Graphs With Many Valencies and Few Eigenvalues

Transcription

1 Graphs With Many Valencies and Few Eigenvalues By Edwin van Dam, Jack H. Koolen, Zheng-Jiang Xia ELA Vol Vicente Valle Martinez Math 595 Spectral Graph Theory Apr 14, 2017

2 V. Valle Martinez (Math 595) 2 of 21 Outline Introduction Definitions Basic results Bipartite Graphs with Five Eigenvalues Strongly Regular Graphs Graphs with four eigenvalues

3 V. Valle Martinez (Math 595) 3 of 21 Notation, Q1 Let G be a graph: Let γ denote the number of distinct eigenvalues in Spec(A G ) Let ν denote the number of distinct valencies of vertices in G Question #1: (Dom de Caen) Do connected graphs with γ = 3 have ν 3?

4 V. Valle Martinez (Math 595) 3 of 21 Notation, Q1 Let G be a graph: Let γ denote the number of distinct eigenvalues in Spec(A G ) Let ν denote the number of distinct valencies of vertices in G Question #1: (Dom de Caen) Do connected graphs with γ = 3 have ν 3? Paraphrasing: We do something else in this paper...

5 V. Valle Martinez (Math 595) 4 of 21 Something else? Construct graphs with four and five distinct eigenvalues with arbitrary number of distinct valencies. Explain how γ = 4 graphs are obtained from regular two-graphs. Characterize disconnected graphs with γ = 3 in the switching class of a regular two-graph.

6 V. Valle Martinez (Math 595) 5 of 21 Definitions Definition A two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple form X contains an even number of triples of the two-graph. A two-graph is regular if it has the property that every pair of vertices lies in the same number of triples of the two-graph.

7 V. Valle Martinez (Math 595) 5 of 21 Definitions Definition A two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple form X contains an even number of triples of the two-graph. A two-graph is regular if it has the property that every pair of vertices lies in the same number of triples of the two-graph. E.g.: X = {1, 2, 3, 4, 5, 6} V (G) = {{1, 2, 3}, {1, 2, 4}, {1, 3, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 6}, {2, 4, 5}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6}}

8 V. Valle Martinez (Math 595) 5 of 21 Definitions Definition A two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple form X contains an even number of triples of the two-graph. A two-graph is regular if it has the property that every pair of vertices lies in the same number of triples of the two-graph. E.g.: X = {1, 2, 3, 4, 5, 6} V (G) = {{1, 2, 3}, {1, 2, 4}, {1, 3, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 6}, {2, 4, 5}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6}} Any quadruple from X contains an even number of triples.

9 V. Valle Martinez (Math 595) 5 of 21 Definitions Definition A two-graph is a set of (unordered) triples chosen from a finite vertex set X, such that every (unordered) quadruple form X contains an even number of triples of the two-graph. A two-graph is regular if it has the property that every pair of vertices lies in the same number of triples of the two-graph. E.g.: X = {1, 2, 3, 4, 5, 6} V (G) = {{1, 2, 3}, {1, 2, 4}, {1, 3, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 6}, {2, 4, 5}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6}} Any quadruple from X contains an even number of triples. Warning: A two-graph is not a graph.

10 V. Valle Martinez (Math 595) 6 of 21 Definitions A two-graph turns out to be equivalent to a (Seidel) switching class of graphs. Why do we require that the graph is connected?

11 V. Valle Martinez (Math 595) 6 of 21 Definitions A two-graph turns out to be equivalent to a (Seidel) switching class of graphs. Why do we require that the graph is connected? Question #2 Give an example of a graph with γ = 3 and ν 3.

12 V. Valle Martinez (Math 595) 6 of 21 Definitions A two-graph turns out to be equivalent to a (Seidel) switching class of graphs. Why do we require that the graph is connected? Question #2 Give an example of a graph with γ = 3 and ν 3. Answer: Let G fancy be the disjoint union of complete bipartite graphs on 2 i + 2 t i vertices (i = 0, 1,...t). Recall, the eigenvalues for K m,n are {0 (n+m 2), ± mn (1) }. G has three distinct eigenvalues, t + 1 valencies. Done. Note: Everything in this paper is in terms of adjacency eigenvalues. A question arises on how these results extend to other matrices.

13 V. Valle Martinez (Math 595) 7 of 21 Basic Results How would you solve this? Question #3 Prove that if the number of distinct eigenvalues is 7, the number of valencies of a connected graph can be made arbitrarily large. (Hint: Spectral proof involves minor details.)

14 V. Valle Martinez (Math 595) 7 of 21 Basic Results How would you solve this? Question #3 Prove that if the number of distinct eigenvalues is 7, the number of valencies of a connected graph can be made arbitrarily large. (Hint: Spectral proof involves minor details.)

15 V. Valle Martinez (Math 595) 7 of 21 Basic Results How would you solve this? Question #3 Prove that if the number of distinct eigenvalues is 7, the number of valencies of a connected graph can be made arbitrarily large. (Hint: Spectral proof involves minor details.) Answer: Use G fancy and add a vertex that is connected to all of its components. By interlacing of eigenvalues, γ 7. The number of valencies can be varied by choice of edges. Add salt to taste.

16 V. Valle Martinez (Math 595) 7 of 21 Basic Results How would you solve this? Question #3 Prove that if the number of distinct eigenvalues is 7, the number of valencies of a connected graph can be made arbitrarily large. (Hint: Spectral proof involves minor details.) Answer: Use G fancy and add a vertex that is connected to all of its components. By interlacing of eigenvalues, γ 7. The number of valencies can be varied by choice of edges. Add salt to taste. This recipe motivates the following...

17 V. Valle Martinez (Math 595) 8 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N K m,n : γ = 5, ν t Proof.

18 V. Valle Martinez (Math 595) 8 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N K m,n : γ = 5, ν t Proof. Consider the disjoint union of f mutually non-isomorphic bipartite graphs with e edges. This graph has spectrum {± e (f ), 0 (g) }.

19 V. Valle Martinez (Math 595) 8 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N K m,n : γ = 5, ν t Proof. Consider the disjoint union of f mutually non-isomorphic bipartite graphs with e edges. This graph has spectrum {± e (f ), 0 (g) }. Add a vertex and connect it in an arbitrary way to all of the components as we did for G fancy.

20 V. Valle Martinez (Math 595) 8 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N K m,n : γ = 5, ν t Proof. Consider the disjoint union of f mutually non-isomorphic bipartite graphs with e edges. This graph has spectrum {± e (f ), 0 (g) }. Add a vertex and connect it in an arbitrary way to all of the components as we did for G fancy.the spectrum becomes {θ 0, e (f 1), θ n, 0 (g 1), θ n f, e (f 1), θ n }. But we can say more about the multiplicity of zero as an eigenvalue (...)

21 V. Valle Martinez (Math 595) 9 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N K m,n : γ = 5, ν t Proof. (...) If we restrict the edges from the new vertex to only one of the color classes (the result is a blow-up graph of the spider graph), the rank of the adjacency matrix is reduced.

22 Bipartite Graphs with Five Eigenvalues Theorem t N K m,n : γ = 5, ν t Proof. (...) If we restrict the edges from the new vertex to only one of the color classes (the result is a blow-up graph of the spider graph), the rank of the adjacency matrix is reduced. This is easily seen: Spider fancy = a 1 a 2 a 3 b 1 b 2 b 3 new a a a b b b new V. Valle Martinez (Math 595) 9 of 21

23 V. Valle Martinez (Math 595) 10 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N K m,n : γ = 5, ν t Proof. (...) {θ 0, e (f 1), θ n, 0 (g 1), θ n f, e (f 1), θ n } becomes {θ 0, e (f 1), 0 (g+1), e (f 1), θ n }

24 V. Valle Martinez (Math 595) 10 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N K m,n : γ = 5, ν t Proof. (...) {θ 0, e (f 1), θ n, 0 (g 1), θ n f, e (f 1), θ n } becomes {θ 0, e (f 1), 0 (g+1), e (f 1), θ n } Fun fact: γ = 4 bipartite are known to be incidence graphs of uniform multiplicative designs. Examples are known for ν 4.

25 V. Valle Martinez (Math 595) 11 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N G K m,n : γ = 5, ν = t Proof.

26 V. Valle Martinez (Math 595) 11 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N G K m,n : γ = 5, ν = t Proof. Omitted.

27 V. Valle Martinez (Math 595) 11 of 21 Bipartite Graphs with Five Eigenvalues Theorem t N G K m,n : γ = 5, ν = t Proof. Omitted. For t 3: Taking the complement of G fancy and calculating the eigenvalues. For t = 1, 2 use the Hamming graph H(4, 3) and 2K 1,2.

28 V. Valle Martinez (Math 595) 12 of 21 Regular Two-Graphs Definition The Seidel matrix for graph Γ is defined through the adjacency matrix S(A(Γ)) = J I 2A(Γ) Definition Let π = {U, W } be a partition of V (Γ). We say a graph Γ π has been obtained by Seidel Switching Γ with respect to π if two distinct vertices x and y are adjacent in Γ π precisely if x and y are adjacent in Γ and either both are in U or both are in W, or if they are not adjacent in Γ and one of them is in U and the other one is in W. Furthermore, the switching class is defined as follows: [Γ] = {Γ π : π is a two-partition of the V (Γ)}

29 V. Valle Martinez (Math 595) 13 of 21 Seidel Switching Observations Switching is an equivalence relation. Spec(S) = Spec(A) There is a bijection between switching classes and two-graphs. Theorem Let Γ a graph in a regular two-graph with v vertices and Seidel eigenvalues 1 2σ, 1 2τ with multiplicities m σ, and m τ, respectively. If e = E(Γ) then the spectrum is {β (1) 1, β(1) 2, σ(mσ 1), τ (mτ 1) }, and the following hold: m σ + mτ = v and m σ σ + m τ τ = v 2 and m σσ 2 + m τ τ 2 = v 2 4 and β 1 + β 2 = σ + τ + v/2 = 2στ and β1 2 + β2 2 = σ2 + τ 2 + 2e v 2 4

30 V. Valle Martinez (Math 595) 14 of 21 Seidel Switching TL;DR The point is that within [Γ] the spectrum is determined by the number of edges in the graph. Proof by pigeon overflow: e ( v 2) whereas [Γ] = 2 v 1. This does not keep track of isomorphism but it is addressed at the end of the paper. Note If [Γ] is a two-graph, Γ π is {v} SRG(v, k = 2µ, λ, µ). No other disconnected graphs in [Γ].

31 V. Valle Martinez (Math 595) 15 of 21 Some results on SRG s Proposition If Γ is a disconnected graph in a nontrivial regular two-graph with Seidel eigenvalues 1 2σ and 1 2τ, then Γ = SRG( (2σ + 1)(2τ + 1), 2στ, σ + τ τσ, τσ) Proposition Let Γ be a graph with at most three distinct eigenvalues in a non-trivial regular two-graph, with Seidel eigenvalues 1 2σ and 1 2τ. Then Γ = SRG( (2σ + 1)(2τ + 1) + 1, τ(2σ + 1), σ(1 τ), τ(σ + 1)) or Γ = SRG( (2σ + 1)(2τ + 1) + 1, σ(2τ + 1), τ(1 σ), σ(τ + 1)) The proof uses identities related to the parameters of SRG s and linear algebra.

32 V. Valle Martinez (Math 595) 16 of 21 Some results on SRG s Proposition Let Γ be a nontrivial regular two-graph. Then Γ is not bipartite. Proof. BWOC suppose it is. Then {β (1) 1, β(1) 2, σ(mσ 1), τ (mτ 1) } is a symmetric spectrum about zero. The last two propositions give Γ is connected and γ = 4. If m τ or m σ > 2 then σ = τ so β 1 = β 2 contradicting the system of equations seen before. The only case is then m σ = 2 = mτ but the only regular two-graphs on four vertices is bipartite.

33 V. Valle Martinez (Math 595) 17 of 21 Graphs with four eigenvalues Let r be a positive integer. Let V = GF (2) 2r. Let <, > denote a symplectic bilinear form (a map ω : V V F that is bilinear, alternating: ω(v, v) = 0 v, and nondegenerate: v, ω(u, v) = 0 u = 0). Construct Γ = (V, u v < u, v > 0) [Γ] is known as the symplectic two-graph. Regular with σ, τ = ±2 r 1. 0 is an isolated vertex. The other component is the symplectic graph Sp(2r) and is SRG(2 2r 1, 2 2r 1, 2 2r 2, 2 2r 2 ). Sp(2r) has every graph on at most 2r 1 vertices as an induced subgraph (!).

34 V. Valle Martinez (Math 595) 18 of 21 Graphs with four eigenvalues Proposition Let be a graph on n vertices with γ ( ) = t 3. Then There exists a connected graph G on at most 2 n+2 vertices with γ (G) = 4 and ν 2, having as an induced subgraph. Proof. Seidel switching on the symplectic two-graph [Γ]. induced by U on Sp(2r) with r = n Switch Γ wrt. π = {U, V \ U} u U has valency d = d (u) then u has 2 2r 1 d neighbors in V \ U and thus 2 2r 2 wr 1 + 2d valency in Γ π

35 V. Valle Martinez (Math 595) 19 of 21 Graphs with four eigenvalues Proposition For every t N, connected non-bipatite graph with γ = 4 and ν t. Proof. Instead of symplectic graphs, use the same process as the previous theorem on regular two-graphs corresponding to Paley graphs of large order (which also have all graphs on a given number of vertices as induced subgraphs Paley graphs are constructed from a suitable finite field by connecting pairs of elements that differ by a quadratic residue).

36 V. Valle Martinez (Math 595) 20 of 21 Back to the start γ = 3, ν = 3? n=24,36,43,46,97 γ = 3, ν = 4? It is known that γ 3 lets you partition according to vertices of the same degree in an equitable fashion. But not clear if this is true for ν > 3.

37 Thank You V. Valle Martinez (Math 595) 21 of 21