3D Interactive Directed Graph with Spectral Analysis

Graph Layout

Vertices:

Radius:

Force-Directed Layout

Speed:

3D forces

💡 Shift+Click node to pin/unpin

🎨 Display Options

Solid Faces (Polyhedron View)

Quick Templates

Select a template

Skew-symmetric (unidirectional)

Edit Mode

VIEW MODE

Edge Operations

Vertex Operations

View: Drag to rotate, scroll to zoom

📷 View Projection

Arrange Layout

Rows: Cols:

Spacing:

📐 Snap Grid

Grid Size:

Snap to Grid

Auto Generate Edges

Count:

🌊 DYNAMICS ANIMATION

Visualize ẋ = Ax (skew-symmetric)

① VISUALIZATION MODE

Displacement Power Energy

Node color shows state magnitude: cyan (+) / magenta (−). Arrows show xᵢ·xⱼ between connected nodes.

② ANIMATION CONTROLS

Speed: 1.0×

Arrow Scale: 30

Base Length: 8

③ SIMULATION STATUS

0.00 TIME Elapsed (s)

0.00 MAX |xᵢ| Peak state

0.00 MAX FLOW Peak xᵢxⱼ

④ ENERGY Glow

T (kinetic)

0.00

V (potential)

0.00

H = T + V

0.00

✓ ΔH: 0.00% — Conserved

⑤ PHASE DIAGRAM

Plot:

Nodes: vs

Ellipse shape reveals phase relationship between nodes

⑥ ADVANCED SETTINGS ▼

Integrator:

Time Step:

Rodrigues: Exact matrix exponential. Energy-conserving.

Cayley: Fast (I+A/2)(I−A/2)⁻¹ approximation.

Trapezoidal: Implicit midpoint rule.

⚛️ Linear System Realizability

① CONFIGURE PARTITION

N (absolute states):

Click to toggle: p (momentum) ↔ q (displacement)

p: - q: -

Auto

② AUDIT RESULTS

Click Audit to check

N (absolute) -

M (relative) -

Violations -

Column Analysis

③ HAMILTONIAN DYNAMICS

State Vector x = [p | q]ᵀ

p (momenta)

q (displacements)

H(t) = T + V

T = 0.000 V = 0.000 H = 0.000

━ H (total) ━ T (kinetic) ━ V (potential)

t = 0.00s Energy Conserved ✓

System Matrix A = [0, B; -Bᵀ, 0]

Graph Info

Unknown

Vertices 0

Edges 0

Skew-Symmetric Eigenvalues

Eigenvalues are purely imaginary: ±iλ

Numerical Values (click to animate)

Symmetric Eigenvalues

Generate a graph to see eigenvalues

Numerical Values (click to animate)

📊 Complex Plane (All Eigenvalues)

System eigenvalues: λ = −α ± i·β·b where b is graph eigenvalue

α (damping): 1.0

β (coupling): 1.0

Stable Bounds λ values

0 eigenvalues Stable

Characteristic Polynomial

Adjacency Matrices

Directed (Skew-Symmetric)

Undirected (Symmetric)

📐 Point Graph Topology

Detect from graph

Physical Parameters

These sync with ANALYZE tab's complex plane

α (damping): 1.0

β (coupling): 1.0

📊 See ANALYZE tab for complex plane visualization

Eigenvalue Bounds (Table I)

Bounds update as α,β change

Stability Analysis

Based on Main Results 1 & 2

🔬 Compare with Numerical

Shows how tight the bounds are

⊕ Direct Sum

Combine two graph spectra: G = G₁ ⊕ G₂

G₁:

G₂:

📖 Reference (Table I & Appendix)

Graph	b₁/β
Path Pₙ	2cos(π/(N+1))
Cycle Cₙ	2
Star Sₙ	√(N-1)
S'ₚ (N=2p+1)	√(p+1)
S²ₚ (N=3p+1)	√((p/2+1)+√((p/2)²+1))
Complete Kₙ	cot(π/2N)
Grid m×n	2cos(π/(m+1))+2cos(π/(n+1))
General Ladder m×n	2cos(π/(n+1))+√(m+1)
Simple Ladder Lₙ	2cos(π/(n+1))+1
Circular Ladder	3
Hypercube Qₐ	d
Wheel Wₙ	≈ √(N-1)+1
General Tree	≤ √(N-1)
General	≤ cot(π/2N)

🔗 Graph Products

Combine graphs with known eigenspectra

Graph A:

Graph B:

n(B):

Result: Current □ P₃ (extrusion) Vertices: depends on current graph λᵢⱼ = λᵢ(A) + μⱼ(B)

📐 Parameterized Families

Build graphs with explicit eigenvalue formulas

Family:

Depth d: 2

Branches p: 4

N = d·p + 1 = 9 λ = ±√(p+1), ±1 (×p-1), 0

Step set S:

C(8, {1,2}): 8 vertices, 4-regular

GP(5,2) = Petersen: 10 vertices

Prism Y₅: 10 vertices

🔍 Find Analytic Graphs

Hierarchical search with product graph optimization

Vertices: n ≤ 8

Connected only faster

📖 Product Eigenvalue Formulas

Product	Eigenvalues
G □ H (Cartesian)	λᵢ(G) + μⱼ(H)
G ⊗ H (Tensor)	λᵢ(G) · μⱼ(H)
G ⊠ H (Strong)	(1+λᵢ)(1+μⱼ) - 1

Extrusion: 2D mesh × Pₙ = 3D extruded mesh
Example: Cycle × Path = Cylinder surface

📚 Analytic Graph Library

Persistent storage for graphs with closed-form eigenvalues

0 Graphs

0 Families

🚀 Universe Navigation

📷 View Projection

Mouse Controls:
• Drag: rotate (3D) / pan (2D views)
• Shift+drag or middle-drag: pan
• Scroll: zoom
• Right-drag: rubber band zoom
• Double-click: load graph

	Name	n	E	Actions

📤 Actions

⚠️ Danger Zone

This cannot be undone!

Graph Layout

Force-Directed Layout

🎨 Display Options

Quick Templates

🌌 Universe Integration

Edit Mode

Edge Operations

Vertex Operations

📷 View Projection

Arrange Layout

📐 Snap Grid

Auto Generate Edges

🌊 DYNAMICS ANIMATION

⚛️ Linear System Realizability

Graph Info

Skew-Symmetric Eigenvalues 📋

Symmetric Eigenvalues 📋

📊 Complex Plane (All Eigenvalues)

Characteristic Polynomial

Adjacency Matrices

Directed (Skew-Symmetric) 📋

Undirected (Symmetric) 📋

📐 Point Graph Topology

Physical Parameters

Eigenvalue Bounds (Table I)

Stability Analysis

🔬 Compare with Numerical

⊕ Direct Sum

📖 Reference (Table I & Appendix)

🔗 Graph Products

📐 Parameterized Families

🔍 Find Analytic Graphs

📖 Product Eigenvalue Formulas

📚 Analytic Graph Library

🚀 Universe Navigation

📷 View Projection

📤 Actions

⚠️ Danger Zone

Skew-Symmetric Eigenvalues

Symmetric Eigenvalues

Directed (Skew-Symmetric)

Undirected (Symmetric)