# Graphs with three eigenvalues

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1 Graphs with three eigenvalues Jack Koolen Joint work with Ximing Cheng and it is work in progress School of Mathematical Sciences, University of Science and Technology of China Villanova University, June 2, 2014

2 Outline 1 Introduction Definitions History 2 Theory Basic Theory 3 Our results Bound Neumaier s result

3 Definitions Let Γ = (V, E) be a graph. The distance d(x, y) between two vertices x and y is the length of a shortest path connecting them. The maximum distance between two vertices in Γ is the diameter D = D(Γ). The valency k x of x is the number of vertices adjacent to it. A graph is regular with valency k if each vertex has k neighbours.

4 Definitions Let Γ = (V, E) be a graph. The distance d(x, y) between two vertices x and y is the length of a shortest path connecting them. The maximum distance between two vertices in Γ is the diameter D = D(Γ). The valency k x of x is the number of vertices adjacent to it. A graph is regular with valency k if each vertex has k neighbours. The adjacency matrix A of Γ is the matrix whose rows and columns are indexed by the vertices of Γ and the (x, y)-entry is 1 whenever x and y are adjacent and 0 otherwise. The eigenvalues of the graph Γ are the eigenvalues of A.

5 Strongly regular graphs A strongly regular graph (SRG) with parameters (n, k, λ, µ) is a k-regular graph on n vertices such that each pair of adjacent vertices have λ common neighbours; each pair of distinct non-adjacent vertices have µ common neighbours

6 Strongly regular graphs A strongly regular graph (SRG) with parameters (n, k, λ, µ) is a k-regular graph on n vertices such that each pair of adjacent vertices have λ common neighbours; each pair of distinct non-adjacent vertices have µ common neighbours Examples The Petersen graph is a strongly regular graph with parameters (10, 3, 0, 1). The line graph of a complete graph on t vertices L(K t ) is a SRG (t(t 1)/2, 2(t 2), t 2, 4).

7 Strongly regular graphs A strongly regular graph (SRG) with parameters (n, k, λ, µ) is a k-regular graph on n vertices such that each pair of adjacent vertices have λ common neighbours; each pair of distinct non-adjacent vertices have µ common neighbours Examples The Petersen graph is a strongly regular graph with parameters (10, 3, 0, 1). The line graph of a complete graph on t vertices L(K t ) is a SRG (t(t 1)/2, 2(t 2), t 2, 4). The line graph of a complete bipartite graph K t,t, L(K t,t ), is a SRG (t 2, 2(t 1), t 2, 2). There are many more examples, coming from all parts in combinatorics.

8 Strongly regular graphs 2 A strongly regular graph has at most diameter two, and has at most three distinct eigenvalues. We can characterize the strongly regular graphs by this property. Theorem A connected regular graph Γ has at most three eigenvalues if and only if it is strongly regular.

9 Small number of distinct eigenvalues Now we will discuss graphs with a small number of distinct eigenvalues. If Γ is a connected graph with t distinct eigenvalues then the diameter of Γ is bounded by t 1. So a connected graph with at most two distinct eigenvalues is just a complete graph and hence is regular.

10 Small number of distinct eigenvalues Now we will discuss graphs with a small number of distinct eigenvalues. If Γ is a connected graph with t distinct eigenvalues then the diameter of Γ is bounded by t 1. So a connected graph with at most two distinct eigenvalues is just a complete graph and hence is regular. But connected graphs with three distinct eigenvalues do not have to be regular. For example the complete bipartite graph K s,t has distinct eigenvalues ± st and 0.

11 Small number of distinct eigenvalues Now we will discuss graphs with a small number of distinct eigenvalues. If Γ is a connected graph with t distinct eigenvalues then the diameter of Γ is bounded by t 1. So a connected graph with at most two distinct eigenvalues is just a complete graph and hence is regular. But connected graphs with three distinct eigenvalues do not have to be regular. For example the complete bipartite graph K s,t has distinct eigenvalues ± st and 0. Also the cone over the Petersen graph (i.e. you add a new vertex and join the new vertex with all the other vertices) is a non-regular graph with exactly three distinct eigenvalues.

12 Outline 1 Introduction Definitions History 2 Theory Basic Theory 3 Our results Bound Neumaier s result

13 History In 1970 M. Doob asked to study graphs with a small number of distinct eigenvalues.

14 History In 1970 M. Doob asked to study graphs with a small number of distinct eigenvalues. In 1979 and 1981 Bridges and Mena constructed infinite many examples of graphs with exactly three distinct eigenvalues. They constructed mainly cones over strongly regular graphs.

15 History In 1970 M. Doob asked to study graphs with a small number of distinct eigenvalues. In 1979 and 1981 Bridges and Mena constructed infinite many examples of graphs with exactly three distinct eigenvalues. They constructed mainly cones over strongly regular graphs. In 1995 W. Haemers asked to construct new families of connected graphs with exactly three distinct eigenvalues. (He was unaware of the papers by Bridges and Mena).

16 History In 1970 M. Doob asked to study graphs with a small number of distinct eigenvalues. In 1979 and 1981 Bridges and Mena constructed infinite many examples of graphs with exactly three distinct eigenvalues. They constructed mainly cones over strongly regular graphs. In 1995 W. Haemers asked to construct new families of connected graphs with exactly three distinct eigenvalues. (He was unaware of the papers by Bridges and Mena). In 1998 Muzychuk and Klin gave more examples of such graphs.

17 History In 1970 M. Doob asked to study graphs with a small number of distinct eigenvalues. In 1979 and 1981 Bridges and Mena constructed infinite many examples of graphs with exactly three distinct eigenvalues. They constructed mainly cones over strongly regular graphs. In 1995 W. Haemers asked to construct new families of connected graphs with exactly three distinct eigenvalues. (He was unaware of the papers by Bridges and Mena). In 1998 Muzychuk and Klin gave more examples of such graphs. In 1998 E. van Dam gave the basic theory for such graphs, and also give some new examples. Also he classified the graphs with exactly three distinct eigenvalues having smallest eigenvalue at least 2.

18 Outline 1 Introduction Definitions History 2 Theory Basic Theory 3 Our results Bound Neumaier s result

19 Basic theory Our motivation is how much of the theory for strongly graphs can be generalised to connected graphs with exactly three distinct eigenvalues. We start with some basic theory.

20 Basic theory Our motivation is how much of the theory for strongly graphs can be generalised to connected graphs with exactly three distinct eigenvalues. We start with some basic theory. Let Γ be a connected graph with exactly three distinct eigenvalues θ 0 > θ 1 > θ 2.

21 Basic theory Our motivation is how much of the theory for strongly graphs can be generalised to connected graphs with exactly three distinct eigenvalues. We start with some basic theory. Let Γ be a connected graph with exactly three distinct eigenvalues θ 0 > θ 1 > θ 2. Then by the Perron-Frobenius Theorem θ 0 has multiplicity one.

22 Basic theory Our motivation is how much of the theory for strongly graphs can be generalised to connected graphs with exactly three distinct eigenvalues. We start with some basic theory. Let Γ be a connected graph with exactly three distinct eigenvalues θ 0 > θ 1 > θ 2. Then by the Perron-Frobenius Theorem θ 0 has multiplicity one. Let A be the adjacency matrix of Γ. As B := (A θ 1 I )(A θ 2 I ) has rank 1 and is positive semi-definite we have B = xx T for some eigenvector x of A corresponding to eigenvalue θ 0.

23 Basic theory Our motivation is how much of the theory for strongly graphs can be generalised to connected graphs with exactly three distinct eigenvalues. We start with some basic theory. Let Γ be a connected graph with exactly three distinct eigenvalues θ 0 > θ 1 > θ 2. Then by the Perron-Frobenius Theorem θ 0 has multiplicity one. Let A be the adjacency matrix of Γ. As B := (A θ 1 I )(A θ 2 I ) has rank 1 and is positive semi-definite we have B = xx T for some eigenvector x of A corresponding to eigenvalue θ 0. By looking at the uv entries of B, this gives k u = θ 1 θ 2 + x 2 u for u a vertex, λ uv = θ 1 + θ 2 + x u x v, for u v, µ xy = x u x v for u and v non-adjacent.

24 A result of Van Dam Theorem (Van Dam) Let Γ be a connected non-regular graph with three distinct eigenvalues θ 0 > θ 1 > θ 2. Then the following hold: Γ has diameter two.

25 A result of Van Dam Theorem (Van Dam) Let Γ be a connected non-regular graph with three distinct eigenvalues θ 0 > θ 1 > θ 2. Then the following hold: Γ has diameter two. If θ 0 is not an integer, then Γ is complete bipartite.

26 A result of Van Dam Theorem (Van Dam) Let Γ be a connected non-regular graph with three distinct eigenvalues θ 0 > θ 1 > θ 2. Then the following hold: Γ has diameter two. If θ 0 is not an integer, then Γ is complete bipartite. θ 1 0 with equality if and only if Γ is complete bipartite. θ 2 2 with equality if and only if Γ is the path of length 2.

27 A result of Van Dam Theorem (Van Dam) Let Γ be a connected non-regular graph with three distinct eigenvalues θ 0 > θ 1 > θ 2. Then the following hold: Γ has diameter two. If θ 0 is not an integer, then Γ is complete bipartite. θ 1 0 with equality if and only if Γ is complete bipartite. θ 2 2 with equality if and only if Γ is the path of length 2. From now on we will assume θ 1 > 0 and hence θ 0 is an integer.

28 Outline 1 Introduction Definitions History 2 Theory Basic Theory 3 Our results Bound Neumaier s result

29 A bound on the number of vertices Lemma Let Γ be a non-regular connected graph with exactly three distinct eigenvalues θ 0 > θ 1 > θ 2. Let u v with k u < k v. Then k u λ uv + 1. This gives x v 1 x u (x v x u ) θ 1 θ 2 + θ 1 + θ 2, and hence x v (θ 1 + 1)(θ 2 + 1).

30 A bound on the number of vertices Lemma Let Γ be a non-regular connected graph with exactly three distinct eigenvalues θ 0 > θ 1 > θ 2. Let u v with k u < k v. Then k u λ uv + 1. This gives x v 1 x u (x v x u ) θ 1 θ 2 + θ 1 + θ 2, and hence x v (θ 1 + 1)(θ 2 + 1). This implies: Proposition Let Γ be a non-regular connected graph on n vertices with three distinct eigenvalues θ 0 > θ 1 > θ 2 with respective multiplicities 1, m 1, m 2. Let be the maximal valency in Γ and let l := min{1 (θ 1 + 1)(θ 2 + 1), θ 1 θ 2 + 1}. Then the following hold: 1 (1 (θ 1 + 1)(θ 2 + 1)) 2 θ 1 θ 2 ) ; 2 If n 1 and θ 1 0, then l 2 θ 1 θ 2 ; 3 n max{(l 2 θ 1 θ 2 1) 2 + 1, (1 (θ 1 + 1)(θ 2 + 1)) 2 θ 1 θ 2 + 1}.

31 Outline 1 Introduction Definitions History 2 Theory Basic Theory 3 Our results Bound Neumaier s result

32 Neumaier s Theorem Neumaier (1979) showed the following result. Neumaier s Theorem Let m be a positive integer. Let Γ be a connected and coconnected (i.e the complement is connected) strongly regular graph with minimal eigenvalue m. Then either the number of vertices is bounded by a function in m, or Γ belongs to one of two infinite (one parameter) families of strongly regular graphs (and we know how to construct all of them if the number of vertices is large enough)

33 Neumaier s Theorem Neumaier (1979) showed the following result. Neumaier s Theorem Let m be a positive integer. Let Γ be a connected and coconnected (i.e the complement is connected) strongly regular graph with minimal eigenvalue m. Then either the number of vertices is bounded by a function in m, or Γ belongs to one of two infinite (one parameter) families of strongly regular graphs (and we know how to construct all of them if the number of vertices is large enough) How can we generalize this result to graphs with three distinct eigenvalues?

34 Neumaier s Theorem 2 Question 1: Let m be a positive integer. Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that 0 < θ 1 m? Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that θ 2 m?

35 Neumaier s Theorem 2 Question 1: Let m be a positive integer. Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that 0 < θ 1 m? Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that θ 2 m? Our bound on the number of vertices implies that the conjecture is true if the graphs have a non-integral eigenvalue.

36 Neumaier s Theorem 2 Question 1: Let m be a positive integer. Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that 0 < θ 1 m? Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that θ 2 m? Our bound on the number of vertices implies that the conjecture is true if the graphs have a non-integral eigenvalue. Van Dam showed that (ii) is true for smallest eigenvalue 2.

37 Neumaier s Theorem 2 Question 1: Let m be a positive integer. Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that 0 < θ 1 m? Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that θ 2 m? Our bound on the number of vertices implies that the conjecture is true if the graphs have a non-integral eigenvalue. Van Dam showed that (ii) is true for smallest eigenvalue 2. We were able to show that the answer to the first part of the question for non-regular graphs with exactly three distinct eigenvalues and exactly two different valencies is positive.

38 Neumaier s Theorem 2 Question 1: Let m be a positive integer. Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that 0 < θ 1 m? Are there only finitely many non-regular connected graphs with distinct eigenvalues θ 0 > θ 1 > θ 2 such that θ 2 m? Our bound on the number of vertices implies that the conjecture is true if the graphs have a non-integral eigenvalue. Van Dam showed that (ii) is true for smallest eigenvalue 2. We were able to show that the answer to the first part of the question for non-regular graphs with exactly three distinct eigenvalues and exactly two different valencies is positive. Note that the answer for the question (i) is negative if you allow four distinct eigenvalues as the friendship graphs show.

39 Note that we only have a very few examples with more then two valencies and all the known examples have at most three different valencies.

40 Note that we only have a very few examples with more then two valencies and all the known examples have at most three different valencies. This leads to: Question 2: Is it true that a connected graph with exactly three distinct eigenvalues has at most 3 different valencies?

41 Note that we only have a very few examples with more then two valencies and all the known examples have at most three different valencies. This leads to: Question 2: Is it true that a connected graph with exactly three distinct eigenvalues has at most 3 different valencies? This was shown for cones, i.e. graphs with a vertex of valency n 1.

42 Note that we only have a very few examples with more then two valencies and all the known examples have at most three different valencies. This leads to: Question 2: Is it true that a connected graph with exactly three distinct eigenvalues has at most 3 different valencies? This was shown for cones, i.e. graphs with a vertex of valency n 1. We end with a challenge. Challenge: Construct more connected non-regular graphs with three distinct eigenvalues.

43 Thank you for your attention.

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