Geodesic
Inverse problems in the class of Q-polynomial graphs
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 14-22 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In the class of distance-regular graphs $\Gamma$ of diameter 3 with a pseudogeometric graph $\Gamma_3$, feasible intersection arrays for the partial geometry were found for networks by Makhnev, Golubyatnikov, and Guo; for dual networks by Belousov and Makhnev; and for generalized quadrangles by Makhnev and Nirova. These authors obtained four infinite series of feasible intersection arrays of distance-regular graphs: $$\big\{c_2(u^2-m^2)+2c_2m-c_2-1,c_2(u^2-m^2),\ (c_2-1)(u^2-m^2)+2c_2m-c_2;1,c_2,u^2-m^2\big\},$$ $$\{mt,(t+1)(m-1),t+1;1,1,(m-1)t\}\ \ \text{for}\ \ m\le t,$$ $$\{lt,(t-1)(l-1),t+1;1,t-1,(l-1)t\},\ \ \text{and}\ \ \{a(p+1),ap,a+1;1,a,ap\}.$$ We find all feasible intersection arrays of $Q$-polynomial graphs from these series. In particular, we show that, among these infinite families of feasible arrays, only two arrays ($\{7,6,5;1,2,3\}$ (folded 7-cube) and $\{191,156,153;1,4,39\}$) correspond to $Q$-polynomial graphs.
Keywords: distance-regular graph, graph $\Gamma$ with a strongly regular graph $\Gamma_3$.
Mots-clés : $Q$-polynomial graph 
@article{TIMM_2020_26_3_a1,
     author = {I. N. Belousov and A. A. Makhnev},
     title = {Inverse problems in the class of {Q-polynomial} graphs},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {14--22},
     year = {2020},
     volume = {26},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a1/}
}
TY  - JOUR
AU  - I. N. Belousov
AU  - A. A. Makhnev
TI  - Inverse problems in the class of Q-polynomial graphs
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2020
SP  - 14
EP  - 22
VL  - 26
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a1/
LA  - ru
ID  - TIMM_2020_26_3_a1
ER  - 
%0 Journal Article
%A I. N. Belousov
%A A. A. Makhnev
%T Inverse problems in the class of Q-polynomial graphs
%J Trudy Instituta matematiki i mehaniki
%D 2020
%P 14-22
%V 26
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a1/
%G ru
%F TIMM_2020_26_3_a1
I. N. Belousov; A. A. Makhnev. Inverse problems in the class of Q-polynomial graphs. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 14-22. http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a1/

[1] Bang S., Koolen J., “Distance-regular graphs of diameter 3 having eigenvalue -1”, Linear Algebra and Appl., 531 (2017), 38–53 | DOI | MR | Zbl

[2] Iqbal Q., Koolen J., Park J., Rehman M., “Distance-regular graphs with diameter 3 and eigenvalue $a_2-c_3$”, Linear Algebra and Appl., 587 (2020), 271–290 | DOI | MR | Zbl

[3] Makhnev A.A., Golubyatnikov M.P., Guo Wenbin, “Inverse problems in distance-regular graphs: nets”, Communications in Mathematics and Statistics, 7:1 (2019), 69–83 | DOI | MR | Zbl

[4] Makhnev A.A., Nirova M.S., “Inverse problems in distance-regular graphs: generalized quadrangles”, Sibirean Electr. Math. Reports, 15 (2018), 927–934 | DOI | MR | Zbl

[5] Brouwer A.E., Cohen A.M., Neumaier A., Distance-Regular Graphs, Springer-Verlag, Berlin; Heidelberg; N Y, 1989, 495 pp. | MR | Zbl

[6] Terwilliger P., “A new unequality for distance-regular graphs”, Discrete Math., 137 (1995), 319–332 | DOI | MR | Zbl

[7] Jurisic A., Vidali J., “Extremal 1-codes in distance-regular graphs of diameter 3”, Des. Codes Cryptogr., 65 (2012), 29–47 | DOI | MR | Zbl