Minimal Euclidean representations of graphs
A simple graph is representable in a real vector space of dimension if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values or , with distance if the vertices are adjacent and distance otherwise. The Euclidean representation number of is the smallest dimension in which is representable. In this note, we bound the Euclidean representation number of a graph using multiplicities of the eigenvalues of the adjacency matrix. We also give an exact formula for the Euclidean representation number using the main angles of the graph.
A simple graph is representable in if there is an embedding of the vertex set in and distinct positive constants and such that for all vertices and ,
(Here .) We will call the smallest such that is representable in the Euclidean representation number of and denote it . The complete graph and its complement, the empty graph, are representable in via a regular simplex. By a bound on the size of a Euclidean -distance set [2, 5], they are not representable in smaller dimensions, so . Nguyen Van Thé  showed that if is a graph on vertices that is not the complete graph or empty graph, then . Here, we use the multiplicities of the smallest and second smallest eigenvalues of the adjacency matrix to give upper and lower bounds for the representation number. The main result is the following.
Let be be a graph on vertices which is not complete or empty. If or its complement is the disjoint union of complete graphs, of which are of maximal size, then
Otherwise, let and respectively denote the multiplicity of the smallest and second smallest eigenvalue of the adjacency matrix of , and similarly define and for the complement of . Then
If is regular, then
To get a precise characterization for irregular graphs, we must consider not just the eigenvalues but the actual eigenspaces, or at the very least, the main angles of eigenspaces. An exact formula for the representation number is given below in Theorem 7.
It is not surprising that the Euclidean representation number of a graph is closely related to the multiplicity of the smallest eigenvalue. If is the smallest eigenvalue of , then the positive semidefinite matrix is the Gram matrix of a set of vectors in . This technique is perhaps the most common method of embedding a graph in a vector space and it plays a critical role in, for example, the characterization of graphs with least eigenvalue and the theory of two-graphs and equiangular lines [6, 10, 12].
Any representation of a graph in is by definition a Euclidean -distance set: a set of vectors such that only two nontrivial distances occur between vectors [2, 5]. Conversely, the distance relation of any Euclidean -distance set of size defines an -vertex graph. The problem of finding the largest -distance set (and its corresponding graph) in a given dimension has been well studied, but to our knowledge the issue in this paper — finding the smallest dimension for a given graph — has not previously been resolved. On the other hand, the main tool used in this paper, namely the Euclidean distance matrix, has been applied to -distance sets quite often (see for example Larman, Rogers, and Seidel ). Moreover, there is a related problem in graph network theory that has received considerable attention: given a graph, and specifying the length of each edge, find an embedding of the graph in . That problem is NP-hard .
Theorem 1 says in particular that a graph with all simple eigenvalues (eigenvalues with multiplicity ) has Euclidean representation number or . Since eigenvalues of a random graph are almost surely simple, Nguyen Van Thé’s upper bound of is often tight. (For the distribution of eigenvalues in random graphs see Chung, Lu & Vu .) In fact, Theorem 7 shows that almost surely, the representation number of a random graph is . It seems that only special graphs, such as line graphs or graphs with a high degree of regularity or symmetry, can be represented in smaller dimensions.
2 General graphs
We characterize representations of using Euclidean distance matrices. An matrix is a Euclidean distance matrix of a set of vectors in if the rows and columns of are indexed by the vectors, and . Thus every Euclidean distance matrix is nonnegative with zero diagonal.
Let denote the adjacency matrix of , let denote the all-ones matrix and let be the adjacency matrix of the complement of . Then up to scaling, the Euclidean distance matrix of a representation of has the form , for some and . The smallest dimension such that is the distance matrix of a set of points in is called the embedding dimension of or dimensionality of [14, 15]. Therefore the Euclidean representation number of is the smallest embedding dimension of a Euclidean distance matrix representing .
We will use two characterizations of Euclidean distance matrices: the first is due to Schoenberg . Let denote the all-ones vector and let , the projection matrix for the space .
Let be a symmetric matrix zero diagonal and positive off-diagonal entries. Then is a Euclidean distance matrix if and only if is negative semidefinite (that is, is negative semidefinite on ). The embedding dimension of is the rank of .
Let be a symmetric matrix zero diagonal and positive off-diagonal entries. For any real vector such that , let
Then is a Euclidean distance matrix if and only if is negative semidefinite. The embedding dimension of is the rank of .
In order to give a complete description of the representation number, we will also use the main angles [11, Chapter 5] of a graph . It will be convenient to order the distinct eigenvalues of from smallest to largest as . Given an eigenvalue , with an eigenspace and projection matrix onto that eigenspace, the main angle of is
Note that , and if and only if . In particular , as the largest eigenvalue has an eigenvector with nonnegative entries by the Perron-Frobenius Theorem (see [10, Theorem 0.2] or [16, Theorem 8.2.11]). Also note that if has vertices, then
If , then is called a main eigenvalue of .
We now consider Euclidean distance matrices of the form . If , then , which is not the distance matrix of a valid representation of a graph unless is the complete or empty graph. Therefore there are two separate cases: and . Without loss of generality, we may assume that , provided that we also consider Euclidean representations of the complement of .
Let be a graph which is not the disjoint union of complete graphs and let be the smallest eigenvalue of , with multiplicity and main value . If , then is a Euclidean distance matrix. The embedding dimension of is if and otherwise.
If and , then . Let
Since , has positive off-diagonal entries. Therefore is a Euclidean distance matrix if and only if is negative semidefinite. Instead of , consider
has largest eigenvalue . Therefore is negative semidefinite, so is also negative semidefinite. Thus is a Euclidean distance matrix.
By Theorem 2, the embedding dimension of is the rank of . Since is negative semidefinite, say , the rank of is the rank of . The null space of is , the eigenspace of . Therefore the null space of is . The dimension of this space is if and otherwise. ∎
Consider , the complement of the cycle graph on six vertices. The eigenvalues of this graph are , so . Moreover, is an eigenvector for the eigenvalue , so is contained in and . By Lemma 4, is a Euclidean distance matrix of embedding dimension . On the other hand, consider , the disjoint union of 2 paths of 3 vertices each. The eigenvalues are , so again . However, is not contained in , so and the embedding dimension of is .
In addition to , there is a second choice of that sometimes results in a distance matrix with small embedding dimension: .
Let denote the -th smallest distinct eigenvalue of , with multiplicity and main angle . Assume , and let and . Then is a Euclidean distance matrix if and only if , , and
Moreover, the embedding dimension of is either if equality holds in (2) and otherwise.
Let . Since is the -th smallest eigenvalue of , is the -th largest eigenvalue of . Let denote the eigenspace of .
Since and , we have for every in , with equality if and only if is in . If is a Euclidean distance matrix, then is negative semidefinite on . So, any satisfies and is therefore in . That is, . Since the dimension of is at least , it follows that . Therefore , , and . In other words, , , and .
Now, suppose , , and . Let be the eigenvalue in , normalized so that . By Theorem 3, is a Euclidean distance matrix if and only if
is negative semidefinite. Note that , and for any , . Therefore is suffices to consider if for of the form , with . Without loss of generality, assume and , so . Then
Using the main values of as in (1), the smallest value of () is . So, is negative semidefinite if and only if
is nonpositive. This quadratic optimization problem (see for example [28, Chapter 9]) obtains its maximum at , where is a normalization constant . Plugging this maximum into (3), we find that is a Euclidean distance matrix if and only if
from which (2) follows.
If equality holds in (2), then the choice of with equality satisfies and therefore . In this case, the null space of contains , , and , so the embedding dimension is . Otherwise, the null space of is and the embedding dimension is . ∎
Consider in Figure 1. The eigenvalues of are , , with multiplicities and main values respectively. It follows that inequality (2) holds without equality, so by Lemma 5, is a Euclidean distance matrix of embedding dimension . On the other hand, consider in Figure 1. The eigenvalues are , , with multiplicities and main values
respectively. Equality holds in (2), so is a Euclidean distance matrix of embedding dimension .
Let and let be a Euclidean distance matrix. If denotes the second smallest distinct eigenvalue of and , then . Moreover, the embedding dimension of is at least , where .
Let and . Since and are the smallest and second smallest eigenvalues of , and are largest and second largest eigenvalues of . Now is a Euclidean distance matrix, so is negative semidefinite on . Since is a space of dimension , it follows that can have at most one positive eigenvalue. Thus , which implies .
The embedding dimension of is the rank of . Since has rank , we see that and . ∎
We combine the results of Lemmas 4, 5 and 6 to find the choice of in that gives the best representation of . Nguyen Van Thé  showed that . Lemma 6 shows that in order of find a smaller representation, we must choose such that has less than full rank. But only if , where is some eigenvalue of . Moreover, by Lemma 6, is a Euclidean distance matrix with only if . So the only possible choices of are and . When , Lemma 4 shows that is a Euclidean distance matrix and the embedding dimension is either or . When , Lemma 5 shows that is sometimes a Euclidean distance matrix, with embedding dimension of is either or .
By considering both and its complement, this gives a complete description of Euclidean representation number of , which we now summarize as a theorem.
Recall that for any graph, the smallest eigenvalue is at most , with equality if and only if is a disjoint union of complete graphs. Now consider the largest component of such a disjoint union (that is, the complete subgraph with the most number of vertices). If has such components of largest size, then the largest eigenvalue of has multiplicity and the smallest eigenvalue of has multiplicity . Moreover, it is not difficult to verify that the eigenspace of is orthogonal to , so the Euclidean distance matrix given in Lemma 4 for the complement of has embedding dimension . This representation is optimal provided that . When is not a disjoint union of complete graphs, the optimal representation either has dimension or occurs when , , for either or its complement.
Let be a graph on vertices and edges. If or its complement is the complete graph, then
If or its complement is the disjoint union of at least complete graphs, of which are of largest size, then
Otherwise, let be the -th smallest distinct eigenvalue of , with multiplicity and main value . Define
Similarly define and for the complement of . Then
As an application of Theorem 7, consider line graphs. If is a graph with vertices and edges, then the line graph is the graph whose vertices are the edges of , with two edges adjacent if and only if they share a common vertex in . The line graph has vertices and has smallest eigenvalue , with multiplicity at least . (This result is due to Sachs ; see also [4, Theorem 3.8].) If follows that is representable in . More precisely, let denote the unoriented incidence matrix of (the matrix in which if vertex is incident with edge and otherwise). Then the adjacency matrix of is , so . Moreover, the rank of the incidence matrix of is easy compute: if denotes the number of connected components of that are bipartite, then the rank of is (see [12, Theorem 8.2.1]). Finally, the -eigenspace of is orthogonal to [10, Corollary to Theorem 3.38], so the representation number of is , provided that . We have:
Let be a graph with vertices, edges, and bipartite connected components. Then
If , then .
Theorem 7 gives the Euclidean representation number of in terms of the eigenspaces of and its complement. In general, there is a relationship between the eigenspace of for and the eigenspace of for : their dimensions differ by at most one (see  or [10, Theorem 2.5]). More generally, the spectrum of the complement of is determined by the eigenvalues, multiplicities, and main angles of . The following observation is due to Cvetkovic and Doob (, see also [11, Proposition 4.5.2]).
Let be a graph with distinct eigenvalues , multiplicities , and main angles , so the characteristic polynomial of is . Then
3 Regular graphs
A graph is regular if every vertex has the same degree (number of neighbours). For regular graphs, the formula for the Euclidean representation number of a graph in Theorem 7 depends only on the multiplicities of the eigenvalues rather than both the multiplicities and the main angles.
If is regular and the disjoint union of complete graphs, then . Here and its complement have representation number (each is represented in as a regular simplex, and each component is embedded into a distinct orthogonal subspace of .) Otherwise, assume is regular of degree . Then , so is an eigenvector of . It follows that the eigenspace of is contained in and . From Theorem 7, we see that the representation number any regular which is not a disjoint union of complete graphs or its complement is
Moreover, the eigenvalues of precisely determine the eigenvalues of . The following now standard result is due to Sachs . (The result follows from Lemma 9, but for a combinatorial proof, see [10, Theorem 2.6]).
Let be an -vertex -regular graph with (not necessarily distinct) eigenvalues , where . Then the eigenvalues of are ,,…,.
As before, denote the distinct eigenvalues of by with multiplicities respectively. If is connected, then the multiplicity of the smallest eigenvalue of is exactly , the multiplicity of the second largest eigenvalue of . If is disconnected, then , where is the multiplicity of and is also the number of components in . Combining these observations, we have:
Let be be a regular graph on vertices. If () or its complement, then
Otherwise, let and be the multiplicities of the smallest and second largest distinct eigenvalues of . If is connected, then
If is disconnected with components, then
For more information about the smallest and second largest eigenvalues, see the survey by Seidel .
By way of application, consider strongly regular graphs; for background, see , , or . A strongly regular graph with parameters is a graph with vertices and valency such that two distinct vertices have common neighbours if they are adjacent and common neighbours otherwise. Such a graph has only two eigenvalues other than , which have multiplicities
The only disconnected strongly regular graph is , which has representation number . Excluding and its complement, we have the following:
Let be a strongly-regular graph which is not a complete multipartite graph or its complement, and has parameters . Then
The Petersen graph has eigenvalues , , and , with multiplicities , , and respectively. Therefore it is representable in .
More generally, given the intersection array of a distance-regular graph , one may readily compute the eigenvalues and multiplicities of and use Theorem 11 to find the Euclidean representation number of . For example, , the cycle on vertices, is a distance-regular graph whose second largest eigenvalue has multiplicity . So for , is representable in .
As another application, consider the following theorem of Petersdorf and Sachs (, see also [4, Proposition 16.6]). A graph is vertex-transitive is its automorphism group acts transitively on the vertices.
Let be a vertex-transitive graph of degree and vertices, and let be a simple eigenvalue of . If is odd, then . If is even, then , where is an integer between and .
If is a vertex-transitive graph other than the complete or empty graph, and has an odd number of vertices , then is representable in .
In a similar vein, a graph is symmetric if for all vertices , if and , then there is a an automorphism of mapping to and to . Every symmetric graph is vertex-transitive. The following theorem due to Biggs [4, Proposition 16.7].
If is a symmetric graph of degree and is a simple eigenvalue of , then .
The eigenvalue occurs in a -regular graph if and only if the graph is bipartite.
If is a non-bipartite symmetric graph on vertices, and is not or , then is representable in .
The author would like to thank Lionel Nguyen Van Thé, Chris Godsil, and an anonymous referee for their helpful input. This research is supported by NSERC, MITACS, PIMS, iCORE, and the University of Calgary Department of Mathematics and Statistics Postdoctoral Program.
- A. Y. Alfakih, A. Khandani, and H. Wolkowicz, Solving Euclidean distance matrix completion problems via semidefinite programming, Comput. Optim. Appl., 12 (1999), 13–30.
- E. Bannai, E. Bannai, and D. Stanton, An upper bound for the cardinality of an -distance subset in real Euclidean space. II, Combinatorica, 3 (1983), 147–152.
- L. W. Beineke and R. J. Wilson, eds., Topics in algebraic graph theory, vol. 102 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2004.
- N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, second ed., 1993.
- A. Blokhuis, A new upper bound for the cardinality of -distance sets in Euclidean space, in Convexity and graph theory (Jerusalem, 1981), vol. 87 of North-Holland Math. Stud., North-Holland, Amsterdam, 1984, 65–66.
- P. J. Cameron and J. H. van Lint, Designs, graphs, codes and their links, vol. 22 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1991.
- F. Chung, L. Lu, and V. Vu, Spectra of random graphs with given expected degrees, Proc. Natl. Acad. Sci. USA, 100 (2003), 6313–6318 (electronic).
- D. Cvetković, Graphs and their spectra, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., (1971), 1–50.
- D. Cvetković and M. Doob, Developments in the theory of graph spectra, Linear and Multilinear Algebra, 18 (1985), 153–181.
- D. Cvetković, M. Doob, and H. Sachs, Spectra of graphs, Johann Ambrosius Barth, Heidelberg, third ed., 1995.
- D. Cvetković, P. Rowlinson, and S. Simić, Eigenspaces of graphs, Cambridge University Press, Cambridge, 1997.
- C. D. Godsil and G. F. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001.
- J. C. Gower, Euclidean distance geometry, Math. Sci., 7 (1982), 1–14.
- , Properties of Euclidean and non-Euclidean distance matrices, Linear Algebra Appl., 67 (1985), 81–97.
- T. L. Hayden, J. Wells, W. M. Liu, and P. Tarazaga, The cone of distance matrices, Linear Algebra Appl., 144 (1991), 153–169.
- R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985.
- D. G. Larman, C. A. Rogers, and J. J. Seidel, On two-distance sets in Euclidean space, Bull. London Math. Soc., 9 (1977), 261–267.
- P. Lisoněk, New maximal two-distance sets, J. Combin. Theory Ser. A, 77 (1997), 318–338.
- A. Neumaier, Distance matrices, dimension, and conference graphs, Nederl. Akad. Wetensch. Indag. Math., 43 (1981), 385–391.
- L. Nguyen Van Thé, On a problem of Specker about Euclidean representations of finite graphs, (2008).
- M. Petersdorf and H. Sachs, Spektrum und Automorphismengruppe eines Graphen, in Combinatorial Theory and its Applications, III (Proc. Colloq., Balatonfüred, 1969), North-Holland, Amsterdam, 1970, 891–907.
- M. Pouzet, Note sur le problème de Ulam, J. Combin. Theory Ser. B, 27 (1979), 231–236.
- H. Sachs, Über selbstkomplementäre Graphen, Publ. Math. Debrecen, 9 (1962), 270–288.
- H. Sachs, Über Teiler, Faktoren und charakteristische Polynome von Graphen. II, Wiss. Z. Techn. Hochsch. Ilmenau, 13 (1967), 405–412.
- J. B. Saxe, Embeddability of weighted graphs in k-space is strongly np-hard, in Proceedings of the Seventeenth Annual Allerton Conference on Communication, Control, and Computing held in Monticello, Ill., October 10–12, 1979, Urbana, 1979, University of Illinois Department of Electrical Engineering, 480–489.
- I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc., 44 (1938), 522–536.
- J. J. Seidel, Graphs and their spectra, in Combinatorics and graph theory (Warsaw, 1987), vol. 25 of Banach Center Publ., PWN, Warsaw, 1989, 147–162.
- D. A. Wismer and R. Chattergy, Introduction to nonlinear optimization: a problem solving approach, North-Holland Publishing Co., New York, 1978.