Inverse problems in the class of Q-polynomial graphs

I. N. Belousov ; A. A. Makhnev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 14-22

Résumé

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.

Détail

Keywords: distance-regular graph, graph $\Gamma$ with a strongly regular graph $\Gamma_3$.
Mots-clés : $Q$-polynomial graph