On bipartite ▫$Q$▫-polynomial distance-regular graphs
Miklavič, Štefko
Naj bo ▫$\Gamma$▫ dvodelen ▫$Q$▫-polinomski razdaljno-regularen graf z množico vozlišč ▫$X$▫, premerom ▫$d \ge 3$▫ in stopnjo ▫$k \ge 3$▫. Naj bo ▫${\mathbb{R}}^X$▫ vektorski prostor nad ...▫${\mathbb{R}}$▫, ki ga sestavljajo vsi stolpčni vektorji z realnimi komponentami, katerih vrstice so indeksirane z ▫$X$▫. Za vsak ▫$z \in X$▫ naj bo ▫$\hat{z}$▫ vektor iz ▫${\mathbb{R}}^X$▫, ki ima ▫$z$▫-koordinato enako 1, vse ostale koordinate pa enake 0. Za poljubni vozlišči ▫$x,y \in X$▫ z ▫$\partial(x,y)$▫ označimo njuno medsebojno razdaljo. Naj bosta ▫$x,y \in X$▫ taki vozlišči grafa ▫$\Gamma$▫, za katera je ▫$\partial(x,y) = 2$▫. Za ▫$0 \le i,j\le d$▫ definirajmo ▫$w_{ij} = \sum \hat{z}$▫, kjer seštevamo po vseh ▫$z \in X$▫, za katere je ▫$\partial(x,z) = i$▫ in ▫$\partial(y,z) = j$▫. Definirajmo ▫$W = \textrm{span} \{w_{ij} \vert 0 \le i,j \le d\}$▫. V tem članku obravnavamo vektorski prostor ▫$MW = \textrm{span}\{mw \vert m \in M, w \in W\}$▫, kjer je ▫$M$▫ Bose-Mesnerjeva algebra grafa ▫$\Gamma$▫. Hitro lahko vidimo, da je ▫$MW$▫ minimalen ▫$A$▫-invarianten podprostor v ▫${\mathbb{R}}^X$▫, ki vsebuje ▫$W$▫, kjer je ▫$A$▫ matrika sosednosti grafa ▫$\Gamma$▫. Poiščemo bazo podprostora ▫$MW$▫, ki je ortogonalna glede na običajni skalarni produkt. Podamo delovanje matrike ▫$A$▫ na tej bazi. Pokažemo, da je dimenzija podprostora ▫$MW$▫ enaka ▫$3d-3$▫, če je graf ▫$\Gamma$▫ 2-homogen, ▫$3d-1$▫, če je graf ▫$\Gamma$▫ antipodni kvocient ▫$2d$▫-kocke, ter ▫$4d-4$▫ sicer. Rezultate dokažemo s pomočjo Terwilligerjeve karakterizacije ▫$Q$▫-polinomskih razdaljno-regularnih grafov z uravnoteženimi množicami.
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