Geodesic
On Cayley representations of finite graphs over Abelian $ p$-groups
Algebra i analiz, Tome 32 (2020) no. 1, pp. 94-120.

Voir la notice de l'article provenant de la source Math-Net.Ru

A polynomial-time algorithm is constructed that, given a graph $ \Gamma $, finds the full set of nonequivalent Cayley representations of $ \Gamma $ over the group $ D\cong C_p\times C_{p^k}$, where $ p\in \{2,3\}$ and $ k\geq 1$. This result implies that the recognition and isomorphism problems for Cayley graphs over $ D$ can be solved in polynomial time.
Keywords: coherent configurations, Cayley graphs, Cayley graph isomorphism problem. 
@article{AA_2020_32_1_a5,
     author = {G. K. Ryabov},
     title = {On {Cayley} representations of finite graphs over {Abelian} $ p$-groups},
     journal = {Algebra i analiz},
     pages = {94--120},
     publisher = {mathdoc},
     volume = {32},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2020_32_1_a5/}
}
TY  - JOUR
AU  - G. K. Ryabov
TI  - On Cayley representations of finite graphs over Abelian $ p$-groups
JO  - Algebra i analiz
PY  - 2020
SP  - 94
EP  - 120
VL  - 32
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2020_32_1_a5/
LA  - ru
ID  - AA_2020_32_1_a5
ER  - 
%0 Journal Article
%A G. K. Ryabov
%T On Cayley representations of finite graphs over Abelian $ p$-groups
%J Algebra i analiz
%D 2020
%P 94-120
%V 32
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2020_32_1_a5/
%G ru
%F AA_2020_32_1_a5
G. K. Ryabov. On Cayley representations of finite graphs over Abelian $ p$-groups. Algebra i analiz, Tome 32 (2020) no. 1, pp. 94-120. http://geodesic.mathdoc.fr/item/AA_2020_32_1_a5/

[1] Veisfeiler B., Leman A., “Privedenie grafa k kanonicheskomu vidu i voznikayuschaya pri etom algebra”, NTI. Ser. 2. Inform. analiz, 1968, no. 9, 12–16

[2] Evdokimov S., Ponomarenko I., “Raspoznavanie i proverka izomorfizma tsirkulyantnykh grafov za polinomialnoe vremya”, Algebra i analiz, 15:6 (2003), 1–34

[3] Evdokimov S., Ponomarenko I., “Shurovost $S$-kolets nad tsiklicheskoi gruppoi i obobschennoe spletenie grupp perestanovok”, Algebra i analiz, 24:3 (2012), 84–127

[4] Muzychuk M., Ponomarenko I., “O $2$-gruppakh Shura”, Zap. nauch. semin. POMI, 435, 2015, 113–162

[5] Ryabov G., “Ob otdelimosti kolets Shura nad abelevymi $p$-gruppami”, Algebra i logika, 57:1 (2018), 73–101 | MR | Zbl

[6] Babai L., “Isomorphism problem for a class of point symmetric structures”, Acta Math. Acad. Sci. Hungar., 29:3-4 (1977), 329–336 | DOI | MR | Zbl

[7] Chen G., Ponomarenko I., Lectures on coherent configurations, 2019 http://www.pdmi.ras.ru/ĩnp/ccNOTES.pdf

[8] Evdokimov S., Ponomarenko I., “Permutation group approach to association schemes”, European J. Combin., 30:6 (2009), 1456–1476 | DOI | MR | Zbl

[9] Evdokimov S., Muzychuk M., Ponomarenko I., “A family of permutation groups with exponentially many non-conjugated regular elementary abelian subgroups”, Algebra i analiz, 29:4 (2017), 46–53 | MR

[10] Klin M., Pech C., Reichard S., COCO2P — a GAP package, 0.14, 07.02.2015 http://www.math.tu-dresden.de/p̃ech/COCO2P

[11] Hanaki A., Miyamoto I., Classification of association schemes with small number of vertices, 2016 http://math.shinshu-u.ac.jp/h̃anaki/as/

[12] Muzychuk M., “On the isomorphism problem for cyclic combinatorial objects”, Discrete Math., 197/198 (1999), 589–606 | DOI | MR | Zbl

[13] Muzychuk M., “A solution of the isomorphism problem for circulant graphs”, Proc. Lond. Math. Soc. (3), 88:1 (2004), 1–41 | DOI | MR | Zbl

[14] Nedela R., Ponomarenko I., Recognizing and testing isomorphism of Cayley graphs over an abelian group of order $4p$ in polynomial time, 2017, 22 pp., arXiv: 1706.06145 [math.CO] | MR

[15] Ryabov G., “On Schur $p$-groups of odd order”, J. Algebra Appl., 16:3 (2017), 1750045 | DOI | MR | Zbl

[16] Seress A., Permutation group algorithms, Cambridge Tracts Math., 152, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl

[17] Weisfeiler B. (ed.), On construction and identification of graphs, Lecture Notes in Math., 558, Springer-Verlag, Berlin etc., 1976 | DOI | MR | Zbl