Lecture 10: Miscellanea: Some research tasks in AGT

Coherent ConfigurationsLecture 10:Miscellanea:

Some research tasks in AGT

Mikhail Klin (Ben-Gurion University)

September 1–5, 2014

M. Klin (BGU) Miscellanea 1 / 67

Preamble

We will discuss a number of research tasks

of different level and significance.

Most of them are formulated in terms of

color graphs, typically CCs or even ASs.


Preamble

As a rule, the tasks are related tocomputations:

to perform some computations;to explain (interpret) computer results;to provide theoretical generalizations ofcomputer aided activities.


Preamble

A list of related publications will be

provided at the end of lecture.

Participants are welcome to communicate

with MK, regarding extra clarifications and

comments (personally or via e-mail).


Part A. Nice interpretation isrequested

As a rule, an interpretation is expected,

though helpful explanation might serve as a

suitable palliative.


A.1: SRGs on 25 and 26 vertices

Up to isomorphism, there exist 15 SRGs

with parameters (25, 12, 5, 6) and 10 SRGs

for (26, 10, 3, 4).

First case, when the use of computers for

full classification is essential.

Also first time, when graphs may have

trivial automorphism group (of order 1).



The results were obtained independently by

a few groups (1973–1975).

There are four remarkable geometric

graphs, which are coming from Latin

squares of order 5 and Steiner triple systems

on 13 points.



All other graphs are obtained from

geometrical ones, using procedure of

switching (in the sense of J.J. Seidel).

Two-graphs are playing essential role in

classification.



Long-standing challenge: Repeate full

classification without the (essential) use of

a computer.

Our first attempts did not succeed: new

ideas are requested!


A.2: Partial geometries pg(5,7,3)

We use notation, traditional in design

theory (due to R.C.Bose), oppositely to the

so-called Belgian notation.

This is the smallest case when classification

of partial geometries is becoming non-trivial.



Up to isomorphism there exist two

pg(5,7,3), they have 45 points and 63

blocks.

The result belongs to R.Mathon (1981) via

exhaustive computer search.

The automorphism groups have orders 1512

and 216.



MK, SR and A.Woldar obtained nice

interpretation of both geometries with the

aid of so-called overlarge sets of affine

designs with 8 points.

This is, indeed, an interpretation.

Prove without the use of a computer, that

there are just two geometries.


A.3: An SRG on 1716 vertices

There is a number of SRGs, which appear

via mergings of classes of Johnson

association scheme.

All were discovered by MK (1974) via the

use of a computer.

Some are Schurian, some not.



The largest one Γ has parameters

(1716, 833, 400, 306), Aut(Γ) ∼= S13.

No nice computer free interpretation of

appearance of Γ has been done.

Probably, pseudo M13 puzzle (John

Conway) might be of help (aka pseudogroup

extension of Mathieu group M12).



Conjecture (MK)All mergings of classes of Johnson schemes are

known.

Partial confirmation by Misha Muzychuk

and Vasyl Ustimenko (theory and

computer).

Extremely difficult to get more progress.


Part B: Partial results are known

Here we consider a few problems, where

first success was achieved, starting with a

computer, and using also theoretical

arguments.

Next steps are welcome, again via

amalgamation of computation and theory.


B.1: Partial geometries pg(8,9,4)

Partial geometries pg(8,9,4) have 120

points and 135 blocks.

Both point and block graphs have classical

parameters.



An example of a partial geometry pg(8,9,4)was known for a long time, though in a fewdifferent incarnations:

F.De Clerck, R.H. Dye, J.A. Thas (1980);A.M.Cohen (1981);W.Haemers, J.H. van Lint (1982).



Relying on ideas of W.M. Kantor and using

a computer, V.D. Tonchev (1984) proved

that all constructions provide the same (up

to isomorphism) incidence structure.



For a while it was suspected that other

examples do not exist.New splash of activities around 1997:

MK and SR;R. Mathon and A.P. Street.



7 more geometries were discovered

(computer), four of them have the same

point graphs.

This common point graph was known

before (A. Brouwer, A.V. Ivanov, MK).



Thus, altogether there are known 8

geometries with different point graphs (one

of them is rank 3 graph).

Are there more pg(8,9,4)?


B.2: Highly symmetrical properloops

To every quasigroup (and particular loop) Q

of order n we associate an SRG Γ(Q) with

the parameters (n2, 3(n − 1), n, 6).

We say that loops Q1 and Q2 belong to

different main classes if Γ(Q1) and Γ(Q2)

are not isomorphic.



Numbers of main classes for small n:n 1 2 3 4 5 6 7

# 1 1 1 2 2 12 147

“Nice” loop (or Latin square) Q

m

Γ = Γ(Q) has a “rich” group Aut(Γ).



Classical result, due to E. Schonhardt

(1930):

If Q is a group, then Aut(Γ(Q)) is transitive and

its structure is (Q2 : Aut(Q)).S3.



Moreover, in this case G = Aut(Γ(Q))

contains a regular subgroup (of order n2).

In a more specific case we speak about

highly symmetric loop (consider direction

preserving subgroup of G ).



Each group is a highly symmetric loop.

Highly symmetric loop, which does not

belong to any main class of a group, is

called proper.

Nice concept on the edge between AGT and

algebra.



No proper hs-loops for orders up to 5.

Just one such loop for order n = 6.

Really reasonably rare structures.



A.Heinze and MK: investigation of infinite

series of proper hs-loops of order 2p, p a

prime.

MK, N.Kriger and A. Woldar get full

classification of such loops of order 2p:

just one class, so-called Wilson loops.



Next interesting case is n = 3p, or, more

generally, n = pq, p and q distinct odd

primes.

Repeat similar classification.


B.3: Elusive groups andpolycirculant conjecture

A permutation g 6= e is called

semiregular, if all its cycles have the same

length (at least 2).

“Most” of transitive groups contain a

semiregular permutation.



For example, automorphism group of

Petersen graph contains an automorphism

of order 5 (two cycles of length 5).

Compact depiction of the famous diagram:

1 2

l = 5



A transitive permutation group G of degree

n is called elusive if it does not contain a

non-identity semiregular permutation.

Smallest degree of elusive group is n = 12:

Famous M11 in transitive action of degree

12.



They are indeed elusive: very restricted

classes of examples.

Recall that a permutation group (G ,Ω) is

called 2-closed, if G = Aut(Γ) for a

suitable color graph with vertex set Ω.



I. Faradzev, MK, O. Korsunskaya (1988)

inspected, using a computer, many

examples of 2-closed permutation groups.

Each time we found a semiregular

automorphism and depicted compact

diagram.



Conjecture (MK, generalizing Dragan

Marusic):

Any 2-closed permutation group is not elusive.

It is usually called the polycirculantconjecture.



A lot of theoretical results confirm this

conjecture for many particular cases.

Typically, CFSG is used.

See recent surveys by Michael Giudici.



Why not to try to confirm this conjecture,

exploiting more essentially graph-theoretical

techniques?

New ideas are requested.


Part C: Open problems

Most problems below rely on some concrete

objects in AGT with prescribed parameters.

A few times we simply consider the smallest

members of some infinite series of

parameter sets.


C.1: Packing of 7 copies of HoSi inK50

Recall that the Hoffman-Singleton graph is

an SRG(50,7,0,1).

50− 1 = 7 · 7.

We can think of color graph: each of 7

graphs is isomorphic to HoSi.


C.1: Packing of 7 copies of HoSi inK50

M. Meszka and J. Siagiova found 5 disjoint

copies of HoSi, using favourite voltage

graph methods.

Prove that packing with 7 copies is

impossible.

Remark: Corresponding negative result for

Petersen graph is known.


C.2: Symmetric associationschemes with 3 classes

Edwin van Dam developed elements of the

theory of 3-class association schemes.

He also did a lot of enumeration.

The results are presented in helpful tables.



New parameter sets (up to 100 vertices)

remain open for about 15 years (no

examples, no killing).



A few open cases (small sample):n valencies spectrum

52 17,17,17 17 + 3 irrational

55 18,18,18 18,−410 +2 irrational

60 21,14,24 21, 332,−424,−73

66 20,40,5 20, 244,−210,−811

96 30,45,20 30, 630,−245,−620


C.3: Pair of dual SRGs on 96vertices

Consider open parameter set of SRGs on 96

vertices:

n = 96, k = 45, l = 50, λ = 21, µ = 18

with spectrum 45, 920,−375.

Assume in addition, that there exists a

spectrally-dual SRG on 96 vertices (with

respect to a suitable commutative fission).


C.3: Pair of dual SRGs on 96vertices

Prove or disprove existence of such a pair.

Remark: In fact, this is one of the smallest

possible cases for a new kind of duality in

comparison with known ones.


C.4: SRG on 100 vertices

The only open parameter set for SRGs on

strictly 100 vertices:

n = 100, k = 33, λ = 8, µ = 12.

According to van Dam, such a graph might

be embedded into 3-class scheme with

valencies 22,33,44 (two different parameter

sets).


C.5: tfSRG on 162 vertices

The parameter set with n = 162, k = 21,

l = 140, λ = 0, µ = 3 is the smallest open

case for a possible primitive tfSRG.

Covering of this case is an ambitious,

though not absolutely hopeless research

task.


C.6: Non-Schurian example of a4-isoregular graph

Let Γ = (V ,E ) be a graph. Valencyval(S) of a subset S ⊆ V is the amount of

vertices in V , which are neighbours of each

x ∈ S .

Graph Γ is called t-isoregular if for each

S ⊆, |S | ≤ t, val(S) depends only in the

isomorphism type of the subgraph, induced

by S .M. Klin (BGU) Miscellanea 49 / 67


1-isoregular graph = regular.

2-isoregular = strongly regular.

Remark: this definition was suggested by

Ja.Ju.Gol’fand (born 1947), killed in

Moscow at the end of 1990th.



Absolutely regular graph is

|V |-isoregular.

Theorem (MK)All absolutely regular graphs are expired (up to

complement) by n Km (imprimitive SRGs) and

the graph L2(3).



Figure : Γ = L2(3).

Note that Γ ∼= Γ.M. Klin (BGU) Miscellanea 52 / 67


Theorem 10.1 [Gol’fand, P.J.Cameron]Each 5-isoregular graph is absolutely regular.

Remark.Cameron is using different terminology.


Theorem 10.2: [Gol’fand,unpublished]

Parameters of any proper 4-isoregular graph

belong to the following series M(r):

v = (2r + 1)(2r 2(2r + 3)− 1),

k = 2r 3(2r + 3),

l = 2(r + 1)3(2r − 1),

λ = r(2r − 1)(r 2 + r − 1),

µ = r 3(2r + 3).



Graph M(1) with parameters

(27, 10, 16, 1, 5) exists and is unique.

It is the famous Schlafli graph.

In certain strict sense the most beautiful

non-trivial graph.



Graph M(2) with parameters

(275, 112, 162, 30, 56) also exists and is

unique.

It is famous McLaughlin graph, related to a

sporadic simple group.



Graph M(3) with the parameters

(1127, 486, 640, 165, 243) does not exist

(easily follows from a result of A. Makhnev,

2002).

The case M(4) with parameters

(3159, 1408, 532, 704) was also considered

by Makhnev (2011). Some partial negative

results were obtained.M. Klin (BGU) Miscellanea 57 / 67


A.Munemasa and B.Venkov obtained (in

absolutely different context of tight

spherical designs) results, which eliminate

existence of M(r) for infinitely many values

of r .

Not visible explicitely for a non-perplexed

reader.



Conjecture.Graphs M(r) do not exist for r ≥ 3.

A very ambitious problem!


Acknowledgements

Thanks again to

Stefan Gyurki,

Danny Kalmanovich,

Christian Pech,

Sven Reichard,

Matan Ziv-Av

for crucial assistance in the preparation of the

lectures.


References

Brouwer, A.E.; Haemers, W.H.: Spectra of

graphs. Universitext. Springer, New York,

2012. xiv+250 pp.

ISBN: 978-1-4614-1938-9.

Cameron, P.J.; Giudici, M.; Jones, G.A.;

Kantor, W.M.; Klin, M. H.; Marusic, D.;

Nowitz, L.A.: Transitive permutation groups

without semiregular subgroups. J. London

Math. Soc. (2) 66 (2002), no. 2, 325–333.


References

De Clerck, F.: Partial and semipartial

geometries: an update. Combinatorics 2000

(Gaeta). Discrete Math. 267 (2003), no.

1–3, 75–86.

van Dam, E.R.: Three-class association

schemes. J. Algebraic Combin. 10 (1999),

no. 1, 69–107.


References

Faradzev, I. A.; Klin, M. H.; Muzichuk, M.

E.: Cellular rings and groups of

automorphisms of graphs. Investigations in

algebraic theory of combinatorial objects,

1–152, Math. Appl. (Soviet Ser.), 84,

Kluwer Acad. Publ., Dordrecht, 1994.


References

Klin, M.; Muzychuk, M.; Ziv-Av, M.:

Higmanian rank-5 association schemes on

40 points. Michigan Math. J. 58 (2009),

no. 1, 255–284.

Klin, M.; Kriger, N.; Woldar, A.:

Classification of highly symmetrical

translation loops of order 2p, p prime.

Beitr. Algebra Geom. 55 (2014), no. 1,

253–276.M. Klin (BGU) Miscellanea 64 / 67

References

Klin, M. Ch.; Poschel, R.; Rosenbaum, K.:

Angewandte Algebra fur Mathematiker und

Informatiker. (German) [Applied algebra for

mathematicians and information scientists]

Einfuhrung in

gruppentheoretisch-kombinatorische

Methoden. [Introduction to

group-theoretical combinatorial methods]

Friedr. Vieweg & Sohn, Braunschweig,

1988. 208 pp. ISBN: 3-528-08985-7M. Klin (BGU) Miscellanea 65 / 67

References

Klin, M.; Pech, C.; Reichard, S.; Woldar,

A.; Ziv-Av, M.: Examples of computer

experimentation in algebraic combinatorics.

Ars Math. Contemp. 3 (2010), no. 2,

237–258.

Siagiova, J.; Meszka, M.: A covering

construction for packing disjoint copies of

the Hoffman-Singleton graph into K50. J.

Combin. Des. 11 (2003), no. 6, 408–412.


Acknowledgements

Thank you very much for interest andattention to this series of lectures!


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Lecture 10: Miscellanea: Some research tasks in AGT