On distance-regular graphs with c2 = 2
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Alexandr A. Makhnev
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Marina S. Nirova
Abstract
Let Γ be a distance-regular graph of diameter 3 with c2 = 2 (any two vertices with distance 2 between them have exactly two common neighbors). Then the neighborhood Δ of the vertex w in Γ is a partial line space. In view of the Brouwer–Neumaier result either Δ is the union of isolated (λ + 1)-cliques or the degrees of vertices k ≥ λ(λ + 3)/2, and in the case of equality k = 5, λ = 2 and Γ is the icosahedron graph. A. A. Makhnev, M. P. Golubyatnikov and Wenbin Guo have investigated distance-regular graphs Γ of diameter 3 such that Γ3 is the pseudo-geometrical network graph. They have found a new infinite set {2u2 −2m2 + 4m − 3,2u2 −2m2,u2 − m2 + 4m −2;1, 2, u2 − m2} of feasible intersection arrays for such graphs with c2 = 2. Here we prove that some distance-regular graphs from this set do not exist. It is proved also that distance-regular graph with intersection array {22, 16, 5; 1, 2, 20} does not exist.
Note: Originally published in Diskretnaya Matematika (2020) 32,№1, 74–80 (in Russian).
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Funding: The study was supported by the project № 18-1-1-17 of the Ural branch of RAS and by the agreement between Ministry of Education and Science of the Russian Federation and Ural Federal university of 27.08.2013, № 02.A03.21.0006.
References
[1] Brouwer A.E., Cohen A.M., Neumaier A., Distance-Regular Graphs, Springer-Verlag, Berlin Heidelberg New York, 1989.10.1007/978-3-642-74341-2Suche in Google Scholar
[2] Gavrilyuk A.L., Makhnev A.A., “On automorphisms of a distance-regular graph with intersection array {56,45,1; 1,9,56}”, Doklady Mathematics, 81:3 (2010), 439–442, https://doi.org/10.1134/S1064562410030282.Suche in Google Scholar
[3] Brouwer A. E., Neumaier A., “A remark on partial linear spaces with girth 5 with an application to strongly regular graphs”, Combinatorica, 8 (1998), 57–61.10.1007/BF02122552Suche in Google Scholar
[4] Brouwer A. E., Sumaloj S., Worawannotai C., “The nonexistence of distance-regular graphs with intersection arrays {27, 20, 10; 1, 2, 18} and {36,28,4; 1, 2, 24}”, Australasian J. Comb., 66:2 (2016), 330–332.Suche in Google Scholar
[5] Makhnev A. A., Golubyatnikov M. P., “A Shilla graph with intersection array {12,10,2; 1,2,8} does not exist”, Math. Notes, 106:5 (2019), 850–853.10.1134/S0001434619110191Suche in Google Scholar
[6] Belousova V. I., Makhnev A. A., “Automorphisms of distance regular graph with intersection array {30,27,24; 1,2,10}”, Siberian Elect. Math. Report, 16 (2019), 493–500 (in Russian).10.33048/semi.2019.16.031Suche in Google Scholar
[7] Makhnev A. A., Khamgokova M. M., “On automorphisms of a distance-regular graph with intersections array {39,36,22; 1,2,18}”, Siberian Elect. Math. Report, 16 (2019), 638–647 (in Russain).10.33048/semi.2019.16.041Suche in Google Scholar
[8] Makhnev A. A., Tokbaeva A. A., “On a distance-regular graph with intersection array {35,28,6; 1,2,30}”, Vladikavkaz. Mat. Zh., 21:2 (2019), 27–37 (in Russian).10.33048/semi.2019.16.031Suche in Google Scholar
[9] Makhnev A. A., Golubyatnikov M. P., Guo Wenbin, “Inverse problems in graph theory: nets”, Comm. Math. Statist., 7:1 (2019), 69–83.10.1007/s40304-018-0159-4Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A method of construction of differentially 4-uniform permutations over Vm for even m
- Ergodicity of the probabilistic converter, a serial connection of two automata
- On distance-regular graphs with c2 = 2
- Minimal contact circuits for characteristic functions of spheres
- Approximation of restrictions of q-valued logic functions to linear manifolds by affine analogues
- Multiaffine polynomials over a finite field
- Large deviations of branching process in a random environment. II
Artikel in diesem Heft
- Frontmatter
- A method of construction of differentially 4-uniform permutations over Vm for even m
- Ergodicity of the probabilistic converter, a serial connection of two automata
- On distance-regular graphs with c2 = 2
- Minimal contact circuits for characteristic functions of spheres
- Approximation of restrictions of q-valued logic functions to linear manifolds by affine analogues
- Multiaffine polynomials over a finite field
- Large deviations of branching process in a random environment. II
