ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P2.04
https://doi.org/10.26493/2590-9770.1271.e54
(Also available at http://adam-journal.eu)
On the Terwilliger algebra of a certain family of
bipartite distance-regular graphs with ∆2 = 0
Štefko Miklavič* , Safet Penjić†
University of Primorska, Andrej Marušič Institute,
Muzejski trg 2, 6000 Koper, Slovenia
Received 27 September 2018, accepted 4 January 2019, published online 10 August 2020
Abstract
Let Γ denote a bipartite distance-regular graph with diameterD ≥ 4 and valency k ≥ 3.
Let X denote the vertex set of Γ, and let Ai (0 ≤ i ≤ D) denote the distance matrices of
Γ. We abbreviate A := A1. For x ∈ X and for 0 ≤ i ≤ D, let Γi(x) denote the set of
vertices in X that are distance i from vertex x.
Fix x ∈ X and let T = T (x) denote the subalgebra of MatX(C) generated by
A,E∗0 , E
∗
1 , . . . , E
∗
D, where for 0 ≤ i ≤ D, E∗i represents the projection onto the ith
subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with
respect to x. By the endpoint of an irreducible T -module W we mean min{i | E∗iW 6= 0}.
In this paper we assume Γ has the property that for 2 ≤ i ≤ D− 1, there exist complex
scalars αi, βi such that for all y, z ∈ X with ∂(x, y) = 2, ∂(x, z) = i, ∂(y, z) = i, we
have αi + βi|Γ1(x) ∩ Γ1(y) ∩ Γi−1(z)| = |Γi−1(x) ∩ Γi−1(y) ∩ Γ1(z)|.
We study the structure of irreducible T -modules of endpoint 2. Let W denote an irre-
ducible T -module with endpoint 2, and let v denote a nonzero vector in E∗2W . We show
that W = span
(
{E∗i Ai−2E∗2v | 2 ≤ i ≤ D} ∪ {E∗i Ai+2E∗2v | 2 ≤ i ≤ D − 2}
)
.
It turns out that, except for a particular family of bipartite distance-regular graphs with
D = 5, this result is already known in the literature. Assume now that Γ is a member of
this particular family of graphs. We show that if Γ is not almost 2-homogeneous, then up
to isomorphism there exists exactly one irreducible T -module with endpoint 2 and it is not
thin. We give a basis for this T -module.
Keywords: Distance-regular graphs, Terwilliger algebra, irreducible modules.
Math. Subj. Class.: 05E30, 05C50
*The author acknowledge the financial support from the Slovenian Research Agency (research core funding
No. P1-0285 and research projects N1-0032, N1-0038, N1-0062, J1-5433, J1-6720, J1-7051, J1-9108, J1-9110).
†The author acknowledges the financial support from the Slovenian Research Agency (research core funding
No. P1-0285 and Young Researchers Grant).
E-mail addresses: stefko.miklavic@upr.si (Štefko Miklavič), safet.penjic@iam.upr.si (Safet Penjić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P2.04
https://doi.org/10.26493/2590-9770.1271.e54
(Dostopno tudi na http://adam-journal.eu)
O Terwilligerjevi algebri določene družine
dvodelnih razdaljno-regularnih grafov z ∆2 = 0
Štefko Miklavič* , Safet Penjić†
Univerza na Primorskem, Inštitut Andrej Marušič,
Muzejski trg 2, 6000 Koper, Slovenia
Prejeto 27. septembra 2018, sprejeto 4. januarja 2019, objavljeno na spletu 10. avgusta 2020
Povzetek
Naj bo Γ dvodelni razdaljno-regularni graf s premerom D ≥ 4 in valenco k ≥ 3. Naj
bo X množica točk grafa Γ, in naj bodo Ai (0 ≤ i ≤ D) razdaljne matrike grafa Γ.
Uporabljamo okrajšavo A := A1. Za x ∈ X in za 0 ≤ i ≤ D, naj bo Γi(x) množica tistih
točk iz X , katerih razdalja od točke x je i.
Za dani x ∈ X naj bo T = T (x) podalgebra algebre MatX(C), generirana zA,E∗0 , E∗1 ,
. . . , E∗D, kjer je E
∗
i projekcija na i-to podkonstituento grafa Γ glede na x, in to za vsak i,
ki ustreza pogoju 0 ≤ i ≤ D. Tedaj je T Terwilligerjeva algebra grafa Γ glede na točko x.
Krajišče ireducibilnega T -modula W definiramo kot min{i | E∗iW 6= 0}.
V tem članku privzamemo, da ima Γ lastnost, da za vsak i, ki ustreza pogoju 2 ≤
i ≤ D − 1, obstajata kompleksna skalarja αi, βi, tako da za vse y, z ∈ X , ki ustrezajo
pogoju ∂(x, y) = 2, ∂(x, z) = i, ∂(y, z) = i, velja αi + βi|Γ1(x) ∩ Γ1(y) ∩ Γi−1(z)| =
|Γi−1(x) ∩ Γi−1(y) ∩ Γ1(z)|.
Raziskujemo strukturo ireducibilnih T -modulov s krajiščem 2. Naj bo W ireducibilni
T -modul s krajiščem 2, in naj bo v neničeln vektor v E∗2W . Dokažemo, da je tedaj W =
span
(
{E∗i Ai−2E∗2v | 2 ≤ i ≤ D} ∪ {E∗i Ai+2E∗2v | 2 ≤ i ≤ D − 2}
)
.
Izkaže se, da je, razen za določeno družino dvodelnih razdaljno-regularnih grafov s
premerom D = 5, ta rezultat že znan v literaturi. Privzemimo zdaj, da je Γ član te družine
grafov. Dokažemo, da če Γ ni skoraj 2-homogen, potem obstaja, do izomorfizma natančno,
en sam ireducibilen T -modul s krajiščem 2, in ta modul ni tanek. Predstavimo bazo tega
T -modula.
Ključne besede: Razdaljno-regularni grafi, Terwilligerjeva algebra, ireducibilni moduli.
Math. Subj. Class.: 05E30, 05C50
*Avtor priznava finančno podporo s strani Javne agencije za raziskovalno dejavnost Republike Slovenije (os-
novno financiranje raziskav št. P1-0285 in raziskovalni projekti N1-0032, N1-0038, N1-0062, J1-5433, J1-6720,
J1-7051, J1-9108, J1-9110).
†Avtor priznava finančno podporo s strani Javne agencije za raziskovalno dejavnost Republike Slovenije (os-
novno financiranje raziskav št. P1-0285 in dotacija za mlade raziskovalce).
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/
E-poštni naslovi: stefko.miklavic@upr.si (Štefko Miklavič), safet.penjic@iam.upr.si (Safet Penjić)