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RUP
FAMNIT - Faculty of Mathematics, Science and Information Technologies
FHŠ - Faculty of Humanities
FM - Faculty of Management
FTŠ Turistica - Turistica – College of Tourism Portorož
FVZ - Faculty of Health Sciences
IAM - Andrej Marušič Institute
PEF - Faculty of Education
UPR - University of Primorska
ZUP - University of Primorska Press
COBISS
University of Primorska, University Library - all departments
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Title:
On bipartite Q-polynominal distance-regular graphs
Authors:
ID
Miklavič, Štefko
(Author)
Files:
http://dx.doi.org/10.1016/j.ejc.2005.09.003
Language:
English
Work type:
Not categorized
Typology:
1.01 - Original Scientific Article
Organization:
IAM - Andrej Marušič Institute
Abstract:
Let
Γ
denote a bipartite
Q
-polynomial distance-regular graph with vertex set
X
, diameter
d
≥
3
and valency
k
≥
3
. Let
R
X
denote the vector space over
R
consisting of column vectors with entries in
r
and rows indexed by
X
. For
z
∈
X
, let
ˆ
z
denote the vector in
R
X
with a 1 in the
z
-coordinate, and 0 in all other coordinates. Fix
x
,
y
∈
X
such that
∂
(
x
,
y
)
=
2
▫
,
w
h
e
r
e
▫
$
∂
denotes the path-length distance. For
0
≤
i
,
j
≤
d
define
w
i
j
=
∑
ˆ
z
, where the sum is over all
z
∈
X
such that
∂
(
x
,
z
)
=
i
and
∂
(
y
,
z
)
=
j
▫
$
.
W
e
d
e
f
i
n
e
▫
$
W
=
span
{
w
i
j
|
0
≤
i
,
j
≤
d
}
. In this paper we consider the space
M
W
=
span
{
m
w
|
m
∈
M
,
w
∈
W
\l
}
, where
M
is the Bose-Mesner algebra of
Γ
. We observe that
M
W
is the minimal
A
-invariant subspace of
R
X
which contains
W
, where
A
is the adjacency matrix of
Γ
. We display a basis for
M
W
that is orthogonal with respect to the dot product. We give the action of
A
on this basis. We show that the dimension of
M
W
is
3
d
−
3
if
Γ
is 2-homogeneous,
3
d
−
1
if
Γ
is the antipodal quotient of the
2
d
-cube, and
4
d
−
4
otherwise. We obtain our main result using Terwilliger's "balanced set" characterization of the
Q
-polynomial property.
Keywords:
mathematics
,
graph theory
,
distance-regular graphs
,
Q
-polynominal property
,
Bose-Mesner algebra
,
balanced set characterization of the Q-polynominal property
Year of publishing:
2007
Number of pages:
str. 94-110
Numbering:
Vol. 28, no. 1
PID:
20.500.12556/RUP-3312
ISSN:
0195-6698
UDC:
519.17
COBISS.SI-ID:
1796823
Publication date in RUP:
15.10.2013
Views:
5024
Downloads:
29
Metadata:
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Vancouver
:
MIKLAVIČ, Štefko, 2007, On bipartite Q-polynominal distance-regular graphs. [online]. 2007. Vol. 28, no. 1, p. 94–110. [Accessed 19 March 2025]. Retrieved from: http://dx.doi.org/10.1016/j.ejc.2005.09.003
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Secondary language
Language:
English
Keywords:
matematika
,
teorija grafov
,
razdaljno regularni grafi
,
Q
-polinomska lastnost
,
Bose-Mesnerjeva algebra
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