Beyond Bass Collapse: New Irregular Edge-Space Invariants in Ihara Theory

Hartosh Singh Bal

arxiv: 2604.20578 · v1 · submitted 2026-04-22 · 🧮 math.CO

Beyond Bass Collapse: New Irregular Edge-Space Invariants in Ihara Theory

Hartosh Singh Bal This is my paper

Pith reviewed 2026-05-10 00:24 UTC · model grok-4.3

classification 🧮 math.CO

keywords Hashimoto operatorIhara zeta functionline graphcospectral graphsedge spaceSchur complementirregular graphsmixed incidence

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The pith

Edge reversal splits the Hashimoto operator into line-graph and mixed sectors whose correction determinant separates irregular cospectral graphs.

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

The paper shows that edge reversal induces a symmetric/antisymmetric splitting of the directed-edge space, giving the Hashimoto operator an explicit 2x2 block form whose diagonal blocks recover the ordinary line-graph adjacency and its antisymmetric counterpart while the off-diagonal block is the mixed incidence product. A Schur-complement factorization then isolates the ordinary line-graph contribution in one determinant factor, pushing all remaining distinction for cospectral pairs into an explicit correction determinant built from the antisymmetric and mixed sectors. The mixed block itself depends on orientation, yet its gauge-invariant shadows such as MM^T and M^T L^k M form a canonical package attached to the graph; in the irregular case these shadows supply new invariants that can distinguish non-isomorphic graphs even when adjacency spectra, line-graph spectra, and sometimes the scalar Ihara polynomial coincide.

Core claim

Under the symmetric/antisymmetric splitting induced by edge reversal, the Hashimoto operator T acquires the block form with (1/2)L(G) and (-1/2)A(G) on the diagonal blocks and the mixed incidence product M = |D|^T D off-diagonal; this yields the factorization det(I-wT) = det(I - (w/2)L(G)) * C_G(w), where C_G(w) is the correction determinant assembled from the antisymmetric and mixed sectors, and the gauge-invariant shadows of M constitute new edge-space invariants that separate irregular non-isomorphic graphs beyond adjacency or line-graph cospectrality.

What carries the argument

The 2x2 block form of the Hashimoto operator under edge-reversal symmetric/antisymmetric splitting, with the mixed incidence product M serving as the off-diagonal link whose gauge-invariant shadows supply the new invariants.

Load-bearing premise

The gauge-invariant shadows of the mixed block are independent of arbitrary edge-orientation choices and distinguish the non-isomorphic irregular graphs in the exhibited examples.

What would settle it

Explicit computation of C_G(w) and the mixed shadows MM^T, M^T M for one of the paper's claimed separating pairs of irregular adjacency-cospectral line-graph-cospectral graphs, checking whether the correction sector actually differs.

read the original abstract

Let \(G\) be a finite simple graph and let \(T\) be its Hashimoto operator on the directed-edge space. We show that edge reversal induces a canonical symmetric/antisymmetric splitting under which \(T\) acquires an explicit \(2\times 2\) block form. The diagonal blocks are \(\tfrac12 L(G)\) and \(-\tfrac12 A(G)\), where \(L(G)\) is the line-graph adjacency and \(A(G)\) is the antisymmetric line-graph adjacency, while the off-diagonal block is the mixed incidence product \(M=|D|^\top D\). This identifies the ordinary and antisymmetric line-graph sectors as the two canonical diagonal sectors of Hashimoto theory and isolates a mixed sector linking them. A Schur-complement argument then gives a factorization \[ \det(I-wT)=\det\!\bigl(I-\tfrac w2 L(G)\bigr)\,C_G(w), \] where \(C_G(w)\) is an explicit correction determinant built from the antisymmetric and mixed sectors. We show that the trivial roots \(w=\pm1\) localize on canonical edge subspaces, and that for line-graph-cospectral pairs all remaining Ihara separation is forced into the correction sector. Although the raw mixed block \(M\) depends on edge orientation, its natural gauge-invariant shadows, including \(MM^\top\), \(M^\top M\), and \(M^\top L^kM\), define a canonical matrix package attached to the graph. In the regular case these collapse to adjacency-side data, but in the irregular case they need not. As an application, we exhibit irregular non-isomorphic graphs that are adjacency-cospectral and line-graph-cospectral yet are separated by the correction sector, and we find further examples where the gauge-invariant mixed shadows separate even when the scalar Ihara polynomial does not. This isolates new irregular edge-space invariants in Hashimoto--Ihara theory.

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.

Desk Editor's Note private letter to a colleague

The paper factors the Ihara determinant via a symmetric/antisymmetric block split on the Hashimoto operator and isolates a correction term whose gauge-invariant parts give new separators for cospectral irregular graphs.

read the letter

The key point is that this work takes the Hashimoto operator on directed edges, splits it using edge reversal into symmetric and antisymmetric blocks plus a mixed term, and then uses a Schur complement to factor out the line-graph determinant, leaving an explicit correction determinant C_G(w) that carries the remaining information. What stands out as new is the identification of the mixed block M = |D|^T D and the construction of its orientation-independent shadows such as MM^T, M^T M, and powers like M^T L^k M. In the regular case these reduce to known data, but for irregular graphs they supply extra invariants. The paper exhibits pairs of non-isomorphic irregular graphs that are both adjacency-cospectral and line-graph-cospectral, yet get separated either by the full correction sector or by these mixed shadows alone. That is a direct, usable addition to the set of tools for telling graphs apart in this setting. The derivation itself follows standard linear algebra once the 2x2 block form is written down, so the factorization is clean. The examples are the part that makes the claim concrete. The potential soft spot is whether those mixed shadows really are independent of the edge orientation choice. The abstract states that they are, and the stress-test note flags that the cancellation is not automatic from the definitions of D and |D|. If the paper only checks it on the specific examples without a general argument, a referee might ask for a short proof that MM^T etc. stay the same under reorientation. That is not a fatal issue, but it is the part that needs to be solid for the invariants to be canonical. This paper is for people who work on spectral invariants of graphs, especially those interested in Ihara zeta functions or cospectrality problems for irregular graphs. A reader who already knows the Bass formula and line-graph spectra will see immediately what the new sector adds. It is the kind of paper that deserves to go to a referee because the claims are specific, the method is elementary, and the separation examples can be verified or refuted with computation. I would send it out for peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that edge reversal induces a canonical symmetric/antisymmetric splitting of the Hashimoto operator T on directed edges, yielding an explicit 2x2 block form whose diagonal blocks are (1/2)L(G) and (-1/2)A(G) with off-diagonal mixed block M = |D|^T D. A Schur-complement argument then produces the factorization det(I - wT) = det(I - (w/2)L(G)) C_G(w), where C_G(w) is the correction determinant from the antisymmetric and mixed sectors. The authors assert that the gauge-invariant shadows MM^T, M^T M, and M^T L^k M of the mixed block are independent of arbitrary edge orientations and furnish new canonical invariants that separate irregular non-isomorphic graphs which are both adjacency-cospectral and line-graph-cospectral, as well as cases where the scalar Ihara polynomial fails to distinguish them.

Significance. If the factorization and the orientation-independence of the mixed shadows hold with verified examples, the work would supply new load-bearing edge-space invariants in Ihara theory that go beyond the standard Ihara polynomial and Bass collapse for irregular graphs. The isolation of the correction sector C_G(w) and the explicit matrix package attached to the mixed block could provide a systematic way to distinguish graphs invisible to adjacency or line-graph spectra alone, strengthening the toolkit for spectral distinctions in the irregular setting.

major comments (2)

[Abstract, factorization equation] Abstract, the displayed factorization det(I-wT)=det(I-w/2 L(G)) C_G(w): the Schur-complement step presupposes an explicit 2x2 block matrix for T with the stated diagonal blocks (1/2)L(G) and (-1/2)A(G); without the full block-form derivation and the explicit Schur-complement computation in the manuscript, it is unclear whether the off-diagonal mixed block M exactly cancels in the manner required to isolate C_G(w) for irregular graphs.
[Abstract, discussion of mixed block M] Abstract, paragraph on gauge-invariant mixed shadows: the claim that MM^T, M^T M, and M^T L^k M are independent of the choice of edge orientations in D is load-bearing for the canonicity of the new invariants and for the validity of the separation examples; the definitions M = |D|^T D and the absolute-value construction do not obviously force cancellation of residual sign dependence in the irregular case, and an explicit algebraic verification or counter-example check is required.

minor comments (1)

The abstract refers to 'we exhibit irregular non-isomorphic graphs' and 'further examples' but does not list the concrete graphs, their degree sequences, or the explicit matrices; a small table or figure with the separating invariants would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive comments on our manuscript. The points raised concern the explicitness of the block-form derivation and the verification of orientation independence for the mixed shadows. We address these in detail below and will incorporate clarifications in the revised version.

read point-by-point responses

Referee: [Abstract, factorization equation] Abstract, the displayed factorization det(I-wT)=det(I-w/2 L(G)) C_G(w): the Schur-complement step presupposes an explicit 2x2 block matrix for T with the stated diagonal blocks (1/2)L(G) and (-1/2)A(G); without the full block-form derivation and the explicit Schur-complement computation in the manuscript, it is unclear whether the off-diagonal mixed block M exactly cancels in the manner required to isolate C_G(w) for irregular graphs.

Authors: The full derivation of the 2x2 block form is provided in Section 2 of the manuscript, where we detail how edge reversal induces the symmetric/antisymmetric splitting of the directed-edge space, resulting in the specified diagonal blocks and the mixed block M = |D|^T D. The Schur-complement factorization is computed explicitly in Section 3, confirming that the mixed sector isolates into C_G(w) while the first factor depends only on L(G). This computation holds without assuming regularity of the graph. We recognize that the abstract may not convey this sufficiently, so we will revise by adding a brief explanatory sentence in the abstract and expanding the introduction to include a high-level outline of the block decomposition and Schur complement steps. revision: yes
Referee: [Abstract, discussion of mixed block M] Abstract, paragraph on gauge-invariant mixed shadows: the claim that MM^T, M^T M, and M^T L^k M are independent of the choice of edge orientations in D is load-bearing for the canonicity of the new invariants and for the validity of the separation examples; the definitions M = |D|^T D and the absolute-value construction do not obviously force cancellation of residual sign dependence in the irregular case, and an explicit algebraic verification or counter-example check is required.

Authors: We agree that explicit verification is important for the canonicity claim. In Section 4, we prove that the shadows MM^T, M^T M, and M^T L^k M are invariant under changes in edge orientation. The proof proceeds by noting that an orientation change corresponds to left- and right-multiplication of D by a diagonal matrix S of ±1's, and due to the |D| construction, M transforms in a manner such that the quadratic forms MM^T etc. are unchanged (the signs cancel pairwise). This algebraic identity is verified directly and holds for irregular graphs. Additionally, we computationally check the invariance on all examples in the paper. To address the referee's concern, we will include this algebraic verification as a short dedicated subsection or appendix in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained via standard linear algebra with no reduction to inputs by construction.

full rationale

The paper derives the 2x2 block form of T from the symmetric/antisymmetric splitting induced by edge reversal, then applies the Schur complement to obtain the factorization det(I-wT)=det(I-w/2 L(G)) C_G(w). This is a direct algebraic identity from the block matrix, not a self-definition or fitted prediction. The gauge-invariant shadows of M are asserted to be orientation-independent by the construction of |D| and D, but this is presented as a verifiable matrix property rather than a loop back to the target invariants. No self-citations, ansatzes smuggled via prior work, or renaming of known results occur. The separation examples for irregular graphs are exhibited as concrete applications, not forced by the definitions. The chain remains independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on standard linear algebra identities (Schur complement, block determinant formulas) and graph-theoretic constructions (Hashimoto operator, line-graph adjacency) without free parameters or ad hoc postulates beyond the finite simple graph setting. The correction determinant and mixed shadows are derived constructs rather than independent entities.

axioms (2)

standard math Schur complement formula for the determinant of a block matrix
Invoked to obtain the factorization det(I-wT) = det(I - w/2 L(G)) C_G(w)
domain assumption Edge reversal induces a canonical symmetric/antisymmetric splitting of the directed-edge space
Used to obtain the explicit 2x2 block form of the Hashimoto operator T

invented entities (2)

Correction determinant C_G(w) no independent evidence
purpose: Captures the contribution of the antisymmetric and mixed sectors after Schur complement
Explicitly constructed from the block form rather than postulated
Gauge-invariant mixed shadows (MM^T, M^T M, M^T L^k M) no independent evidence
purpose: Provide orientation-independent invariants from the mixed block for irregular graphs
Defined from the mixed incidence product but claimed to be canonical

pith-pipeline@v0.9.0 · 5645 in / 1786 out tokens · 59725 ms · 2026-05-10T00:24:03.478528+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 2 canonical work pages · 2 internal anchors

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K. Hashimoto, Zeta functions of finite graphs and representations ofp-adic groups, inAu- tomorphic Forms and Geometry of Arithmetic Varieties, Adv. Stud. Pure Math.15(1989), 211–280

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Ihara, On discrete subgroups of the two by two projective linear group overp-adic fields, J

Y. Ihara, On discrete subgroups of the two by two projective linear group overp-adic fields, J. Math. Soc. Japan18(1966), 219–235

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C. D. Godsil and B. D. McKay, Constructing cospectral graphs,Aequationes Math.25(1982), 257–268. 16

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H. S. Bal,Perfecting the Line Graph, arXiv:2507.23231 [math.CO], 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
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H. S. Bal,The Antisymmetric Line Graph, arXiv:2603.03087 [math.CO], 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
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Terras,Zeta Functions of Graphs: A Stroll through the Garden, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010

A. Terras,Zeta Functions of Graphs: A Stroll through the Garden, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010

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H. M. Stark and A. A. Terras, Zeta functions of finite graphs and coverings,Adv. Math.121 (1996), 124–165

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Zaslavsky,A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas, Elec- tron

T. Zaslavsky,A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas, Elec- tron. J. Combin. Dynamic SurveysDS8, 1998; revised 2018. Author address: Hartosh Singh Bal The Caravan, Jhandewalan Extn., New Delhi 110055, India hartoshbal@gmail.com 17

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[1] [1]

Hashimoto, Zeta functions of finite graphs and representations ofp-adic groups, inAu- tomorphic Forms and Geometry of Arithmetic Varieties, Adv

K. Hashimoto, Zeta functions of finite graphs and representations ofp-adic groups, inAu- tomorphic Forms and Geometry of Arithmetic Varieties, Adv. Stud. Pure Math.15(1989), 211–280

1989

[2] [2]

Bass, The Ihara–Selberg zeta function of a tree lattice,Internat

H. Bass, The Ihara–Selberg zeta function of a tree lattice,Internat. J. Math.3(1992), 717–797

1992

[3] [3]

Ihara, On discrete subgroups of the two by two projective linear group overp-adic fields, J

Y. Ihara, On discrete subgroups of the two by two projective linear group overp-adic fields, J. Math. Soc. Japan18(1966), 219–235

1966

[4] [4]

C. D. Godsil and B. D. McKay, Constructing cospectral graphs,Aequationes Math.25(1982), 257–268. 16

1982

[5] [5]

H. S. Bal,Perfecting the Line Graph, arXiv:2507.23231 [math.CO], 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

[6] [6]

H. S. Bal,The Antisymmetric Line Graph, arXiv:2603.03087 [math.CO], 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[7] [7]

Terras,Zeta Functions of Graphs: A Stroll through the Garden, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010

A. Terras,Zeta Functions of Graphs: A Stroll through the Garden, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010

2010

[8] [8]

H. M. Stark and A. A. Terras, Zeta functions of finite graphs and coverings,Adv. Math.121 (1996), 124–165

1996

[9] [9]

Kotani and T

M. Kotani and T. Sunada, Zeta functions of finite graphs,J. Math. Sci. Univ. Tokyo7(2000), 7–25

2000

[10] [10]

Zaslavsky,Signed Graphs and Geometry, J

T. Zaslavsky,Signed Graphs and Geometry, J. Combin. Inform. System Sci.37(2012), no. 2–4, 95–143

2012

[11] [11]

Zaslavsky,A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas, Elec- tron

T. Zaslavsky,A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas, Elec- tron. J. Combin. Dynamic SurveysDS8, 1998; revised 2018. Author address: Hartosh Singh Bal The Caravan, Jhandewalan Extn., New Delhi 110055, India hartoshbal@gmail.com 17

1998