bibkey: harithachithra2022distanceseidel
authors: T. Haritha and A. V. Chithra
year: 2022
title: Distance Seidel matrix of a connected graph
doi: 10.1080/09728600.2025.2479147
url: https://arxiv.org/abs/2210.05940v2
claim: "Problem 5.1 asks whether deleting any edge of K_{a,b}, with a,b > 1, strictly increases the distance Seidel energy."
strata_touched:
  - D5/S3/Combinatorics/Graph/KyFanGap/HarithaChithraDistanceSeidelEnergy
license: citation-only
triage: anchor

Distance Seidel energy of complete bipartite graphs

Verified locator

DOI: 10.1080/09728600.2025.2479147

Source: https://arxiv.org/abs/2210.05940v2

The arXiv version is v2. The journal version is titled “The distance Seidel matrix of connected graphs”. The quotations and page numbers below refer to arXiv v2.

Source definitions

Page 2:

D^S(G) = J − I − 2D = −1 if v_r is adjacent to v_t, 1 − 2d_G(v_r, v_t) otherwise.

The distance Seidel energy E_{D^S}(G) of a graph G is defined as E_{D^S}(G) = Σ_{r=1}^n |∂_r^S|.

The equation J − I − 2D has diagonal zero, while the casewise expression has diagonal one. The Lean definition uses the matrix equation. Its energy sums absolute real characteristic roots with multiplicity; the distance matrices in the theorem are real symmetric.

Problem 5.1

Page 13:

Let G = K_{a,b}, with a, b > 1. Is E_{D^S}(K_{a,b} − e) > E_{D^S}(K_{a,b}), for any edge e of K_{a,b}?

The module proves the affirmative statement for every a,b ≥ 2 and every edge. Its supporting trace functional is unchanged by the rank-two edge perturbation; noncommutation makes the trace inequality strict. Propositions 5.2–5.3 give the cases with first part of size two and three. The universal Lean theorem uses the zero-diagonal matrix; the diagonal-one reading has only finite experiment evidence here.

Literature boundary

The literature screen is partial; completeness is ASSUMED-UNVERIFIED. The preregistration is https://github.com/the-omega-institute/trureturing/issues/19532. The source and later works reached by that screen did not supply a general settlement.

Finite comparison

The finite comparison for 2 ≤ a ≤ b ≤ 12 is in trureturing-experiments, commit 23a1280fffd81c90b7f5f6981dd13a24a390c3c3, at docs/reports/haritha-chithra-2022-distance-seidel-energy-problem-5-1/check.py (SHA-256 d98dd9d8b190718d24a193f64dc33e4f018260d706052b646d596a275a797405). Running /tmp/op-lwc-venv/bin/python /Users/auric/trureturing-experiments/docs/reports/haritha-chithra-2022-distance-seidel-energy-problem-5-1/check.py exits 0 with RESULT PASS. This comparison checks both diagonal readings on that finite range; the universal Lean theorem concerns J − I − 2D.