Some observations on the smallest adjacency eigenvalue of a graph

12/09/2019
by   Sebastian M. Cioabă, et al.
University of Delaware
0

In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds.

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1 Introduction

Our graph notation is standard, see [11] for undefined terms or notation. The eigenvalues of a graph are the eigenvalues of its adjacency matrix . For a graph with vertices and , denote by the -th greatest eigenvalue of and let be its -th smallest eigenvalue. Let denote the smallest eigenvalue . The smallest eigenvalue of a graph is closely related to its chromatic number and independence number [11, 22]. Since the spectrum of a connected graph is symmetric if and only if the graph is bipartite, it is natural to think of as a measure of how bipartite is. It is therefore not surprising that the smallest eigenvalue has close connections to the max-cut [5, 10, 24, 26]. There are several methods to obtain upper bounds for . Using Rayleigh quotients, it is well known that

(1)

Depending on the context, choosing appropriate vectors can yield useful upper bounds on

such as the ones involving the max-cut [5, 24] or Hoffman’s ratio bound on the independence number (see [11, Section 3.5] for example). The connection between eigenvalues and Rayleigh quotients also yields important interlacing results such as Cauchy interlacing or Haemers interlacing [11, Section 3.5]. In each case, the eigenvalues of a smaller matrix (principal submatrix of in the case of Cauchy interlacing or quotient matrix in the case of Haemers interlacing) interlace the eigenvalues of and therefore, is bounded from above by the smallest eigenvalue of this smaller matrix. Again, these important methods yield interesting consequences such Cvetkovic’s inertia bound for the independence number [11, Theorem 3.5.1] or Hoffman’s ratio bound for the chromatic number [11, Theorem 3.6.2] to name just a few. In other situations, manipulations of the trace of powers of the adjacency matrix of a graph (see [15] for example) or edge perturbations in graphs (see [6, 7]) can yield upper bounds for .

In this paper, we are interested in the finding lower bounds for the smallest eigenvalue using graph decompositions. We apply our methods to various situations and we describe their successes and limitations. In general, lower bounds on the smallest eigenvalue of a graph are not easy to obtain. In [5], Alon and Sudakov show that for a nonbipartite simple graph with maximum degree and diameter (see also [16, 32] for small improvements). Trevisan [34] obtained interesting connections between and the bipartiteness ratio which is defined as , where the minimum is taken over all subsets of and all partitions of , denotes the number of edges in the subgraph induced by (similar definition for ) and denotes the number of edges with exactly one endpoint in . Trevisan’s results are similar to the ones relating the second largest eigenvalue of a graph to its expansion/isoperimetric constant (see [1, 3, 34]) and for a -regular graph, give the following interesting lower bound: . However, we have not been able to use this bound for the graphs considered in this paper as the parameter does not seem easy to calculate.

In Section 2, we use weighted graph decompositions of the edge set of a graph to bound the spectrum of a graph from below. Our results are similar and have been obtained independently from the recent work of Knox and Mohar [27].

In Section 3, we specialize these decompositions to clique decompositions and give examples when the bounds are tight and when they are not. It is not surprising that for line graphs, generalized line graphs and point-line graphs of finite geometries, our bounds are tight, but there are many graphs where our methods do not yield tight bounds.

In Section 4, we discuss the smallest eigenvalue of -free graphs. Linial [30] asked whether the property of the eigenvalues of line graphs to be bounded from below by an absolute constant also holds for claw-free simple graphs. In [14], the first author showed that the answer is negative by describing a family of regular claw-free simple graphs with arbitrarily negative eigenvalues. Recently, motivated by problems in topological combinatorics, Aharoni, Alon and Berger [4] studied the largest eigenvalue of the Laplacian of -free graphs which when restricted to regular graphs, is equivalent to studying the smallest eigenvalue of regular -free graphs. In Section 4, we describe their results and remark that their proof actually gives a more general lower bound for the smallest eigenvalue of graphs with dense neighborhoods. In [4], the authors also constructed examples of -regular -free graphs with very negative by taking clique blowups of bipartite -regular graphs. Their construction works when for and . In the case of claw-free graphs (), their construction works for and . In this section, we also show that every cubic claw-free graph has which slightly improves the lower bound of from [4].

2 Smallest eigenvalue and graph decompositions

In this section, we introduce graph decompositions as a means of obtaining lower bounds on the least adjacency eigenvalue of a graph. Like many such general bounds, there are cases where the estimates are strong and others where they are weak. The advantage of using decompositions lies in the flexibility of choice: for graphs with many triangles, decompositions by complete graphs are often successful, while for graphs with few triangles, decompositions allowing paths and cycles may be more fruitful. Also, taking decompositions of an odd power of a graph can sometimes improve a lower bound.

We begin with a simple observation on matrix decompositions and then introduce weighted graphs to capture the generality of an arbitrary symmetric matrix. All matrices are assumed to be symmetric and have real entries.

A decomposition of a matrix of order is a collection of matrices such that

Using Rayleigh quotients, we quickly obtain a lower bound on the least eigenvalue of :

(2)

The support of a matrix is the set of all row indices of such that for some column index of . Arbitrary real symmetric matrices may be regarded as adjacency matrices of edge-weighted graphs and, in our application to graph decompositions, often have small support. In such cases, the simple estimate (2) can be improved. We require the following notation.

A weighted graph is a graph with vertex set together with a function that assigns a (possibly negative) real number to each unordered vertex pair . Also, if . This implies that if there is no loop on in .

We say that a set of weighted graphs is a decomposition of a weighted graph and write

if for and for all unordered pairs of vertices of . Here, we take if either or is not in . The (weighted) adjacency matrix of a weighted graph is the symmetric matrix , indexed by the vertices in , with entry equal to . Let denote the set of graphs in that contain vertex .

Theorem 2.1

Let be a decomposition of a weighted graph . For each vertex , let be the sum of the minimum eigenvalues of the graphs of that contain vertex . Then

(3)

Let . Then equality holds in (3) if and only if there is a vector of real numbers indexed by the vertices in such that

  1. whenever ; and,

  2. For each , the restriction of to

    is either a zero vector or an eigenvector of

    with eigenvalue .

Proof. Let be the adjacency matrix of and, for , let be the adjacency matrix of augmented by zero rows and columns indexed by vertices of not in . Because , we have . Let be the (0,1)-diagonal matrix with entry equal to 1 if vertex is in and 0, otherwise. The matrix is a diagonal matrix with entry . Let . Then is positive semidefinite because each of its summands is. Therefore, letting , it follows that the matrix

is positive semidefinite. Thus, with equality if and only if is an eigenvector of ; equivalently, with equality if and only if for some vector .

Since is a positive semidefinite matrix, for some matrix . Thus if and only if . Substituting for and recalling that each of its summands is positive semidefinite, it follows that if and only if and, for each , . The first condition holds if and only if whenever . Because the matrices involved are positive semidefinite, the second condition holds if and only if for .

The Cartesian product of the simple graphs , is the simple graph on the vertex set with two vertices adjacent if there is an index such that in and for all . Using the Hadamard product , we state the well-known fact that the adjacency matrix is the sum of the products of the form , where the -th term is a product of identity matrices (of orders equal to those of the corresponding graphs) together with in the -th position. From this it follows that if is an -eigenvector of , , then is an eigenvector of . As an illustration, in the following example, we also obtain the least eigenvalue of using Theorem 2.1.

Example 2.2

(Cartesian products and Hamming graphs) Let . Taking the induced subgraphs of on sets of vertices where all but one of the coordinates is fixed, we obtain a decomposition consisting of copies of , . Since each vertex of is contained in one copy of for each , we have by (3). However, finding the eigenvector that satisfies the conditions sufficient for equality in Theorem 2.1 would be difficult without appealing to the above form of the adjacency matrix for .

A Cartesian product of complete graphs, each of order at least 2, is called a Hamming graph. Thus, if is a Hamming graph whose vertices are -tuples, then .

We frequently require the following graphs in decompositions. The loop graph of order has a loop at each vertex and no other edges. We use the symbol

for both it and its adjacency matrix (the identity matrix of order

). The looped complete graph of order is obtained by adding a loop to each vertex of . We use the symbol for both it and its adjacency matrix (the all-one matrix). The simple complete graph of order has an edge between each pair of distinct vertices. We use the symbol to denote both it and its adjacency matrix, . The graph has no edges and is not used in decompositions. The graph is called a single loop and may appear in decompositions. Note that although when , we have .

For a graph and real number (possibly negative), we write for the weighted graph that has constant weight function on the edges (and 0 on the non-edges). In particular, each graph may be regarded as a weighted graph with edge weights all while is the graph with edge weights all . Thus, and . For , , but . Also, for all . Because for and , we have while .

The next three examples use a multigraph formed from a simple graph ; that is, a multigraph with adjacency matrix where is a positive integer. Thus, the number of edges in with endpoints is the number of -walks of length in . For example, if is a simple graph and are adjacent vertices in with degrees and neighbour sets then the number of edges in with endpoints equals if and equals the number of edges between and if . If is odd, the least eigenvalues of and are related by the equation .

Example 2.3

(The 5-cycle) If is the 5-cycle , then

Let . Then and so, by Theorem 2.1,

Thus, . In Example 2.5, we shall see that equality holds.

Example 2.4

(The Petersen graph) If is the Petersen graph, then

Let . Then and so, by Theorem 2.1,

Thus, . Also we see that , by noticing that the 6-cycle is an induced subgraph of .

Of course, the eigenvalues for the 5-cycle and for the Petersen graph, indeed, for any strongly regular graph , may be found immediately from the equation on the adjacency matrix (see [23, p.218]):

(4)

For, by multiplying (4) by an eigenvector orthogonal to it follows that the only eigenvalues of other than are the roots and of the quadratic equation

(5)

In particular, for the Petersen graph, , .

Example 2.5

(Strongly regular graphs ) For a strongly regular graph , taking in Theorem 2.1 often leads to the exact value of . To see this note that multiplying (4) by and substituting (4) for gives an equation of the form

for some nonnegative integers depending on . Thus

By Theorem 2.1, if , then

Thus, is at least as large as the minimum root of the cubic . Two of the roots are and , inherited from (5), and the other is necessarily since the coefficient of is 0. Thus, . Therefore, taking in Theorem 2.1 gives when is a strongly regular graph such that . In particular, when , a condition that must be satisfied by at least one of a strongly regular graph and its complement.

Example 2.6

(The dodecahedral graph ) Let be the plane graph whose vertices and edges are those of the dodecahedron. Then is 3-regular and the 20 face 5-cycles of constitute a decomposition of with precisely 3 cycles through each vertex. Thus by Theorem 2.1, . The exact value is .

If some weighted graph in a decomposition is disconnected, it is clear that replacing in by its set of weighted components cannot weaken the estimate in Theorem 2.1. Thus, there is no loss in restricting the weighted graphs in a decomposition to be connected. In Example 2.5, we could replace each loop graph by its separate individual loops .

By the type of a weighted graph, we mean its underlying unweighted graph. Each choice of types for the weighted graphs in in (3) leads to a lower bound on by maximizing over all weighted graphs of that type and so, when applied to a simple graph , yields a new graph parameter. Of course, equality holds if all types of weighted graph are allowed (take to be itself). The trick is to pick a family of graphs that are easy to deal with and that often yield good lower bounds in (3) when maximized over all weightings. The conditions for equality in Theorem 2.1 suggest that the bound (3) might often be best for decompositions of a weighted graph that employ weightings of connected graphs whose minimum eigenvalues have large multiplicity and small absolute value. Such are the simple complete graphs and looped complete graphs on subsets of . For when , has least eigenvalue with multiplicity , while has least eigenvalue with multiplicity . (In both cases, the eigenvectors associated with the minimum eigenvector are the nonzero vectors such that .) Because has no edges, it is never used in a decomposition. The graph will be called a loop. It may be used in a decomposition, noting carefully that its least eigenvalue is . We are therefore led to the following definition.

A complete graph decomposition of a weighted graph is a decomposition of consisting of scalar multiples of complete graphs, looped or simple. Because negative weights are allowed, cancellation may occur (as in Example 2.8), so the graphs or in need not be subgraphs of . Taking in Theorem 2.1, we obtain the following corollary.

Corollary 2.7

Let be a complete graph decomposition of a weighted graph and, for each vertex , let equal the sum of the minimum eigenvalues of the complete graphs in that contain the vertex . Then

(6)

Let . Then equality holds in (6) if and only if there is a vector of real numbers assigned to the vertices of such that

  1. whenever ;

  2. is constant on each vertex set for which and, for each vertex set of order greater than 1 for which .

Example 2.8

(Complete multipartite graphs ) Let be the complete multipartite graph with vertex parts of orders . Then

where is the looped complete graph of order and, for , is the negatively weighted looped complete graph on . Since and , we have by Corollary 2.7. Moreover, it is straightforward to check that conditions 1 and 2 imply that equality holds if and only if . Of course, using the characteristic polynomial of [18, p.74], it follows that has precisely negative eigenvalues and that they interlace . Also, has one positive eigenvalue (in fact, this characterizes the complete multipartite graphs [18, p.163]). All remaining eigenvalues are equal to 0.

If the values of the weight function of a weighted graph are nonnegative integers, then may be regarded as a multigraph with distinct unweighted edges between each unordered pair of vertices of . We call the multiplicity of . To emphasize this distinction, we use the notation for multigraphs and continue to use for graphs (simple or looped) and for weighted graphs.

Example 2.9

(Line graphs of multigraphs) Let be a multigraph with maximum edge multiplicity . The line graph of has the edges of as vertices. Two edge vertices of are adjacent if they have precisely one common end vertex in . Thus edges in with the same two endpoints are nonadjacent as vertices in . Note that is a simple graph. Also, if a loop at in is replaced by an edge with one end at and the other at an additional new vertex, then the line graph is not changed. Thus we may assume that has no loops.

The line graph has a natural decomposition into complete multipartite graphs. To see this, for each vertex of , let be the subgraph of induced by the claw at , that is by the edges incident to in . The subgraph of is a complete multipartite graph and, because has no loops, the part sizes of are equal to the multiplicities of the edges incident to in . Thus, by Example 2.8, . Because adjacent edge vertices of have precisely one vertex in common in , it follows that the graphs decompose . Also since each edge of has two distinct endpoints, each edge vertex of is in precisely two graphs of the decomposition. Thus, by Theorem 2.1, if is a loopless multigraph with maximum edge multiplicity , then

(7)

Equality can be attained in (7). To see this, note first that if is a simple graph and is the adjacency matrix of , then the adjacency matrix of is the Hadamard product . Thus, Therefore, if has maximum edge multiplicity and is an induced subgraph of with (see Example 2.10), then and so equality is attained in (7). We leave it as an problem to figure out if it is possible to characterize the multigraphs for which equality (7) is attained.

Example 2.10

(Twig replication and generalized line graphs ) There are interesting cases where the lower bound (7) can be improved. Suppose that a loopless multigraph is formed from a connected simple graph by optionally increasing the multiplicity of twigs of , that is, of edges of (if any) that have an end vertex of degree 1. Decompose by the complete multipartite graphs as in Example 2.9. If is a vertex of degree 1 in , then will have no edges in and may be omitted. Now further decompose each subgraph by graphs and of appropriate orders as in Example 2.8. Because of the construction, the vertex parts of size 2 or more that occur in the graphs will be vertex disjoint, and so, each vertex of will be in at most one graph of order 2 or more. Also, each edge vertex of will be in at most two graphs in the decomposition since it is in at most two ’s. Thus, by Theorem 2.1, if is a multigraph formed from a simple graph by replicating twigs, then

(8)

where is the maximum twig multiplicity in .

When , we have and where is the usual line graph of a simple graph . In this case, each vertex of is in precisely two complete graphs, , and the conditions for equality can be shown to imply a result of Doob (see, for example, [20, p.29]) which states that if and only if each component of is either a tree or is odd-unicyclic.

When , we again have . The graphs with are the generalized line graphs of Hoffman [25] (see also [18, 21] or [20, p.6]). The usual proofs that for a generalized line graph employ modifications of the vertex-edge incidence matrix of [25], [20, p.6].

Let be the best possible estimate of that can be obtained in (6); that is, let

where the supremum is taken over all complete graph decompositions of . To see that the supremum is attained, we show that

is the optimal value of a linear programming problem.

Let be the incidence matrix with rows indexed by all of the (unordered) vertex pairs and columns indexed by all complete graphs, looped and simple, with vertex sets contained in . (Note that because cancellation may occur, all complete graphs must be taken, whether or not they are subgraphs of .) Let be the weight vector determined by the given weight function on ; that is, for each unordered vertex pair . Then a vector indexed by the complete graphs specifies a complete graph decomposition of with weights if and only if . Now let be incidence matrix with rows indexed by the vertices and columns by the complete graphs and let be the diagonal matrix with diagonal entries equal to the minimum eigenvalues of all of the complete graphs with vertex sets contained in . Then is the smallest number such that . Thus is the optimal value of the following linear programming problem in the variables , .

Minimize
Subject to (9)

Thus, is attained for some complete graph decomposition of . Moreover, if is rational valued, then the optimal value is rational and an optimal vector giving equality in (9) may be chosen to have rational entries. Consequently, there is a positive integer such that is an integer and has integer entries. When is rational valued, this observation allows us to work with decompositions consisting of integer multiples of complete graphs, looped or simple, as long as multiples of the weighted graph are employed.

Thus, for a simple graph , the graph parameter has the following equivalent definition:

where the minimum is taken over all positive integers and all decompositions of by integer multiples (positive or negative) of complete graphs or .

It is perhaps impossible to classify the simple graphs

for which the parameter equals the least eigenvalue but there are a few simple observations that limit the graphs for which equality holds. Because the characteristic polynomial of a graph (or multigraph ) is monic with integer coefficients, every rational root is an integer. Therefore, if the rational number is not an integer, then . Also, if happens to be irrational (as for example, for the 5-cycle), then .

3 Smallest eigenvalue and clique partitions

In this section, we further restrict the type of decompositions in (2.7) to a special type that often appear in the literature, clique partitions.

A clique in a multigraph is a simple complete subgraph. A clique partition of a (necessarily loopless) multigraph is a collection of cliques of whose edge-sets partition the edge-set of . Here we do not weight the cliques, but may take the same clique more than once. Consequently, because all of the cliques in a clique partition have least eigenvalue , for each vertex of , we have the convenient expressions

(10)

where is the number of cliques in that contain the vertex and . Thus,

(11)

We are mainly interested in graphs that are simple. Because we are now only allowing copies of cliques in our partitions, taking scalar multiples can sometimes improve our bound on . We have the following corollary to Theorem 2.1.

Corollary 3.1

Let be a clique partition of a multiple of a simple graph . Then

(12)

with equality if and only if there is a vector of real numbers assigned to the vertices of such that

(13)

The conditions (13) for equality in Corollary 3.1 may be restated in a convenient matrix form. If is a clique partition of a multiple of a simple graph , let be the vertex-clique incidence matrix of with rows indexed by the vertices of and columns by the cliques in . Thus, the -entry of is if and is zero otherwise. Then equality holds in (12) if and only if there is a vector indexed by the vertices of such that and whenever .

Example 3.2

(Line graphs) It is an immediate consequence of Corollary 3.1 that if a simple graph can be edge-partitioned by simple cliques so that each vertex is in at most two of the cliques, then . But we have already encountered these graphs in Example 2.9: by a result of J. Krausz [28], they are precisely the line graphs of simple graphs.

Example 3.3

(Partial geometries) A partial geometry is an incidence structure of points and lines such that any two points are incident with at most one line, every line has points, every point is on lines, and for any line and any point , there are exactly lines through that intersect . Partial geometries were introduced by Bose [9] along with strongly regular graphs. The point graph of a partial geometry is the graph whose vertices are the points of the geometry where two vertices/points are adjacent if there is a line that contains them. It is known (see [35, Problem 21H]) that the point graph of a partial geometry is a strongly regular graph with smallest eigenvalue . Note that the edge set of this graph can be partitioned into cliques (corresponding to the lines of the geometry) such that each vertex is contained in exactly cliques. Corollary 3.1 with implies that the smallest eigenvalue of this graph is at least which is tight. The point graphs of partial geometries also appear in [17] where it is proved that certain random walks on them mix faster than the non-backtracking walks considered in [2].

Example 3.4

(The Johnson graphs) Let be positive integers with . The Johnson graph has the -subsets of a -set as vertices with adjacent if . If is a -subset of , then the set of all -subsets of that contain is the vertex set of a clique in . Each pair of adjacent vertices is in precisely one such clique, the clique . Thus, the family is a clique partition of . Also, for each . By Corollary 3.1, . Moreover, a nonzero vector satisfies the conditions for equality if and only if for each clique , . This is a system of homogeneous linear equations in variables and so has a nontrivial solution. Thus (with multiplicity , since the constraints can be shown to be linearly independent). There are explicit formulas for all of the eigenvalues and multiplicities of the relation graphs of the Johnson schemes and, in particular, for the Johnson graphs [35, p.413].

Corollary 3.1 leads us to a graph parameter based on clique partitions. For a simple graph , let be the best possible estimate of that can be obtained using clique partitions of scalar multiples of ; that is, let

(14)

where the maximum is taken over all positive integers and all clique partitions of . Then

(15)

As in (9) with , a linear programming problem shows that is attained by some and is rational.

Remark 3.5

As with the equality , it may be impossible to classify the simple graphs for which , but there are conditions that restrict the possible simple graphs. Again, because rational roots of monic polynomials are integers, must be an integer if . We also note that we may as well restrict our attention to simple graphs that contain triangles, . For if is free, then the only cliques in are edges and it follows that , the maximum vertex degree in . Thus, if is connected and triangle free, if and only if is a regular bipartite graph. (This can be seen by standard results, or from Remark 3.7 below.)

Another limitation on the equality follows by noting that conditions (13) for equality in Corollary 3.1 can sometimes be extended if for some vertex . Let and let be a vector satisfying conditions (13). If , stop. If , then for all . There may now be a clique that meets in only one vertex , say. Then since for all and . Let be the set of vertices obtained by deleting all vertices for which there is a clique that meets only in . Then for all . Repeat this last step. That is, given , let

Eventually, we obtain a set (possibly empty) such that each clique in is either disjoint from or else meets in two or more vertices. We call the vertices in the -essential vertices. Note that if satisfies conditions (13), then for all . Also, because each clique in is either disjoint from or meets in two or more vertices, if , we must have for each vertex . Thus,

(16)

We now have the following result.

Lemma 3.6

Let be a simple graph with vertex set . Let be a clique partition of and let be the set of -essential vertices. Then

where and is the set of nonempty restrictions of cliques in to . Moreover, if , then and for all .

Proof. Suppose that . Then there is a vector satisfying conditions (13) for equality in Corollary 3.1. Thus is nonempty, otherwise the observations above imply that for all , a contradiction. Let be the restriction of to . Because and for , we have and for each restricted clique . Also, by 16, for . Thus, is a nonzero vector that satisfies the conditions (13) for equality for the clique partition of . Therefore, . Also, since .

Suppose now that . By Corollary 3.1 and (16), Thus, . The reverse inequality holds since is an induced subgraph of . Thus, .

Remark 3.7

For Lemma 3.6 to hold, it is necessary that the clique partition of be obtained by the restrictions of the cliques in . In particular, if there is a clique partition of with , then and so .

In the special case that each clique in is a single edge (in particular, if is bipartite), it follows from Lemma 3.6 and the conditions (13) that if and only if is regular and some component is bipartite. The key observation needed here is that if for each edge in , then the set of vertices is the vertex set of a union of connected components of and the two subsets , are a bipartition.

Example 3.8

Let be a simple -regular bipartite graph and let be a set of vertices disjoint from . Let be a simple graph obtained from by replacing some (or all) of the edges of by cliques in that meet in the vertices only so that:

  1. Each vertex of is in at most of the cliques.

  2. Each pair of distinct vertices in is in at most one clique.

Let be the cliques that replaced the edges together with the edges of that were not replaced. Then , is the edge set of and since .

In Lemma 3.6, we observed that if the set of -essential vertices is nonempty and is the set of nonempty restrictions of cliques in to , then for each vertex . Therefore, when searching for simple graphs for which , we may focus our attention on simple graphs such that some multiple has a clique partition for which is constant and .

Lemma 3.9

Let be a simple graph with maximum vertex degree . If is a clique partition of and is the smallest of the orders of the cliques in , then