Integral Graph History Survey 2002: The Overlooked Clues
- 01. What Is the Integral Graph History Survey 2002?
- 02. Core Findings from the Survey
- 03. Key Historical Milestones in Integral Graph Research
- 04. Integral Graph Classification Data (2002 Survey)
- 05. Why This Survey Matters for Mathematics Education
- 06. FAQ: Integral Graph History Survey 2002
- 07. Connections to Marist Educational Values
What Is the Integral Graph History Survey 2002?
The 2002 Integral Graph Survey is a definitive mathematical reference compiling all known results on integral graphs-graphs whose adjacency matrix eigenvalues are all integers. Published by researchers from Serbia and Poland, it answered the 1973 question posed by F. Harary and A.J. Schwenk about which graphs have integral spectra.
Core Findings from the Survey
- Exactly 13 connected cubic (3-regular) integral graphs exist, including the Petersen graph and K₄
- Exactly 150 connected integral graphs exist up to 10 vertices: 1 (n=1), 1 (n=2), 1 (n=3), 2 (n=4), 3 (n=5), 6 (n=6), 7 (n=7), 22 (n=8), 24 (n=9), 83 (n=10)
- Exactly 13 non-regular, non-bipartite integral graphs with maximum degree 4 were identified
- No balanced integral tree of diameter 7 or diameter 4k+1 exists; diameter 6 trees require perfect square parameters
Key Historical Milestones in Integral Graph Research
- 1973: Harary & Schwenk pose the "which graphs have integral spectra?" question, noting the general problem appears intractable
- 1976: Bussemaker & Cvetković publish "There are exactly 13 connected, cubic, integral graphs"
- 1978: R.L. Graham solves the double-star integral tree problem
- 1986: Radosavljević & Simić identify 13 non-regular non-bipartite integral graphs with max degree 4
- 2002: Balińska et al. publish the comprehensive survey finalizing research on integral graphs up to 12 vertices (325 graphs at n=12)
Integral Graph Classification Data (2002 Survey)
| Category | Count | Key Examples |
|---|---|---|
| Connected cubic integral graphs | 13 | Petersen graph, K₄, K₃,₃ |
| Integral graphs (n≤10) | 150 | K₁, K₂, C₄, C₆, stars K₁,ₚ² |
| Integral graphs (n=11) | 236 | Various non-regular structures |
| Integral graphs (n=12) | 325 | Including 4-regular cases |
| Non-regular, max degree 4, non-bipartite | 13 | Exceptional graphs from E₈ root system |
| 4-regular bipartite avoiding ±3 | 16 | D₁=K₄,₄ (smallest), D₁₁/D₁₆ (largest, 30-32 vertices) |
Why This Survey Matters for Mathematics Education
The survey's proof techniques-spectral moments, Diophantine equations, graph angles, NEPS operations-provide a model for rigorous mathematical inquiry applicable to Catholic educational pedagogy emphasizing truth, logic, and intellectual virtue. Marist schools in Brazil and Latin America can integrate this content into advanced mathematics curricula to demonstrate how educational rigor blends with spiritual mission through disciplined intellectual work.
FAQ: Integral Graph History Survey 2002
Connections to Marist Educational Values
This mathematical survey exemplifies the Marist emphasis on presence, simplicity, and family spirit in collaborative research: five scholars from two countries united by a single purpose-clarifying an intractable problem through systematic investigation. The survey's evidence-based analysis and primary source fidelity align with Marist pedagogy's commitment to intellectual honesty and measurable impact in education across Latin America.
Key concerns and solutions for Integral Graph History Survey 2002 The Overlooked Clues
What is an integral graph?
An integral graph is a graph whose spectrum (eigenvalues of its adjacency matrix) consists entirely of integers. The characteristic polynomial is monic, so rational eigenvalues must be integers.
Who authored the 2002 integral graph survey?
The survey was authored by K. Balińska, D. Cvetković, Z. Radosavljević, S. Simić, and D. Stevanović, published at University of Belgrade's Faculty of Electrical Engineering in 2002.
How many integral graphs exist up to 12 vertices?
There are 325 connected integral graphs on 12 vertices, 236 on 11 vertices, and 83 on 10 vertices, totaling 150 integral graphs up to 10 vertices.
Are there infinitely many integral trees?
Yes. For balanced integral trees, Theorem 5 states that if sequence (nₖ,...,n₁) is integral, then (q²nₖ,...,q²n₁) is also integral for any q∈ℕ, producing infinite families.
What makes cubic integral graphs special?
There are exactly 13 connected cubic integral graphs-a finite, completely classified set. This contrasts with general integral graphs, which are infinite but rare.
