Algebraic Combinatorics and Finite Geometry

An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

Leo Storme

Ghent University Department of Mathematics: Analysis, Logic and Discrete Mathematics Krijgslaan 281 - Building S8 9000 Ghent Belgium

Francqui Foundation, May 5, 2021

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q) ACKNOWLEDGEMENT

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

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q) OUTLINE

1 AN INTRODUCTIONTO ALGEBRAIC GRAPH THEORY

2 ERDOS˝ -KO-RADO RESULTS

3 CAMERON-LIEBLERSETSIN PG(3, q)

4 CAMERON-LIEBLER k-SETSIN PG(n, q)

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q) OUTLINE

1 AN INTRODUCTIONTO ALGEBRAIC GRAPH THEORY

2 ERDOS˝ -KO-RADO RESULTS

3 CAMERON-LIEBLERSETSIN PG(3, q)

4 CAMERON-LIEBLER k-SETSIN PG(n, q)

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

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

We say that Γ has order |X|.

EXAMPLE 5 0 7 2 3 6 4 1

If x ∼ y, then x is a neighbour of y.

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

EXAMPLE

The Petersen Graph is 3-regular.

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

DEFINITION

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

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

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

Let Γ = (X, ∼) be a graph of order n.

DEFINITION The adjacency matrix A of Γ is an (n × n)-matrix over R deﬁned by ( 1 if x ∼ y, Axy = 0 if x 6∼ y.

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

EXAMPLE

 0 1 1 1 0 0 1 0 0   1 0 1 0 1 0 0 1 0     1 1 0 0 0 1 0 0 1     1 0 0 0 1 1 1 0 0     0 1 0 1 0 1 0 1 0     0 0 1 1 1 0 0 0 1     1 0 0 1 0 0 0 1 1     0 1 0 0 1 0 1 0 1  0 0 1 0 0 1 1 1 0

DEFINITION The characteristic polynomial of a graph Γ with adjacency matrix 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: (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.

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

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.

DEFINITION

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

LEMMA For Γ = (X, ∼) with adjacency matrix A, we have:

1 P n (Av)x = x∼y vy for all v ∈ R . 2 (AχY )x = |Γ(x) ∩ Y | for all Y ⊆ X. T 3 (χY ) A(χZ ) = |{(y, z): y ∈ Y , z ∈ Z , y ∼ z}| for all Y , Z ⊆ X.

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 and ci with i ∈ {0,..., d}, named intersection numbers, so that for any 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.

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)| is independent of our choice of x and y.

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

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}.

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

DEFINITION A k-regular graph of order n is strongly regular with parameters (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 satisfying n − 1 > k > 0 and µ > 0 are exactly the distance-regular graphs with diameter 2.

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

EXAMPLE 1 The Petersen Graph (Example 3). Leo Storme Algebraic Combinatorics and Finite Geometry 2 The K3 × K3-graph (Example 7). An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

LEMMA

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

1 Pd We have i=0 Ai = J. 2 We have A1Ai = ci+1Ai+1 + ai Ai + 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 hA0,..., Ad i consists of symmetric matrices, and is closed under matrix multiplication (which is also commutative).

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

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

(a a b if q = 1, = a−i+1 b Qb q −1 q i=1 qi −1 otherwise.

a If not a ≥ b ≥ 0, then we set b q = 0.

LEMMA Let q be a prime power.

1 a a Then b q is the number of b-spaces of Fq. a−c 2 The number of b-spaces through a ﬁxed c-space is . b−c q

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

DEFINITION n Let 0 ≤ k ≤ n. Let X be the set of all k-spaces of Fq. For x, y ∈ X, say that x ∼ y if dim(x ∩ y) = k − 1. Then (X, ∼) is the Grassmann graph Jq(n, k).

The Grassmann graph is also called q-Johnson graph.

THEOREM n The graph Jq(n, k) is distance-regular, and of order k q with diameter min(k, n − k). In particular,

2 i 2i+1 k − i n − k − i ci = , bi = q . 1 q 1 q 1 q

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

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

THE q-KNESERGRAPH Kq(n, k)

The q-Kneser graph Kq(n, k) = The disjointness graph Kq(n, k). n Vertices are k-spaces in Fq ((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 n − k 2 valency = qk . n − 2k q

The minimal eigenvalue is

n − k − 1 − qk(k−1). n − 2k q

We can often bound substructures of graphs with eigenvalues.

A clique is a set of pairwise adjacent vertices.

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

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

THEOREM (HOFFMAN’S BOUND/RATIO BOUND)

Let Γ be a non-empty k-regular graph, k > 0, of order n, with smallest eigenvalue λ. Let C be a coclique of Γ. Then n |C| ≤ . 1 − k/λ

c In case of equality, χC = n j + w, where w is an eigenvector of A for λ. Also in this case, each vertex not in C is adjacent to exactly −λ elements of C.

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q) OUTLINE

1 AN INTRODUCTIONTO ALGEBRAIC GRAPH THEORY

2 ERDOS˝ -KO-RADO RESULTS

3 CAMERON-LIEBLERSETSIN PG(3, q)

4 CAMERON-LIEBLER k-SETSIN PG(n, q)

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

P. Erdos˝

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

C. Ko

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

R. Rado

Problem: What are largest sets of k-sets in n-set, pairwise intersecting 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 n − 1 in at least one element, then |S| ≤ . If 2k + 1 ≤ n, k − 1 then equality only holds if S consists of all k-sets through ﬁxed element of Ω. n = 2k: If n = 2k, other sets with equality: all k-sets in ﬁxed subset of size n − 1 = 2k − 1 of Ω.

q-Analog problem: What are the largest sets of k-spaces in V (n, q), pairwise intersecting 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?

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

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

THEOREM Zij n ≥ 2k > 0. If S is an intersecting set of (k − 1)-spaces in PG(n − 1, q), then n − 1 |S| ≤ . k − 1 q If also n > 2k, then if |S| = n−1 , then S is a point-pencil. k−1 q

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

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

THE q-KNESERGRAPH 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 n − k 2 valency = qk . n − 2k q

The minimal eigenvalue is

n − k − 1 − qk(k−1). n − 2k q

THEOREM (HOFFMAN’S BOUND/RATIO BOUND)

Let Γ be a non-empty k-regular graph, k > 0, of order n, with smallest eigenvalue λ. Let C be a coclique of Γ. Then n |C| ≤ . 1 − k/λ

c In case of equality, χC = n j + w, where w is an eigenvector of A for λ. Also in this case, each vertex not in C is adjacent to exactly −λ elements of C.

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

n k n − 1 |S| ≤ q = . [ n−k ] qk2 k − 1 1 − n−2k q q − n−k−1 qk(k−1) [ n−2k ]q In case of equality, a (k − 1)-space not in the Erdos-Ko-Rado˝ set S is disjoint to

n − k − 1 qk(k−1) n − 2k q

(k − 1)-spaces of S.

R. Rado https: //de.wikipedia.org/wiki/Richard-Rado-Preis 2020 laureate: Lisa Sauermann Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q) OUTLINE

1 AN INTRODUCTIONTO ALGEBRAIC GRAPH THEORY

2 ERDOS˝ -KO-RADO RESULTS

3 CAMERON-LIEBLERSETSIN PG(3, q)

4 CAMERON-LIEBLER k-SETSIN PG(n, q)

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

P.J. Cameron

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

R.A. Liebler

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

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

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

(0 ≤ x ≤ q2 + 1)

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

THEOREM There exist Cameron-Liebler sets in PG(3, q), with parameter x and q2 + 1 − x, for 2+ 1 q 1 x = 2 , with q odd. 2− 2 q 1 x = 2 , 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 incident point-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 disjoint lines ` 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 function χ. Consider incidence matrix A of points and lines of PG(3, q). χ ∈ row(A). χ ∈ ker(AT )⊥.

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

n + n(n − x) ≡ 0 (mod q + 1), 2 where n is the number of lines of L through the point P or contained in the plane π.

Eliminates more than 50% of possible parameters x.

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q) OUTLINE

1 AN INTRODUCTIONTO ALGEBRAIC GRAPH THEORY

2 ERDOS˝ -KO-RADO RESULTS

3 CAMERON-LIEBLERSETSIN PG(3, q)

4 CAMERON-LIEBLER k-SETSIN PG(n, q)

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q) 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 by k-spaces.)

Ai = incidence matrix of relation 0 0 0 Ri = {(π, π )|π, π ∈ Πk , dim(π ∩ π ) = k − i}.

Leads to Grassmann scheme Jq(n + 1, k + 1).

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q) SETTING

Π R k = V0 ⊥ V1 ⊥ · · · ⊥ Vk+1 in common eigenspaces of A0, A1,..., Ak+1.

K := Ak+1 for disjointness matrix.

THEOREM (BLOKHUIS,DE BOECK,D’HAESELEER) Let L be non-empty set of k-spaces in PG(n, q), n ≥ 2k + 1, n with characteristic vector χ, and x so that |L| = x . Then the k following properties are equivalent. 1 χ ∈ row(A). ⊥ 2 χ ∈ ker(AT ) . 3 For every k-space π, number of elements of L disjoint from n−k−1 k 2+k π is (x − χ(π)) k q .

THEOREM qk+1−1 1 = χ − The vector v x qn+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:  " #" # k+1 n−k n − k − 1 k  q −1 i q −1 i(i−1)  (x − 1) k−i+1 + q i q if π ∈ L  q −1 q −1 i − 1 i " #" # .  n − k − 1 k + 1 x qi(i−1) if π∈ / L  i − 1 i

THEOREM (BLOKHUIS,DE BOECK,D’HAESELEER) 0 1 for every pair of conjugate switching sets R and R , |L ∩ R| = |L ∩ R0|. If PG(n, q) has k-spread, then following property is equivalent to the previous ones. 2 |L ∩ S| = x for every k-spread S in PG(n, q).

(Millenaris Park, Budapest)

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

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

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 2 1 n − k − 3k − 3 k − k + 1 p 2 ≤ x ≤ √ q 2 4 4 2 (q − 1) 4 4 2 q2 + q + 1. 8 2

Boolean degree one functions: link to Cameron-Liebler sets of k-spaces in PG(n, q). Boolean functions: {0, 1}-valued functions on a ﬁnite domain Ω.

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

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

θn X + f :Πk → R : π 7→ c + ai Pi (π), i=1

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 a Cameron-Liebler set of k-spaces in PG(n, q) if and only if L = Lf for some Boolean degree one function f on the set of k-spaces in PG(n, q).

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

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

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

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

THEOREM (JOHNSONAND NEWMAN) If Γ and Γ0 are graphs with adjacency matrices A and A0, then the following are equivalent: The graphs Γ and Γ0 are cospectral, and so are their complements. 0 The graphs Γ and Γ are R-cospectral. There exists a regular orthogonal matrix Q such that Qt AQ = A0.

THEOREM (ABIAD, VAN DAM,FIOL) A regular graph Γ with d + 1 distinct eigenvalues and with girth g is distance-regular if and only if either one of the following conditions is valid: g ≥ 2d − 1, g ≥ 2d − 2 and G is bipartite.

These results follow from the spectrum of the graphs.

THEOREM (ABIAD, VAN DAM,FIOL) Let G be a regular graph with d + 1 distinct eigenvalues λ0 > λ1 > ··· > λd and girth ≥ 2d − 2. Then

γd ≥ −(λ1 + ··· + λd )

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

Main tool to construct cospectral graphs: Godsil-McKay switching. Usually (but not always) the newly obtained graph is not isomorphic with the original graph. Abiad, Brouwer and Haemers developed some necessary conditions to guarantee the non-isomorphism after switching for some graph products.

THEOREM The tensor product of the lattice graph L(`, m) (` > m ≥ 2) with a graph Γ having at least one vertex of degree 2 is not determined by its spectrum.

Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3, q) Cameron-Liebler k-sets in PG(n, q)

Thank you very much for your attention

Leo Storme Algebraic Combinatorics and Finite Geometry