On $Q$-polynomial distance-regular graphs $\Gamma$ with strongly regular graphs $\Gamma_2$ and $\Gamma_3$

I. N. Belousov; A. A. Makhnev; M. S. Nirova

Geodesic

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On $Q$-polynomial distance-regular graphs $\Gamma$ with strongly regular graphs $\Gamma_2$ and $\Gamma_3$

I. N. Belousov ; A. A. Makhnev ; M. S. Nirova

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1385-1392

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Résumé

Let $\Gamma$ be a distance-regular graph of diameter 3 with strongly regular graphs $\Gamma_2$ and $\Gamma_3$. Then $\Gamma$ has intersection array $\{t(c_2+1)+a_3,tc_2,a_3+1;1,c_2,t(c_2+1)\}$ (Nirova M.S.) If $\Gamma$ is $Q$-polynomial then either $a_3=0,t=1$ and $\Gamma$ is Taylor graph or $(c_2+1)=a_3(a_3+1)/(t^2-a_3-1)$. We found 4 infinite series feasible intersection arrays in this situation.

Keywords: distance-regular graph
Mots-clés : $Q$-polynomial graph.

@article{SEMR_2019_16_a72,
     author = {I. N. Belousov and A. A. Makhnev and M. S. Nirova},
     title = {On $Q$-polynomial distance-regular graphs $\Gamma$ with strongly regular graphs $\Gamma_2$ and $\Gamma_3$},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1385--1392},
     publisher = {mathdoc},
     volume = {16},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a72/}
}

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I. N. Belousov; A. A. Makhnev; M. S. Nirova. On $Q$-polynomial distance-regular graphs $\Gamma$ with strongly regular graphs $\Gamma_2$ and $\Gamma_3$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1385-1392. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a72/