Geodesic
On a class of vertex-primitive arc-transitive amply regular graphs
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 258-268
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A simple $k$-regular graph with $v$ vertices is an amply regular graph with parameters $(v, k, \lambda, \mu)$ if any two adjacent vertices have exactly $\lambda$ common neighbors and any two vertices which are at distance $2$ in this graph have exactly $\mu$ common neighbors. Let $G$ be a finite group, $H \le G$, ${\mathfrak{H}} = \{H^g \,|\, g \in G \}$ be the corresponding conjugacy class of subgroups of $G$, and $1 \le d $ be an integer. We construct a simple graph $\Gamma(G, H, d)$ as follows. The vertices of $\Gamma(G, H, d)$ are the elements of ${\mathfrak{H}}$, and two vertices $H_1$ and $H_2$ from ${\mathfrak{H}}$ are adjacent in $\Gamma(G, H, d)$ if and only if $|H_1 \cap H_2| = d$. In this paper we prove that if $q$ is a prime power with $13 \le q \equiv 1 \pmod{4}$, $G=SL_2(q)$, and $H$ is a dihedral maximal subgroup of $G$ of order $2(q-1)$, then the graph $\Gamma(G, H, 8)$ is a vertex-primitive arc-transitive amply regular graph with parameters $\left(\dfrac{q(q+1)}{2}, \dfrac{q-1}{2}, 1, 1\right)$ and with ${\rm Aut}(PSL_2(q))\le {\rm Aut}(\Gamma)$. Moreover, we prove that $\Gamma(G, H, 8)$ has a perfect $1$-code, in particular, its diameter is more than $2$.
Keywords: finite simple group; arc-transitive graph; amply regular graph; edge-regular graph; graph of girth 3; Deza graph; perfect 1-code. 
@article{TIMM_2022_28_2_a19,
     author = {M. P. Golubyatnikov and N. V. Maslova},
     title = {On a class of vertex-primitive arc-transitive amply regular graphs},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {258--268},
     year = {2022},
     volume = {28},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a19/}
}
TY  - JOUR
AU  - M. P. Golubyatnikov
AU  - N. V. Maslova
TI  - On a class of vertex-primitive arc-transitive amply regular graphs
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2022
SP  - 258
EP  - 268
VL  - 28
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a19/
LA  - en
ID  - TIMM_2022_28_2_a19
ER  - 
%0 Journal Article
%A M. P. Golubyatnikov
%A N. V. Maslova
%T On a class of vertex-primitive arc-transitive amply regular graphs
%J Trudy Instituta matematiki i mehaniki
%D 2022
%P 258-268
%V 28
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a19/
%G en
%F TIMM_2022_28_2_a19
M. P. Golubyatnikov; N. V. Maslova. On a class of vertex-primitive arc-transitive amply regular graphs. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 258-268. http://geodesic.mathdoc.fr/item/TIMM_2022_28_2_a19/

[1] Akers S.B., Harel D., and Krishnamurthy B., “The star graph: an attractive alternative to the $n$-cube”, Proc. Int. Conf. Parallel Processing, 1987, 393–400

[2] Bray J.N., Holt D.F., Roney-Dougal C.M., The maximal subgroups of the low-dimensional finite classical groups, Cambridge Univ. Press, Cambridge, 2013, 438 pp. | DOI | MR | Zbl

[3] Brouwer A.E., Cohen A.M., Neumaier A., Distance-regular graphs, Springer-Verlag, Berlin etc, 1989, 495 pp. | MR | Zbl

[4] GAP - Groups, Algorithms, Programming - a system for computational discrete algebra, Version 4.10.1, 2019 URL: https://www.gap-system.org

[5] Godsil C., Royle G., Algebraic graph theory, Springer, NY, 2001, 443 pp. | DOI | MR | Zbl

[6] Koblitz N., “Finite fields and quadratic residues”, A course in number theory and cryptography, Graduate Texts in Mathematics, 114, Springer, NY, 1994, 31–53 | DOI | MR

[7] Lakshmivarahan S., Jwo J.-S., Dhall S.K., “Symmetry in interconnection networks based on Cayley graphs of permutation groups: A survey”, Parallel Comput., 19 (1993), 361–407 | DOI | MR | Zbl

[8] Lovasz L., “Combinatorial structures and their applications”, Proc. Calgary Internat. Conf. Calgary (Alberta, 1969), Gordon and Breach, NY, 1970, 243–246

[9] Marusic D. and Scapellato R., “A class of non-Cayley vertex-transitive graphs associated with $PSL(2,p)$”, Discrete Math., 109:1–3 (1992), 161–170 | DOI | MR | Zbl

[10] Maslova N.V., “Classification of maximal subgroups of odd index in finite simple classical groups: Addendum”, Siberian Electron. Math. Reports, 15 (2018), 707–718 | DOI | MR | Zbl

[11] Mulder M., “(0,$\lambda$)-graphs and $n$-cubes”, Discrete Math., 28:2 (1979), 179–188 | DOI | MR | Zbl

[12] Ray B.N.B., “Parallel algorithm for Hermite interpolation on the star graph”, International J. Advanced Research in Comp. Sci. and Soft. Engineering, 5:5 (2015), 1019–1026