square of order n ≥ 5, it computes explicitly that H_1 = 0, H_i = 0 for i ≥ 3, and dim H_2 = (n−1)^3 − I(M) over any field, where I(M) counts intercalates.

Significance. If the results hold, the paper supplies a sharp restriction on when clique homology of SRGs can be nontrivial, tying it directly to eigenvalue decay or specific infinite families. The explicit, field-independent computation of all homology groups for Latin-square graphs (including the precise formula for dim H_2 in terms of intercalates) is a self-contained, verifiable contribution that does not rely on the classification and stands as a concrete advance.

major comments (1)
  1. [Introduction / Main Theorem] Introduction and the statement of the main dichotomy theorem: the assertion that H_1(Cl(G), F) ≠ 0 is possible only for the enumerated families (Petersen, Shrikhande, complete bipartite, conference graphs ≤255 vertices, lattice graphs, E_m) is obtained solely by quoting the completeness of Neumaier's classification of SRGs with smallest integral eigenvalue. No independent combinatorial or spectral argument is supplied that would bound the possible graphs with non-vanishing H_1 when λ_min is bounded away from −∞; consequently the infinite-family dichotomy rests on an external result whose exhaustiveness is not re-proved or verified inside the manuscript.
minor comments (1)
  1. [Abstract] Abstract: the sentence 'H_i(Cl(G),F)=0 for i=1 or i ≥ 3' is ambiguous; it should be rephrased to state separately that the vanishing holds for i=1 and for every i≥3.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We respond point by point to the major comment below.

read point-by-point responses

  1. Referee: [Introduction / Main Theorem] Introduction and the statement of the main dichotomy theorem: the assertion that H_1(Cl(G), F) ≠ 0 is possible only for the enumerated families (Petersen, Shrikhande, complete bipartite, conference graphs ≤255 vertices, lattice graphs, and E_m) is obtained solely by quoting the completeness of Neumaier's classification of SRGs with smallest integral eigenvalue. No independent combinatorial or spectral argument is supplied that would bound the possible graphs with non-vanishing H_1 when λ_min is bounded away from −∞; consequently the infinite-family dichotomy rests on an external result whose exhaustiveness is not re-proved or verified inside the manuscript.

    Authors: We agree that the main dichotomy and the enumeration of cases rest on Neumaier's classification of strongly regular graphs with integral smallest eigenvalue; no independent combinatorial or spectral bound is provided in the manuscript. This classification is a deep, established result in the literature, and the paper's contribution is the application of that classification to determine when the first clique homology can be nontrivial. Re-proving the classification would constitute a separate substantial work outside the paper's scope. To address the concern, we will add a short explanatory paragraph in the introduction that recalls the relevant statements from Neumaier's classification and explicitly indicates how they imply both the finite list of exceptional graphs and the infinite-family dichotomy. This will clarify the logical dependence for readers. revision: partial

Circularity Check

0 steps flagged

No circularity; results rest on external classification and direct computation

full rationale

The derivation applies Neumaier's external classification of SRGs with integral smallest eigenvalue to enumerate the finite list of families where non-vanishing H_1(Cl(G),F) can occur, then uses direct topological arguments to obtain the infinite-family dichotomy (lattice or lambda_min to -infty) and explicit homology formulas for Latin-square graphs. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the cited classification is independent and the computations (e.g., dim H_2 = (n-1)^3 - I(M)) are parameter-free once the graph is fixed.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure-mathematics proof work that invokes one external classification result as a domain assumption; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Neumaier's classification of strongly regular graphs with smallest integral eigenvalue is complete and accurate.
    Used to conclude that non-vanishing first homology can occur only in the enumerated short list of cases.

pith-pipeline@v0.9.1-grok · 5950 in / 1355 out tokens · 60763 ms · 2026-06-26T03:29:19.972764+00:00 · methodology

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Reference graph

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