Povzetek
Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular graph with vertex set ▫$X$▫, diameter ▫$d \ge 3$▫ and valency ▫$k \ge 3$▫. Let ▫${\mathbb{R}}^X$▫ denote the vector space over ▫$\mathbb{R}$▫ consisting of column vectors with entries in ▫$\mathbb{r}$▫ and rows indexed by ▫$X$▫. For ▫$z \in X$▫, let ▫$\hat{z}$▫ denote the vector in ▫${\mathbb{R}}^X$▫ with a 1 in the ▫$z$▫-coordinate, and 0 in all other coordinates. Fix ▫$x,y \in X$▫ such that ▫$\partial(x,y)=2▫, where ▫$\partial$▫ denotes the path-length distance. For ▫$0 \le i,j \le d$▫ define ▫$w_{ij} = \sum\hat{z}$▫, where the sum is over all ▫$z \in X$▫ such that ▫$\partial(x,z) = i$▫ and ▫$\partial(y,z) = j▫$. We define ▫$W = \textrm{span} \{w_{ij}|0 \le i,j \le d\}$▫. In this paper we consider the space ▫$MW = \textrm{span} \{mw |m \in M, w \in W \l\}$▫, where ▫$M$▫ is the Bose-Mesner algebra of ▫$\Gamma$▫. We observe that ▫$MW$▫ is the minimal ▫$A$▫-invariant subspace of ▫${\mathbb{R}}^X$▫ which contains ▫$W$▫, where ▫$A$▫ is the adjacency matrix of ▫$\Gamma$▫. We display a basis for ▫$MW$▫ that is orthogonal with respect to the dot product. We give the action of ▫$A$▫ on this basis. We show that the dimension of ▫$MW$▫ is ▫$3d-3$▫ if ▫$\Gamma$▫ is 2-homogeneous, ▫$3d-1$▫ if ▫$\Gamma$▫ is the antipodal quotient of the ▫$2d$▫-cube, and ▫$4d-4$▫ otherwise. We obtain our main result using Terwilliger's "balanced set" characterization of the ▫$Q$▫-polynomial property.
Ključne besede
matematika;teorija grafov;razdaljno regularni grafi;▫$Q$▫-polinomska lastnost;Bose-Mesnerjeva algebra;mathematics;graph theory;distance-regular graphs;▫$Q$▫-polynominal property;Bose-Mesner algebra;balanced set characterization of the Q-polynominal property;
Podatki
Jezik: |
Angleški jezik |
Leto izida: |
2007 |
Tipologija: |
1.01 - Izvirni znanstveni članek |
Organizacija: |
UP - Univerza na Primorskem |
UDK: |
519.17 |
COBISS: |
1796823
|
ISSN: |
0195-6698 |
Št. ogledov: |
3459 |
Št. prenosov: |
28 |
Ocena: |
0 (0 glasov) |
Metapodatki: |
|
Ostali podatki
Sekundarni jezik: |
Angleški jezik |
Sekundarne ključne besede: |
matematika;teorija grafov;razdaljno regularni grafi;▫$Q$▫-polinomska lastnost;Bose-Mesnerjeva algebra; |
Vrsta dela (COBISS): |
Delo ni kategorizirano |
Strani: |
str. 94-110 |
Letnik: |
ǂVol. ǂ28 |
Zvezek: |
ǂno. ǂ1 |
Čas izdaje: |
2007 |
ID: |
1471768 |