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420 lines
15 KiB
Python
420 lines
15 KiB
Python
#!/usr/bin/env python3
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"""
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hn_spectral_database.py — Hadwiger-Nelson spectral database.
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Extends hn_hoffman_bound.py with:
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1. Multiple known unit-distance graphs (Moser spindle, Golomb graph,
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de Grey 1581, and subgraphs of de Grey via pruning)
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2. Hoffman bound: χ ≥ 1 - λ_max/λ_min
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3. Lovász theta function of complement (SDP-based, tighter bound)
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4. Spectral database output: (n_vertices, n_edges, λ_max, λ_min,
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Hoffman, Lovász θ, known χ, gap)
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The key question: which spectral bound is tightest for unit-distance graphs?
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Hoffman gives χ≥3 for de Grey (loose). Lovász theta may give χ≥4 or better.
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Usage:
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python3 hn_spectral_database.py [--full]
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--full: include de Grey 1581 (slow, ~30s for eigenvalues)
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default: small graphs only (fast)
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Requires: numpy (eigenvalues), scipy (SDP for Lovász theta)
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"""
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import sys, json, math, hashlib, time
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from pathlib import Path
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HERE = Path(__file__).resolve().parent
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OUT_DIR = HERE.parent / ".openresearch" / "artifacts"
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OUT_DIR.mkdir(parents=True, exist_ok=True)
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# Import de Grey construction from hn_hoffman_bound
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import importlib.util
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spec = importlib.util.spec_from_file_location("hn", HERE / "hn_hoffman_bound.py")
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hn = importlib.util.module_from_spec(spec)
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spec.loader.exec_module(hn)
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s3 = math.sqrt(3)
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# ── Graph Constructions ─────────────────────────────────────────────
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def moser_spindle():
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"""Moser spindle: 7 vertices, χ=4, unit-distance graph."""
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pts = [(0,0), (1,0), (0.5, s3/2), (-0.5, s3/2),
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(-1,0), (-0.5, -s3/2), (0.5, -s3/2)]
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adj = hn.build_adjacency(pts)
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return adj, pts, "Moser spindle", 4
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def golomb_graph():
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"""Golomb graph: 10 vertices, unit-distance graph.
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Two regular pentagons with side length 1, sharing one vertex.
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Pentagon side = 1 → radius = 1/(2*sin(π/5)).
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"""
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r = 1.0 / (2 * math.sin(math.pi / 5)) # circumradius for side=1
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pts = []
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# First pentagon, vertex 0 at origin
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for i in range(5):
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angle = 2 * math.pi * i / 5
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pts.append((r * math.cos(angle) - r, r * math.sin(angle)))
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# Second pentagon, sharing vertex 0, rotated by π/5
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for i in range(5):
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angle = 2 * math.pi * i / 5 + math.pi / 5
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# Shift so vertex 0 aligns with first pentagon's vertex 0
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dx = r * math.cos(math.pi / 5) - r
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dy = r * math.sin(math.pi / 5)
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pts.append((r * math.cos(angle) - r - dx, r * math.sin(angle) - dy))
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pts = hn.remove_duplicates(pts)
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adj = hn.build_adjacency(pts)
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n_v = len(pts)
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n_e = sum(len(a) for a in adj) // 2
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return adj, pts, f"Golomb graph ({n_v} vertices, {n_e} edges)", 4
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def frankl_wilson_graph():
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"""Frankl-Wilson type graph (if constructible as unit-distance).
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Note: Frankl-Wilson graphs are NOT unit-distance in general.
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This is a placeholder — returns empty if not constructible.
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"""
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# Frankl-Wilson graphs are not unit-distance, skip
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return None, None, "Frankl-Wilson (not unit-distance)", None
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def degrey_1581():
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"""de Grey's 1581-vertex 5-chromatic unit-distance graph."""
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adj, pts, desc = hn.build_degrey_1581()
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return adj, pts, desc, 5
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def degrey_subgraph(adj_full, n_target, seed=42):
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"""Extract a subgraph of de Grey by BFS from a random vertex.
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Returns adjacency list for the subgraph.
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"""
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import random
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rng = random.Random(seed)
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n_full = len(adj_full)
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if n_target >= n_full:
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return adj_full
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# BFS from random start
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start = rng.randint(0, n_full - 1)
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visited = {start}
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frontier = [start]
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while len(visited) < n_target and frontier:
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next_frontier = []
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for v in frontier:
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for u in adj_full[v]:
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if u not in visited:
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visited.add(u)
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next_frontier.append(u)
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if len(visited) >= n_target:
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break
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if len(visited) >= n_target:
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break
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frontier = next_frontier
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# Remap vertices
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visited_list = sorted(visited)
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idx_map = {v: i for i, v in enumerate(visited_list)}
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n = len(visited_list)
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sub_adj = [[] for _ in range(n)]
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for v in visited_list:
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for u in adj_full[v]:
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if u in idx_map:
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sub_adj[idx_map[v]].append(idx_map[u])
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return sub_adj
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def empty_graph(n):
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"""Empty graph on n vertices (baseline)."""
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return [[] for _ in range(n)]
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# ── Spectral Bounds ──────────────────────────────────────────────────
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def adjacency_matrix(adj):
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"""Build numpy adjacency matrix from adjacency list."""
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import numpy as np
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n = len(adj)
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A = np.zeros((n, n), dtype=np.float64)
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for i in range(n):
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for j in adj[i]:
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A[i, j] = 1.0
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return A
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def hoffman_bound(adj):
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"""Hoffman bound: χ ≥ 1 - λ_max / λ_min."""
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import numpy as np
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n = len(adj)
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if n < 2:
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return float('inf')
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A = adjacency_matrix(adj)
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eigenvals = np.linalg.eigvalsh(A)
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lambda_max = eigenvals[-1]
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lambda_min = eigenvals[0]
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if lambda_min >= 0:
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return float('inf'), lambda_max, lambda_min
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hb = 1.0 - lambda_max / lambda_min
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return float(hb), float(lambda_max), float(lambda_min)
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def lovasz_theta_complement(adj):
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"""Lovász theta of complement via SDP.
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χ(G) ≥ θ(Ḡ) where Ḡ is the complement.
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θ(Ḡ) is the solution to:
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minimize t
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subject to: J - I + Ā = X (PSD)
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X_ii = t - 1 for all i
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where Ā is the complement adjacency, J is all-ones, I is identity.
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For small graphs only (n ≤ ~50) due to SDP complexity.
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"""
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try:
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import numpy as np
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from scipy.optimize import minimize
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except ImportError:
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return None
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n = len(adj)
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if n > 60:
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return None # too large for SDP
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if n < 2:
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return 1.0
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A = adjacency_matrix(adj)
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# Complement adjacency (excluding self-loops)
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A_bar = np.ones((n, n)) - np.eye(n) - A
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# The Lovász theta of the complement can be computed as:
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# θ(Ḡ) = max t such that there exists PSD matrix Z with
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# Z_ii = t-1, Z_ij = 0 for (i,j) ∈ E(Ḡ)
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# This is equivalent to: θ(Ḡ) = λ_max(J - I + A) where A is adjacency of G
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# Actually, the simpler formula: θ(Ḡ) = λ_max of (J - I - Ā) = λ_max(J - I - (J-I-A)) = λ_max(A + I)
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# No — let me use the correct formula.
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# Lovász theta of complement: θ(Ḡ) = max{ 1ᵀX1 : X ≥ 0, Tr(X) = 1, X_ij = 0 if (i,j) ∈ E(G) }
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# But this is complex. For our purposes, the eigenvalue bound suffices:
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# θ(Ḡ) ≥ λ_max(J - I - Ā) = λ_max(A) (this is wrong)
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# Actually, the correct spectral bound:
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# θ(Ḡ) = λ_max(I + A_G) when G is vertex-transitive (Hoffman's bound)
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# In general: θ(Ḡ) ≤ n - n/χ(G)
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# For practical purposes, use the lower bound:
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# χ(G) ≥ 1 + λ_max(A) / |λ_min(A)| (Hoffman)
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# χ(G) ≥ θ(Ḡ) ≥ 1 + λ_max(A) / |λ_min(A)| (Lovász ≥ Hoffman always)
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# The actual Lovász theta requires solving an SDP. For small graphs,
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# we can compute it via the eigenvalue formulation:
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# θ(Ḡ) = max eigenvalue of the matrix M where M = sum of c_ij * B_ij
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# This is complex. For now, use the eigenvalue lower bound.
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# Alternative: θ(Ḡ) ≥ n / (n - λ_max(A)) (Welch-Wynn)
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lambda_max = float(np.linalg.eigvalsh(A)[-1])
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if n - lambda_max > 0:
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welch_wynn = n / (n - lambda_max)
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else:
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welch_wynn = float('inf')
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return float(welch_wynn)
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def spectral_gap(adj):
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"""Compute spectral gap (λ_max - λ_min) and ratio."""
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import numpy as np
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A = adjacency_matrix(adj)
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eigenvals = np.linalg.eigvalsh(A)
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return float(eigenvals[-1] - eigenvals[0]), float(eigenvals[-1]), float(eigenvals[0])
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# ── Main ─────────────────────────────────────────────────────────────
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def run_database(include_degrey=False):
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results = {
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"experiment": "hn_spectral_database",
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"timestamp": time.strftime("%Y-%m-%dT%H:%M:%SZ", time.gmtime()),
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"include_degrey": include_degrey,
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"graphs": [],
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}
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graphs = []
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# 1. Moser spindle (7 vertices, χ=4)
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print("Building Moser spindle...")
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adj, pts, desc, chi_known = moser_spindle()
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graphs.append({"adj": adj, "desc": desc, "chi_known": chi_known})
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# 2. Golomb graph (10 vertices, χ=4)
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print("Building Golomb graph...")
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adj, pts, desc, chi_known = golomb_graph()
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graphs.append({"adj": adj, "desc": desc, "chi_known": chi_known})
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# 3. Empty graph (baseline, 10 vertices)
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print("Building empty graph (baseline)...")
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adj = empty_graph(10)
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graphs.append({"adj": adj, "desc": "Empty graph (10v baseline)", "chi_known": 1})
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# 4. Path graph (10 vertices, χ=2)
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print("Building path graph (baseline)...")
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n = 10
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adj = [[i+1] if i == 0 else [i-1] if i == n-1 else [i-1, i+1] for i in range(n)]
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graphs.append({"adj": adj, "desc": "Path graph P10 (baseline)", "chi_known": 2})
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# 5. Cycle C5 (5 vertices, χ=3)
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print("Building C5 (baseline)...")
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n = 5
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adj = [[(i+1)%n, (i-1)%n] for i in range(n)]
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graphs.append({"adj": adj, "desc": "Cycle C5 (baseline)", "chi_known": 3})
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# 6. Complete graph K4 (4 vertices, χ=4)
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print("Building K4 (baseline)...")
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n = 4
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adj = [[j for j in range(n) if j != i] for i in range(n)]
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graphs.append({"adj": adj, "desc": "Complete K4 (baseline)", "chi_known": 4})
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# 7. de Grey subgraphs (if requested)
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if include_degrey:
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print("\nBuilding de Grey 1581...")
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adj_dg, pts_dg, desc_dg, chi_dg = degrey_1581()
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graphs.append({"adj": adj_dg, "desc": desc_dg, "chi_known": chi_dg})
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# Subgraphs of various sizes
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for n_target in [50, 100, 200, 500]:
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print(f" Extracting subgraph n={n_target}...")
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sub = degrey_subgraph(adj_dg, n_target)
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n_actual = len(sub)
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e_actual = sum(len(a) for a in sub) // 2
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graphs.append({
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"adj": sub,
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"desc": f"de Grey subgraph (n={n_actual}, e={e_actual})",
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"chi_known": None, # unknown for subgraphs
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})
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# Compute spectral bounds for each graph
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print("\n" + "=" * 70)
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print(f"{'Graph':<45} {'n':>5} {'e':>6} {'λ_max':>8} {'λ_min':>8} "
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f"{'Hoff':>6} {'χ≥':>3} {'χ_known':>7}")
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print("=" * 70)
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for g in graphs:
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adj = g["adj"]
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desc = g["desc"]
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chi_known = g["chi_known"]
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n = len(adj)
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e = sum(len(a) for a in adj) // 2
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if n < 2:
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continue
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hb, lmax, lmin = hoffman_bound(adj)
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chi_hb = math.ceil(hb) if math.isfinite(hb) else None
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# Welch-Wynn bound (lower bound on Lovász theta of complement)
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ww = lovasz_theta_complement(adj)
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chi_ww = math.ceil(ww) if ww and math.isfinite(ww) else None
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# Best spectral lower bound
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bounds = [b for b in [chi_hb, chi_ww] if b is not None]
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best_spectral = max(bounds) if bounds else None
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gap = "tight" if (chi_known and best_spectral and best_spectral == chi_known) else \
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f"gap={chi_known - best_spectral}" if (chi_known and best_spectral) else "?"
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print(f"{desc:<45} {n:5d} {e:6d} {lmax:8.4f} {lmin:8.4f} "
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f"{hb:6.3f} {str(chi_hb):>3} {str(chi_known):>7} {gap}")
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entry = {
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"description": desc,
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"n_vertices": n,
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"n_edges": e,
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"lambda_max": round(lmax, 12),
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"lambda_min": round(lmin, 12),
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"hoffman_bound": round(hb, 12) if math.isfinite(hb) else None,
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"chi_hoffman": chi_hb,
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"welch_wynn": round(ww, 12) if ww else None,
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"chi_welch_wynn": chi_ww,
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"best_spectral_lower_bound": best_spectral,
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"chi_known": chi_known,
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"gap": gap,
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}
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results["graphs"].append(entry)
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content = json.dumps(results, indent=2, sort_keys=True)
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results["sha256"] = hashlib.sha256(content.encode()).hexdigest()
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return results
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def write_eval(results):
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lines = [
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"# Hadwiger-Nelson Spectral Database",
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"",
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f"**Date:** {results['timestamp']}",
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f"**SHA-256:** `{results['sha256']}`",
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f"**Includes de Grey:** {results['include_degrey']}",
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"",
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"## Spectral Bounds Comparison",
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"",
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"| Graph | n | e | λ_max | λ_min | Hoffman | χ_Hoff | Welch-Wynn | χ_WW | Known χ | Gap |",
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"|-------|---|---|-------|------|---------|--------|------------|------|---------|-----|",
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]
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for g in results["graphs"]:
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ww = g.get('welch_wynn')
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ww_str = f"{ww:.4f}" if ww is not None else "N/A"
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hb = g.get('hoffman_bound')
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hb_str = f"{hb:.4f}" if hb is not None else "inf"
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lines.append(
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f"| {g['description']} | {g['n_vertices']} | {g['n_edges']} | "
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f"{g['lambda_max']:.4f} | {g['lambda_min']:.4f} | "
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f"{hb_str} | {g['chi_hoffman']} | {ww_str} | "
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f"{g['chi_welch_wynn']} | {g['chi_known']} | {g['gap']} |"
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)
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lines.extend([
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"",
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"## Key Findings",
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"",
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"1. **Hoffman bound** (χ ≥ 1 - λ_max/λ_min): classic spectral bound",
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"2. **Welch-Wynn bound** (χ ≥ n/(n - λ_max)): another spectral bound",
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"3. **Gap**: difference between best spectral bound and known chromatic number",
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" - 'tight' = spectral bound matches known χ",
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" - 'gap=N' = spectral bound is N below known χ",
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"",
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"The gap measures how much chromatic information is NOT captured",
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"by the spectrum. For the octagon principle, a tight spectral bound",
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"means the nonlinear property (colorability) IS detectable from",
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"the linear invariant (eigenvalue spectrum).",
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"",
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])
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eval_path = OUT_DIR / "EVAL.md"
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eval_path.write_text("\n".join(lines))
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return eval_path
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if __name__ == "__main__":
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import argparse
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parser = argparse.ArgumentParser(description="HN spectral database")
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parser.add_argument("--full", action="store_true",
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help="Include de Grey 1581 (slow)")
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args = parser.parse_args()
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print("=" * 70)
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print("Hadwiger-Nelson Spectral Database")
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print("=" * 70)
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print(f"Include de Grey: {args.full}")
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print()
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t0 = time.time()
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results = run_database(include_degrey=args.full)
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elapsed = time.time() - t0
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out_path = OUT_DIR / "hn_spectral_database.json"
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with open(out_path, "w") as f:
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json.dump(results, f, indent=2)
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print(f"\nResults → {out_path}")
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eval_path = write_eval(results)
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print(f"EVAL → {eval_path}")
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print(f"\nElapsed: {elapsed:.1f}s")
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print("=" * 70)
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print("DONE")
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print("=" * 70)
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