Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


In the present paper we suggest an approach for counting Jacobian group of the Δ-graph Δ(n; k, l, m). The notion of Δ-graph arises as a continuation of the families of I-, Y- and H-graphs well-known in the graph theory. In particular, graph Δ(n; 1, 1, 1) is isomorphic to discrete torus C3xCn. It this case, the structure of the Jacobian group will be find explicitly.

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1. M. Baker, S. Norine, Harmonic morphisms and hyperelliptic graphs, Int. Math. Res. Notes. 15 (2009), 2914–2955. Zbl 1178.05031

2. N.L. Biggs, Three remarkable graphs, Canad. J. Math., 25 (1973), 397–411. Zbl 0256.05114

3. N.L. Biggs, Chip-firing and the critical group of a graph, J. Algebraic Comb., 9:1 (1999), 25–45. Zbl 0919.05027

4. R. Bacher, P. de la Harpe, T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France., 125 (1997), 167–198. Zbl 0891.05062

5. F.T. Boesch, H. Prodinger, Spanning tree formulas and Chebyshev polynomials, Graphs Combin., 2:1 (1986), 191–200. Zbl 0651.05028

6. R. Cori, D. Rossin, On the sandpile group of dual graphs, European J. Combin., 21:4 (2000), 447–459. Zbl 0969.05034

7. P.J. Davis, Circulant Matrices, AMS Chelsea Publishing, 1994. Zbl 0418.15017

8. J.D. Horton, I.Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory. Ser. B, 53 (1991), 114–129. Zbl 0689.05046

9. Y.S. Kwon, A.D. Mednykh, I.A. Mednykh, On Jacobian group and complexity of the generalized Petersen graph GP(n,k) through Chebyshev polynomials, Linear Algebra Appl., 529 (2017), 355–373. Zbl 1365.05135

10. Y.S. Kwon, A.D. Mednykh, I.A. Mednykh, Complexity of the circulant foliation over a graph, J. Algebraic Comb., 53 (2021), 115–129. Zbl 1464.05194

11. D. Lorenzini, Smith normal form and Laplacians, J. Combin. Theory Ser. B., 98:6 (2008), 1271–1300. Zbl 1175.05088

12. I.A. Mednykh, M.A. Zindinova, On the structure of Picard group for Moebius ladder, Sib. Electron. Math. Rep., 8 (2011), 54–61. Zbl 1329.05147

13. A.D. Mednykh, I.A. Mednykh, On the structure of the Jacobian group for circulant graphs, Dokl. Math., 94:1 (2016), 445–449. Zbl 1350.05061

14. I.A. Mednykh, On Jacobian group and complexity of the I-graph I(n,k,l) through Chebyshev polynomials, Ars Math. Contemp., 15 (2018), 467–485. Zbl 1411.05126

15. A.D. Mednykh, I.A. Mednykh,Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs, Dokl. Math., 97:2 (2018), 147–151. Zbl 1391.05150

16. A. Mednykh, I. Mednykh, The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic, Discrete Math., 342:6 (2019), 1772–1781. Zbl 1414.05080

17. A.D. Mednykh, I.A. Mednykh, On the structure of the critical group of a circulant graph, Russian Mathematical Surveys, 75:1 (2020), 190–192.

18. S.D. Nikolopoulos, C. Papadopoulos, The number of spanning trees in Kn-complements of quasi-threshold graphs, Graph Combin., 20 (2004), 383–397. Zbl 1054.05058

19. R. Shrock, F.Y. Wu, Spanning trees on graphs and lattices in d-dimensions, J. Phys. A, 33 (2000), 3881–3902. Zbl 0949.05041

20. W. Sun, S. Wang, J. Zhang, Counting spanning trees in prism and anti-prism graphs, J. Appl. Anal. Comput., 6 (2016), 65–75. Zbl 1463.05287

21. Yaoping Hou, Chingwah Woo, Pingge Chen, On the sandpile group of the square cycle, Linear Algebra Appl., 418 (2006), 457–467. Zbl 1108.05062

22. Chen Pingge, Hou Yaoping, On the critical group of the Mobius ladder graph, Austral. J. Combin., 36 (2006), 133–142. Zbl 1104.05032

23. I.N. Yudin, On spectra of poly-circulant graphs, Magister thesis, Gorno-Altaisk State University, Gorno-Altaisk, 2022.

24. Zhang Yuanping, Yong Xuerong, M.J. Golin, The number of spanning trees in circulant graphs, Discrete. Math., 223:1 (2000), 337–350. Zbl 0969.05036

25. Zhang Yuanping, Xuerong Yong, M.J. Golin, Chebyshev polynomials and spanning tree formulas for circulant and related graphs, Discrete Math., 298:1 (2005), 334–364. Zbl 1070.05029

26. Chen Xiebin, Qiuying Lin, Fuji Zhang, The number of spanning trees in odd valent circulant graphs, Discrete Math., 282:1 (2004), 69–79. Zbl 1042.05051



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