Algebraic Combinatorics and Finite Geometry

An Introduction to Algebraic Graph TheoryErdos-Ko-Rado results

Cameron-Liebler sets in PG(3, q)

Cameron-Liebler k -sets in PG(n, q)

Algebraic Combinatorics and Finite Geometry

Leo Storme

Ghent UniversityDepartment of Mathematics: Analysis, Logic and Discrete Mathematics

Krijgslaan 281 - Building S89000 Ghent

Belgium

Francqui Foundation, May 5, 2021

Leo Storme Algebraic Combinatorics and Finite Geometry




ACKNOWLEDGEMENT

Acknowledgement: A big thank you to Ferdinand Ihringer forallowing me to use drawings and latex code of his slidepresentations of his lectures for Capita Selecta in Geometry(Ghent University).





OUTLINE

1 AN INTRODUCTION TO ALGEBRAIC GRAPH THEORY

2 ERDOS-KO-RADO RESULTS

3 CAMERON-LIEBLER SETS IN PG(3,q)

4 CAMERON-LIEBLER k -SETS IN PG(n,q)





OUTLINE









DEFINITION

A graph Γ = (X ,∼) consists of a set of vertices X and ananti-reflexive, symmetric adjacency relation ∼⊆ X × X .

We say that Γ has order |X |.

EXAMPLE

0

1

23

4

5

6

7





Two vertices x , y are adjacent if x ∼ y .

If x ∼ y , then x is a neighbour of y .

A graph is (k -)regular (or: has valency k ) if each vertex hasexactly k neighbours.

EXAMPLE

The Petersen Graph is 3-regular.





DEFINITION

A path of length l , from a vertex v0 to a vertex vl , in agraph Γ is a sequence of (distinct) vertices(v0, v1, v2, . . . , vl−1, vl), such that the vertices vi−1 and viare adjacent for all i ,1 ≤ i ≤ l .The distance d(x , y) between two vertices x and y is theminimal length of a path (v0, . . . , vl) with v0 = x , vl = y .





DEFINITION

For a given vertex v ∈ V , the set of vertices in Γ atdistance i from v is denoted by Γi(v).A graph Γ is connected if there exists a path betweenevery two vertices of Γ.The maximal distance that occurs between two vertices ofa connected graph Γ is called the diameter of the graph.The girth of a graph is the length of the shortest cycle inthe graph.





ADJACENCY MATRIX

Let Γ = (X ,∼) be a graph of order n.

DEFINITION

The adjacency matrix A of Γ is an (n × n)-matrix over Rdefined by

Axy =

1 if x ∼ y ,0 if x 6∼ y .

Note: A is symmetric and has only zeros on the diagonal.





ADJACENCY MATRIX

EXAMPLE

0 1 1 1 0 0 1 0 01 0 1 0 1 0 0 1 01 1 0 0 0 1 0 0 11 0 0 0 1 1 1 0 00 1 0 1 0 1 0 1 00 0 1 1 1 0 0 0 11 0 0 1 0 0 0 1 10 1 0 0 1 0 1 0 10 0 1 0 0 1 1 1 0





CHARACTERISTIC POLYNOMIAL AND EIGENVALUES

DEFINITION

The characteristic polynomial of a graph Γ with adjacencymatrix A in an unknown λ is det(λI − A).

The roots of Γ are the eigenvalues of A.The spectrum of Γ is the eigenvalues of A (with multiplicity).





CHARACTERISTIC POLYNOMIAL AND EIGENVALUES

0 1 1 1 0 0 1 0 01 0 1 0 1 0 0 1 01 1 0 0 0 1 0 0 11 0 0 0 1 1 1 0 00 1 0 1 0 1 0 1 00 0 1 1 1 0 0 0 11 0 0 1 0 0 0 1 10 1 0 0 1 0 1 0 10 0 1 0 0 1 1 1 0

Characteristic Polynomial: (λ− 4)(λ− 1)4(λ+ 2)4.Spectrum: 4,1,1,1,1,−2,−2,−2,−2.

We call v an eigenvector of A if Av = λv , v 6= 0.





THEOREM

Let Γ be a graph of order n with adjacency matrix A.1 The matrix A has n (real) eigenvalues.2 The sum of all eigenvalues is zero.3 The matrix A is diagonalizable and we can find an

orthonormal basis of eigenvectors.





CHARACTERISTIC VECTOR

DEFINITION

For Γ = (X ,∼) and Y ⊆ X , the characteristic vector χY of Yis defined by (χY )x = 1 if x ∈ Y and (χY )x = 0 otherwise.

LEMMA

For Γ = (X ,∼) with adjacency matrix A, we have:1 (Av)x =

∑x∼y vy for all v ∈ Rn.

2 (AχY )x = |Γ(x) ∩ Y | for all Y ⊆ X.3 (χY )T A(χZ ) = |(y , z) : y ∈ Y , z ∈ Z , y ∼ z| for all

Y ,Z ⊆ X.





DISTANCE-REGULAR GRAPH

Let Γ = (X ,∼) be a finite, non-empty graph with diameter d .

DEFINITION

We say that Γ is distance-regular if there are numbers bi andci with i ∈ 0, . . . ,d, named intersection numbers, so that forany x , y ∈ X with d(x , y) = i , we have that

|Γi−1(x) ∩ Γ(y)| = ci for all i ∈ 0, . . . ,d,|Γi+1(x) ∩ Γ(y)| = bi for all i ∈ 0, . . . ,d.

We will always use bi and ci as above.

Note that Γ is k -regular with k = b0.





DISTANCE-REGULAR GRAPH

We have that

k = |Γi−1(x) ∩ Γ1(y)|+ |Γi(x) ∩ Γ1(y)|+ |Γi+1(x) ∩ Γ1(y)|.

Hence, for d(x , y) = i , the number ai = |Γi(x) ∩ Γ1(y)| isindependent of our choice of x and y .





As bd = 0 = c0, one writes the parameters of Γ as

b0,b1, . . . ,bd−1; c1, . . . , cd.

EXAMPLE

The hexagon is an example with parameters 2,1,1; 1,1,2.





DEFINITION

A k -regular graph of order n is strongly regular withparameters (n, k , λ, µ) if

1 two adjacent vertices have exactly λ common neighbours,2 two non-adjacent vertices have exactly µ common

neighbours.

Strongly regular graphs with parameters satisfyingn − 1 > k > 0 and µ > 0 are exactly the distance-regulargraphs with diameter 2.

Its parameters (as a distance-regular graph) arek , k − λ− 1; 1, µ.

EXAMPLE

1 The Petersen Graph (Example 3).2 The K3 × K3-graph (Example 7).Leo Storme Algebraic Combinatorics and Finite Geometry




LEMMA

Let Γ be a distance-regular graph with diameter d. Let Ai be theadjacency matrix of Γi . Then:

1 We have∑d

i=0 Ai = J.2 We have A1Ai = ci+1Ai+1 + aiAi + bi−1Ai−1 for all

i ∈ 0, . . . ,d, where b−1 = cd+1 = 0 and A−1 = Ad+1 = 0.3 Each Ai can be written as a degree i polynomial in A1.4 Each power of A1 is a linear combination of A0, . . . ,Ad .5 The vector space 〈A0, . . . ,Ad〉 consists of symmetric

matrices, and is closed under matrix multiplication (whichis also commutative).





DEFINITION

For a given constant q and a,b integers with a ≥ b ≥ 0, definethe Gaussian coefficient (also: q-binomial coefficient)[

ab

]q

=

(ab

)if q = 1,∏b

i=1qa−i+1−1

qi−1 otherwise.

If not a ≥ b ≥ 0, then we set[a

b

]q = 0.

LEMMA

Let q be a prime power.1 Then

[ab

]q is the number of b-spaces of Fa

q.

2 The number of b-spaces through a fixed c-space is[a−c

b−c

]q.





DEFINITION

Let 0 ≤ k ≤ n. Let X be the set of all k -spaces of Fnq. For

x , y ∈ X , say that x ∼ y if dim(x ∩ y) = k − 1. Then (X ,∼) isthe Grassmann graph Jq(n, k).

The Grassmann graph is also called q-Johnson graph.

THEOREM

The graph Jq(n, k) is distance-regular, and of order[n

k

]q with

diameter min(k ,n − k). In particular,

ci =

[i1

]2

q, bi = q2i+1

[k − i

1

]q

[n − k − i

1

]q.

For two vertices x and y, we have d(x , y) = i if and only ifdim(x ∩ y) = k − i .





THE q-KNESER GRAPH Kq(n, k)

The q-Kneser graph Kq(n, k) = The disjointness graph Kq(n, k).Vertices are k -spaces in Fn

q ((k − 1)-spaces inPG(n − 1,q)).Two vertices adjacent if and only if they are disjoint.

THEOREM

Kq(n, k) is regular with

valency =

[n − kn − 2k

]qqk2

.

The minimal eigenvalue is

−[n − k − 1

n − 2k

]qqk(k−1).





SUBSTRUCTURES IN GRAPHS

We can often bound substructures of graphs witheigenvalues.

A clique is a set of pairwise adjacent vertices.

A coclique is a set of pairwise non-adjacent vertices.





THEOREM (HOFFMAN’S BOUND/RATIO BOUND)

Let Γ be a non-empty k-regular graph, k > 0, of order n, withsmallest eigenvalue λ. Let C be a coclique of Γ. Then

|C| ≤ n1− k/λ

.

In case of equality, χC = cn j + w, where w is an eigenvector of

A for λ. Also in this case, each vertex not in C is adjacent toexactly −λ elements of C.





OUTLINE









P. Erdos





C. Ko





R. Rado





ERDOS-KO-RADO PROBLEM

Problem: What are largest sets of k -sets in n-set, pairwiseintersecting in at least one element?

THEOREM (ERDOS-KO-RADO)If S is set of k-sets in n-set Ω, with 2k ≤ n, pairwise intersecting

in at least one element, then |S| ≤(

n − 1k − 1

). If 2k + 1 ≤ n,

then equality only holds if S consists of all k-sets through fixedelement of Ω.

n = 2k : If n = 2k , other sets with equality: all k -sets in fixedsubset of size n − 1 = 2k − 1 of Ω.





ERDOS-KO-RADO PROBLEM IN VECTOR SPACES

q-Analog problem:What are the largest sets of k -spaces in V (n,q), pairwiseintersecting in at least one dimension?What are the largest sets of (k − 1)-spaces in PG(n− 1,q),pairwise intersecting in at least a projective point?





POINT-PENCIL

All (k − 1)-spaces through fixed projective point P(point-pencil = p.-p.) is Erdos-Ko-Rado set.





THEOREM

Zij n ≥ 2k > 0. If S is an intersecting set of (k − 1)-spaces inPG(n − 1,q), then

|S| ≤[n − 1k − 1

]q.

If also n > 2k, then if |S| =[n−1

k−1

]q, then S is a point-pencil.

PROOF.An intersecting set of (k − 1)-spaces in PG(n − 1,q) is acoclique in the q-Kneser graph Kq(n, k).





THE q-KNESER GRAPH Kq(n, k)

The q-Kneser graph Kq(n, k) = The disjointness graph Kq(n, k)

Vertices are (k − 1)-spaces in PG(n − 1,q).Two vertices adjacent if and only if they are disjoint.

THEOREM

Kq(n, k) is regular with

valency =

[n − kn − 2k

]qqk2

.

The minimal eigenvalue is

−[n − k − 1

n − 2k

]qqk(k−1).





USE HOFFMAN’S BOUND

THEOREM (HOFFMAN’S BOUND/RATIO BOUND)

Let Γ be a non-empty k-regular graph, k > 0, of order n, withsmallest eigenvalue λ. Let C be a coclique of Γ. Then

|C| ≤ n1− k/λ

.

In case of equality, χC = cn j + w, where w is an eigenvector of

A for λ. Also in this case, each vertex not in C is adjacent toexactly −λ elements of C.





USE HOFFMAN’S BOUND

Erdos-Ko-Rado set S of (k − 1)-spaces in PG(n − 1,q) iscoclique in q-Kneser graph Kq(n, k).

|S| ≤

[nk

]q

1−[ n−kn−2k]q

qk2

−[n−k−1n−2k ]

qqk(k−1)

=

[n − 1k − 1

]q.

In case of equality, a (k − 1)-space not in theErdos-Ko-Rado set S is disjoint to[

n − k − 1n − 2k

]qqk(k−1)

(k − 1)-spaces of S.





RICHARD RADO PRIZE IN COMBINATORICS

R. Rado

https://de.wikipedia.org/wiki/Richard-Rado-Preis2020 laureate: Lisa Sauermann


https://de.wikipedia.org/wiki/Richard-Rado-Preis

https://de.wikipedia.org/wiki/Richard-Rado-Preis




OUTLINE









P.J. Cameron





R.A. LieblerLeo Storme Algebraic Combinatorics and Finite Geometry




CAMERON-LIEBLER SETS

Cameron and Liebler introduced specific line classes inPG(3,q) when investigating the orbits of the subgroups ofthe collineation group of PG(3,q).By Block’s Lemma, a collineation group of PG(n,q) has atleast as many orbits on lines as on points.Cameron and Liebler tried to determine which collineationgroups have equally many point and line orbits.Leads to Cameron-Liebler sets in PG(3,q).





EQUIVALENT DEFINITIONS FOR CL-SETS IN PG(3,q)

Spread of PG(3,q): partition of point set of PG(3,q) in q2 + 1lines.

DEFINITION

Cameron-Liebler set of lines L of PG(3,q):There exists integer x such that L shares x lines with everyspread of PG(3,q).There exists integer x such that L shares x lines with everyregular spread of PG(3,q).

(0 ≤ x ≤ q2 + 1)





CLASSICAL EXAMPLES

x = 1: all lines through fixed point P, or all lines in fixedplane π.x = 2: all lines through fixed point P, and all lines in fixedplane π, with P 6∈ π.Complement of these examples: Cameron-Liebler set withparameter x = q2 and x = q2 − 1.

THEOREM

Every Cameron-Liebler set of lines L of PG(3,q), with:x = 1: consists all lines through fixed point P, or all lines infixed plane π.x = 2: all lines through fixed point P, and all lines in fixedplane π, with P 6∈ π.





OTHER EXAMPLES OF CAMERON-LIEBLER SETS

THEOREM

There exist Cameron-Liebler sets in PG(3,q), with parameter xand q2 + 1− x, for

1 x = q2+12 , with q odd.

2 x = q2−12 , with q ≡ 5 or 9 (mod 12).

3 x = (q + 1)2/3 for q ≡ 2 (mod 3).

1. Bruen and Drudge.2. De Beule, Demeyer, Metsch, Rodgers, and Feng, Momihara,Xiang.3. Feng, Momihara, Rodgers, Xiang, Zou.






DEFINITION

Cameron-Liebler set of lines L of PG(3,q):There exists integer x such that for every line ` of PG(3,q):

|m ∈ L \ `|m meets `| = x(q + 1) + (q2 − 1)χ(`).

There exists integer x such that for every incidentpoint-plane pair (P, π) of PG(3,q):

|Star(P)∩L|+ |Line(π)∩L| = x + (q + 1)|Pencil(P, π)∩L|.






DEFINITION

Cameron-Liebler set of lines L of PG(3,q):There exists integer x such that for every pair of disjointlines ` and m of PG(3,q):

|n ∈ L|n meets ` and m| = x + q(χ(`) + χ(m)).

Let L be set of lines in PG(3,q) with characteristic functionχ. Consider incidence matrix A of points and lines ofPG(3,q).χ ∈ row(A).χ ∈ ker(AT )⊥.





MODULAR EQUALITY

THEOREM (GAVRILYUK, METSCH)

If L is a Cameron-Liebler set in PG(3,q) with parameter x, andlet P be a point and let π be a plane of PG(3,q), then(

n2

)+ n(n − x) ≡ 0 (mod q + 1),

where n is the number of lines of L through the point P orcontained in the plane π.

Eliminates more than 50% of possible parameters x .





OUTLINE









SETTING

Πk = set of k -spaces in PG(n,q), 0 ≤ k ≤ n.A = incidence matrix of points and k -spaces of PG(n,q).(rows of A are indexed by points and columns byk -spaces.)Ai = incidence matrix of relationRi = (π, π′)|π, π′ ∈ Πk , dim(π ∩ π′) = k − i.Leads to Grassmann scheme Jq(n + 1, k + 1).





SETTING

RΠk = V0 ⊥ V1 ⊥ · · · ⊥ Vk+1 in common eigenspaces ofA0,A1, . . . ,Ak+1.K := Ak+1 for disjointness matrix.





EQUIVALENT DEFINITIONS

THEOREM (BLOKHUIS, DE BOECK, D’HAESELEER)

Let L be non-empty set of k-spaces in PG(n,q),n ≥ 2k + 1,

with characteristic vector χ, and x so that |L| = x[nk

]. Then the

following properties are equivalent.1 χ ∈ row(A).2 χ ∈ ker(AT )⊥.3 For every k-space π, number of elements of L disjoint fromπ is (x − χ(π))

[n−k−1k

]qk2+k .






THEOREM

1 The vector v = χ− x qk+1−1qn+1−1 j is a vector in V1.

2 χ ∈ V0 ⊥ V1.3 For i ∈ 1, . . . , k + 1 and given k-space π, number of

elements of L, meeting π in (k − i)-space is:(

(x − 1) qk+1−1qk−i+1−1 + q i qn−k−1

q i−1

)q i(i−1)

[n − k − 1

i − 1

][ki

]if π ∈ L

x

[n − k − 1

i − 1

][k + 1

i

]q i(i−1) if π /∈ L

.






THEOREM (BLOKHUIS, DE BOECK, D’HAESELEER)1 for every pair of conjugate switching sets R and R′,|L ∩ R| = |L ∩ R′|.

If PG(n,q) has k-spread, then following property is equivalentto the previous ones.

2 |L ∩ S| = x for every k-spread S in PG(n,q).





GENERATORS ON HYPERBOLIC QUADRIC

(Millenaris Park, Budapest)





CLASSIFICATION RESULTS

THEOREM (BLOKHUIS, DE BOECK, D’HAESELEER)Let L be a Cameron-Liebler set of k-spaces with parameterx = 1 in PG(n,q), n ≥ 2k + 1. Then L is a point-pencil, orn = 2k + 1 and L is the set of all k-spaces in hyperplane ofPG(2k + 1,q).





POINT-PENCIL

All k -spaces through fixed point P (point-pencil = p.-p.) isCameron-Liebler set with x = 1.





CLASSIFICATION RESULTS

THEOREM (BLOKHUIS, DE BOECK, D’HAESELEER)

There are no Cameron-Liebler sets of k-spaces in PG(n,q),n ≥ 3k + 2 and q ≥ 3, with parameter

2 ≤ x ≤ 18√2

qn2−

k24 −

3k4 −

32 (q − 1)

k24 −

k4 + 1

2√

q2 + q + 1.





BOOLEAN DEGREE ONE FUNCTIONS

Boolean degree one functions: link to Cameron-Liebler sets ofk -spaces in PG(n,q).

Boolean functions: 0,1-valued functions on a finitedomain Ω.Boolean function f on Ω = ω1, ω2, . . . , ωn corresponds ton-dimensional 0,1-vector v , such that the i ’th element ofv is equal to f (ωi).In this setting: point in the Grassmann graphJq(n + 1, k + 1). For a coordinate x , we denote thecharacteristic function of x by x+: x+(π) = 1 if the elementx is contained in the object π, and x+(π) = 0 otherwise.






DEFINITION

A Boolean degree one function on the set of k -spaces inPG(n,q) is a 0,1-valued function of the form:

f : Πk → R : π 7→ c +θn∑

i=1

aiP+i (π),

with ai , c ∈ R and Pi | 1 ≤ i ≤ θn the set of points in PG(n,q).






THEOREM

Consider the projective space PG(n,q), then a set L is aCameron-Liebler set of k-spaces in PG(n,q) if and only ifL = Lf for some Boolean degree one function f on the set ofk-spaces in PG(n,q).





RESULTS OF IHRINGER AND FILMUS

THEOREM (IHRINGER, FILMUS)For q = 2,3,4,5, the only Cameron-Liebler sets of k-spaces inPG(n,q) are the trivial Cameron-Liebler sets.





SPECTRAL GRAPH THEORY

Spectral graph theory associates a matrix to a graph.Motivation: to deduce from the eigenvalues of this matrix,structural properties about the graph.





SPECTRAL GRAPH THEORY

Understand which graph properties can be deduced fromits spectrum, or which do not follow from its spectrum.Cospectral graphs: graphs with the same spectrum.

THEOREM

A connected graph Γ with distinct eigenvaluesλ0 > λ1 > · · · > λd is bipartite if and only if λ0 = −λd .





COSPECTRAL GRAPHS

THEOREM (JOHNSON AND NEWMAN)

If Γ and Γ′ are graphs with adjacency matrices A and A′, thenthe following are equivalent:

The graphs Γ and Γ′ are cospectral, and so are theircomplements.The graphs Γ and Γ′ are R-cospectral.There exists a regular orthogonal matrix Q such thatQtAQ = A′.





DISTANCE-REGULAR GRAPHS

THEOREM (ABIAD, VAN DAM, FIOL)A regular graph Γ with d + 1 distinct eigenvalues and with girthg is distance-regular if and only if either one of the followingconditions is valid:

g ≥ 2d − 1,g ≥ 2d − 2 and G is bipartite.

These results follow from the spectrum of the graphs.





DISTANCE-REGULAR GRAPHS

THEOREM (ABIAD, VAN DAM, FIOL)Let G be a regular graph with d + 1 distinct eigenvaluesλ0 > λ1 > · · · > λd and girth ≥ 2d − 2. Then

γd ≥ −(λ1 + · · ·+ λd )

with equality if and only if G is distance-regular and eitherbipartite or a generalized Odd graph.





GODSIL-MCKAY SWITCHING

Main tool to construct cospectral graphs: Godsil-McKayswitching.Usually (but not always) the newly obtained graph is notisomorphic with the original graph.Abiad, Brouwer and Haemers developed some necessaryconditions to guarantee the non-isomorphism afterswitching for some graph products.

THEOREM

The tensor product of the lattice graph L(`,m) (` > m ≥ 2) witha graph Γ having at least one vertex of degree 2 is notdetermined by its spectrum.





Thank you very much for your attention


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