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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) Algebraic Combinatorics and Finite Geometry 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 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) 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 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) 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 jXj. EXAMPLE 5 0 7 2 3 6 4 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) 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 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 2 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. 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) ADJACENCY MATRIX 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. 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) ADJACENCY MATRIX EXAMPLE 0 0 1 1 1 0 0 1 0 0 1 B 1 0 1 0 1 0 0 1 0 C B C B 1 1 0 0 0 1 0 0 1 C B C B 1 0 0 0 1 1 1 0 0 C B C B 0 1 0 1 0 1 0 1 0 C B C B 0 0 1 1 1 0 0 0 1 C B C B 1 0 0 1 0 0 0 1 1 C B C @ 0 1 0 0 1 0 1 0 1 A 0 0 1 0 0 1 1 1 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) CHARACTERISTIC POLYNOMIAL AND EIGENVALUES 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). 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) CHARACTERISTIC POLYNOMIAL AND EIGENVALUES 0 0 1 1 1 0 0 1 0 0 1 B 1 0 1 0 1 0 0 1 0 C B C B 1 1 0 0 0 1 0 0 1 C B C B 1 0 0 0 1 1 1 0 0 C B C B 0 1 0 1 0 1 0 1 0 C B C B 0 0 1 1 1 0 0 0 1 C B C B 1 0 0 1 0 0 0 1 1 C B C @ 0 1 0 0 1 0 1 0 1 A 0 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. 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. 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) CHARACTERISTIC VECTOR DEFINITION For Γ = (X; ∼) and Y ⊆ X, the characteristic vector χY of Y is defined by (χY )x = 1 if x 2 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 2 R . 2 (AχY )x = jΓ(x) \ Y j for all Y ⊆ X. T 3 (χY ) A(χZ ) = jf(y; z): y 2 Y ; z 2 Z ; y ∼ zgj for all Y ; Z ⊆ 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) 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 and ci with i 2 f0;:::; dg, named intersection numbers, so that for any x; y 2 X with d(x; y) = i, we have that jΓi−1(x) \ Γ(y)j = ci for all i 2 f0;:::; dg; jΓi+1(x) \ Γ(y)j = bi for all i 2 f0;:::; dg: We will always use bi and ci as above. Note that Γ is k-regular with k = b0. 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) DISTANCE-REGULAR GRAPH We have that k = jΓi−1(x) \ Γ1(y)j + jΓi (x) \ Γ1(y)j + jΓi+1(x) \ Γ1(y)j: Hence, for d(x; y) = i, the number ai = jΓi (x) \ Γ1(y)j 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 fb0; b1;:::; bd−1; c1;:::; cd g: EXAMPLE The hexagon is an example with parameters f2; 1; 1; 1; 1; 2g. 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.
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