Arc-Transitive Graphs. Chris Godsil, Gordon Royle. Pages PDF · Generalized Polygons and Moore Graphs. Chris Godsil, Gordon Royle. Pages 77 ALGEBRAIC GRAPH THEORY. Second Edition. NORMAN BIGGS. London School of Economics. CAMBRIDGE. EREN UNIVERSITY PRESS. The rapidly expanding area of algebraic graph theory uses two algebra to explore various aspects of graph theory: linear algebra (for.
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This booklet is the collection of abstracts for the Algebraic Graph Theory meeting to be held in Dubrovnik, June , Apart from the rapidly developing field. Algebraic Graph Theory. Chris Godsil & Gordon Royle. During the course of this semester I have read chapters (1), (2), 3, 4, 5, (8),. 12, 13 and. An Introduction to Algebraic Graph Theory. Rob Beezer [email protected] Department of Mathematics and Computer Science. University of.
The chromatic polynomials of all complete multipartite graphs can be found in this way. The smallest graph which is quasi-separable but not separable is shown in Pig. Since T con- tains this complete graph, T has no vertex-colouring with fewer than t colours, and so U t is a factor of C T; u.
For instance, the chromatic polynomial of the graph shown in Fig. The chromatic polynomials of these graphs are: Edge-subgraph expansions The methods developed in the previous chapter for the calcu- lation of chromatic polynomials are of two kinds. The method of deletion and contraction is a simple, if tedious, technique whereby the chromatic polynomial of any given graph can be calculated in a finite number of steps. The other two methods Corollary 9.
I n this chapter we shall introduce a technique whose immediate importance is more theoretical than practical: I n later chapters it will appear that this technique can be refined in such a way as to make it practicable in specific cases.
We begin with a definition which will be used both in this chapter, and in Chapter The chromatic polynomial is a partial evaluation of the rank polynomial. In our proof of this fact we shall develop a general form of expansion which has other interesting consequences.
Let [u]x denote the set of all functions g: The following definition will provide the means of relating the rank polynomial and the chromatic polynomial. The result follows from Proposition 9. Then Lemma We shall apply this procedure without comment in the following exposition. The chromatic polynomial C T;u can be expanded in terms of the weights of the edge-subgraphs of T, as follows: The last step is the expansion of the product in powers of t.
We now consider the double summation. The connection between the chromatic polynomial and the rank polynomial given by Corollary Using Corollary Expressing the coefficients of the chromatic polynomial of K3 3 in terms of the alternating sums of the rows of the rank matrix we find cf.
Further, the only edge-subgraphs of T with rank g — 1 are the forests with g—1 edges these have co-rank zero , and the 7 circuits with g edges these have co-rank 1. To do this, we employ a reduction of Corollary This ordering is to remain fixed throughout our discussion. The next proposition expresses the coefficients of the chromatic polynomial in terms of those edge-subgraphs which contain no broken circuits; clearly such subgraphs contain no circuits, and so they are forests.
Proof Suppose BvB2, Then a partition of the set of all edge-subgraphs of F. We show that, in the expression the total contribution from 2 1?
Thus Whitney's ordering of broken circuits is not the dictionary ordering: Whitney's proof is given wrongly in Ore's book on graph theory, for there the last edge is removed, but the ordering of broken circuits is lexicographic. The present treatment, removing the first edge, seems more natural, and we shall make a similar modification in Chapter They have the same rank, but their co-ranks differ by one, and so their contributions to the alternating sum cancel.
Consequently, we need only consider the contribution of 2 0 to 2 — l jPij.
Then the coefficients ofC T; u alternate in sign. Proof The characterization of Proposition Let us order the edges of F so that the edges in F come first. Thus we have pro- duced the necessary edge-subgraphs, and the result follows. Thus the coefficients of the chromatic polynomial alternate strictly, and then become zero.
Then we have: Edge-subgraph expansions 71 c The chromatic polynomial may be expanded as follows: Nagle The logarithmic transformation In this chapter and the next one we shall investigate expansions of the chromatic polynomial which involve relatively few sub- graphs in comparison with the expansion of Chapter The present chapter is devoted to a transformation of the general expansion of Theorem The original idea appears in the work of Whitney , and it has been independently rediscovered by several people, including Tutte andagroupof physicists who interpret the method as a 'linked-cluster expansion' Baker The simplified version given here is based on a paper by the present author Biggs We begin with some remarks about separable and non- separable graphs.
Let T be a separable graph in which as in Definition 9. I t follows t h a t every general graph is the union of its maximal non-separable edge-subgraphs, called blocks, together with a set possibly empty of isolated vertices, which are sometimes called degenerate blocks.
We shall make the convention that a degenerate block is not a block, although the opposite convention is equally tenable. Throughout the rest of this chapter, pointed brackets refer to edge-subgraphs, unless the contrary is explicitly stated. We shall set up our theory in a general framework. Let Y be a function defined for all general graphs, taking positive real values, and having the following properties: Suppose that Y has no isolated vertices and is separable.
Hence the entire sum above is zero, and we have the result. Finally, a separable edge-subgraph either has no edges, or it has no isolated vertices; consequently Proposition Then, by Theorem In the case of the additive expansion of Theorem Remarkably, when the logarithmic trans- formation is applied to this pair X, Y the resulting multiplica- tive expansion is independent of t. This we now prove. If Y is a graph with more than one edge, then Y Y is independent oft.
That part of this expression which depends apparently on t, is: We now apply the result of Theorem Thus the terms in t cancel, and we have the result. We give only one example, as these functions will be superseded in the next chapter.
We shall also show how to overcome an apparent circularity in the use of Proposition I t will be shown that this seemingly fundamental objection can be surmounted by means of a few simple observations.
The vertex-subgraph expansion The multiplicative expansion of the chromatic polynomial can be modified in such a way that the edge-subgraphs are replaced by vertex-subgraphs. This procedure has two advantages. First, the number of vertex-subgraphs is usually much smaller than the number of edge-subgraphs; and second, the factors which replace the q factors in the notation of Proposition The result of Proposition Accordingly we can assume that the variable u is a complex variable, with the usual convention that the poles of rational functions need not be mentioned explicitly.
The formal details of the transition to vertex-subgraphs are quite straightforward. Proof The factors which appear in Proposition This grouping of factors corresponds precisely to that given in the definition of Q and the resulting expression for C. I n order to prove this we must look again at the machinery of Chapter Recall that q was defined to be the function Y of the pair X, T , related by means of the logarithmic transformation to the following pair X, Y: We can eliminate the logarithmic and exponential functions from the definition of Y, writing Y directly in terms of X: I n fact, it is possible to be more precise: The only satisfactory proof is given by Tutte , and it involves the application of 'tree mappings' to the product formula for q.
As the exposition of this concept would take up a great deal of time and space, we shall refer the reader to Tutte's paper for details. Accepting this result, we can prove the same thing for Q. Proof The function Q is defined to be the product of func- tions q, over a set of graphs with the same number of vertices. Thus the result for q implies the result for Q. I t is possible, using Proposi- tion But, from Proposition Thus the polynomial part of P u is a correct expression for C T; u , except for the coeffi- cients of u and 1.
An example will elucidate this argument. Precisely, suppose A1? Proof We know the C and Q function for all the graphs with at most three vertices. Thus, recursive use of this procedure leads eventually to the chromatic polynomial of T.
The C and Q functions for these graphs are: This means that we could have ignored A 5 completely, both in setting up the matrix N and in the subsequent calculations. Proof We prove this result by induction on the number of vertices of P. The result is true for all quasi-separable graphs with at most four vertices.
For this set contains only one graph the graph of Pig. Now, the expansion of Proposition The result follows.
We now state, as a theorem, the culmination of the theory of Chapters Proof This theorem is the conjunction of Propositions We shall illustrate this idea by reference to the infinite square lattice graph [W, p.
I n this case we choose Y s to be a square portion of Y with s 2 vertices, so that Y s h a s approximately 2s2 edges.
The only other vertex-subgraphs of Y s which are not quasi-separable and have fewer than 9 vertices are the copies of G4 approximately s2 in number and the copies of 0 8 also approximately s 2 in number.
Our approximation gives 1. The Tutte polynomial An expansion of a kind completely different from t h a t discussed in the preceding chapters is t h a t which relates the rank poly- nomial to the spanning trees of a graph. This expansion is the subject of the present chapter. The definition of the rank polynomial depends upon the assign- ment of the ordered pair rank, co-rank of non-negative integers to each edge-subgraph; we shall call such an assignment a bigrading of the set of edge-subgraphs.
If T is connected, the set of edge-subgraphs whose bigrading is r T , 0 is just the set of spanning trees of T. We shall introduce a new bigrading of edge- subgraphs, which has the remarkable property that, if it is given only for the spanning trees of T, then the entire rank polynomial of r is determined.
Our procedure is based initially upon an ordering of the edge-set EY, although a consequence of our main result is the fact t h a t this arbitrary ordering is essentially irrelevant.
Another consequence of the main result is an expansion of the chromatic polynomial in terms of spanning trees; this will be the subject of Chapter We now fix some hypotheses and conventions which will remain in force throughout this chapter. I n what follows we shall exploit the obvious similarity between the two operators by giving proofs only for one of them.
Our first lemma says that the edges which must be added to an edge- subgraph A to get Ax, can be added in any order. Suppose A. Proof Since r i? Suppose X B is not in Ax. This is a contradiction, so our hypothesis was false and A B is in Ax. Our next definition introduces a new bigrading of the subsets of E. The relationship between this bigrading and the A and ju, operators will lead to our main result.
The number of edges which are externally internally active with respect to X is called the external internal activity of X. These concepts may be clarified by studying them in the case of a spanning tree; in that case they are related to the systems of basic circuits and cutsets which we introduced in Chapter 5.
Proof 1 By definition, e is externally active if and only if e is the first edge whose removal decreases the co-rank of T U e. But T U e contains just one circuit, which is cir J7, e , and any edge whose removal decreases the co-rank must belong to this circuit. Remarkably, it will appear as a consequence of our main theorem t h a t T is independent of the chosen ordering. We shall investigate the relationship between the concepts just defined and the following diagram of operators: D We note the analogous result: Proof Suppose t h a t the edge e is externally active with respect to T.
We shall show that the whole of cir T, e belongs to W, whence it follows that e is externally active with respect to W. This contradicts the externally active property of e. Hence cir T, e c W, and e is externally active with respect to W.
Conversely, if e is externally active with respect to W it follows immediately that e is externally active with respect to T. The last definition occurs in Definition Then the Tutte polynomial is related to the modified rank polynomial: By Proposition These subgraphs are obtained by removing from T any set of i edges contained in the k internally active edges of T.
This proves the first identity. For the second identity, we consider ft: By the analogue of Proposition This proves the second identity. Proof This statement follows from Theorem The original proof of Theorem The coefficients av The chromatic polynomial and spanning trees I n this chapter we shall apply the main result of Chapter 13 to the study of the chromatic polynomial of a connected graph. Proof We have only to invoke some identities derived in earlier chapters. The chromatic polynomial is related to the rank polynomial by Corollary The result now follows from the definition of the Tutte polynomial.
For if we are given the incidence matrix of a graph T, then the basic circuits and cutsets associated with each spanning tree T of V can be found by matrix operations, as explained in Chapter 5.
From this information we can compute the internal and external activities of T, using the results of Proposition The method is impracticable for hand calcula- tion, but it is well-adapted to automatic computation in view of the availability of sophisticated programs for carrying out matrix algebra.
Further, e2 is not an edge of T, otherwise both ex and e2 would be internally active; also e1 e cir T, e2 , otherwise e2 would be externally active. I t is suffi- ciently important to warrant a name. The number 6 Y is related to the chromatic polynomial of T, as a consequence of Theorem This relationship can also be used to justify the use of the name 'invariant J for d T. There is strong numerical evidence to support this conjecture, but a proof seems surprisingly elusive.
We do have the following partial result. Supposing that V has n vertices and m edges, we have: We shall take for granted the ele- mentary facts about automorphisms [B, p. This group is called the automorphism group of r , and is written G T.
The study of automorphisms of graphs illuminates both graph theory and group theory. If a graph has a high degree of sym- metry then we may hope to use this fact in the investigation of its graph-theoretical properties; conversely, these properties may be used to determine the structure of its group of automorphisms.
We say t h a t V is vertex-transitive if 0 T acts transitively on VF. Our first proposition is typical of the kind of result which follows directly from the definitions. Since T is edge- transitive and connected any vertex of T belongs either to U or to V. If there is some vertex in both U and V, then V would be vertex-transitive; consequently V n V is empty and every edge of F has one end in U and one end in V. That is, V is bipartite. There are also regular graphs which are edge-transitive but not vertex- transitive, although examples of this type are by no means easy to find Bouwer I t enables us to draw very strong conclusions about the spectra of graphs with 'large' automorphism groups.
These conclusions can then be applied, with the results of earlier chapters, to the solution of graph-theoretical problems. We shall formulate a result of this kind which is a consequence of the following lemma. Proof We can choose x to be a real vector. Thus P x is an eigenvector of A corresponding to the eigen- value A. Proof Suppose that each eigenvalue of Y has multiplicity one. Then, by Lemma Thus, if all the eigenvalues are simple, every non-identity automorphism has order two.
Theorem I n Chapter 20 we shall study the remarkable properties of graphs for which the lower bound is attained. I n its most general form, the question of a relationship between the structure of a graph and the structure of its group of automorphisms has a rather disappointing answer: Frucht was the first to prove that, for every abstract finite group G, there is some graph Y whose auto- morphism group is abstractly isomorphic to G.
His work has been extended by several authors to show t h a t the conclusion remains true, even if we specify in advance t h a t Y must satisfy a number of graph-theoretical conditions. However, if we strengthen the question by asking whether every permutation group is equivalent [B, p.
The example we shall give is of a transitive permutation group which is not equivalent to the group of any vertex-transitive graph. We now introduce some conditions whose implications will be investigated in subsequent chapters. I t should be remarked that the second condition applies only to connected graphs, and t h a t in the case of a connected graph we have a hierarchy of conditions: We shall investigate these conditions in turn, beginning with the weakest one.
Then there are infinitely many graphs V with the properties: General properties of graph automorphisms We have: A homogeneous graph is a distance-transitive graph with diameter at most 2; con- sequently its girth is at most 5. The only known graphs of girth three which are homogeneous are: Vertex-transitive graphs In this chapter we study graphs Y for which the automorphism group G T acts transitively on VY.
We recall the standard results [B, Chapter 1] on transitive permutation groups, among them the following. Then all stabilizer subgroups Gv veVY are conjugate in G9 and con- sequently isomorphic. I n this case, the order of G is equal to the number of vertices. There is a standard construction, due originally to Cayley , which enables us to construct many, but not all, vertex- transitive graphs. We give a streamlined version which has proved to be well-adapted to the needs of algebraic graph theory.
Let G be any abstract finite group, with identity 1; and suppose Q is a set of generators for G, with the properties: Simple verifications show t h a t EY is well-defined, and t h a t r 6r,Q is a connected graph. Then n, regarded as a permutation of the vertices of Y G,Q 9 is a graph automorphism fixing the vertex 1. The set of all g geG constitutes a group 0 isomorphic with G , which is a subgroup of the full group of automorphisms of T G, Q , and acts transitively on the vertices. Since every group automorphism of S3 fixes Q setwise, it follows t h a t the stabilizer of the vertex 1 has order at least 6.
In fact, the order of the stabilizer is Not every vertex-transitive graph is a Cayley graph; the simplest counter-example is Petersen's graph. However, a slightly more general construction will yield all vertex-transitive graphs. There is, in addition, a similar construction which uses, instead of an abstract group and a generating set, a permuta- tion group and one of its orbits [B, p. We begin our study of the hierarchy of symmetry conditions with the minimal case: The next definition will facilitate discussion of this case.
We can easily show that the group Ss admits no graphical regular representation. For, if there were an admissible graph T, then it would be a Cayley graph T 83, O. This example suffices to show that not every permutation group is equivalent to the group of a graph.
Vertex4ransitive graphs I n the case of transitive Abelian groups, precise information is provided by the next proposition. Proof Any transitive Abelian group G is regular. If this automorphism were non-trivial, then part 2 of Proposition We now turn to a discussion of some simple spectral properties of vertex-transitive graphs.
A vertex-transitive graph Y is neces- sarily a regular graph, and so its spectrum has the properties which are stated in Proposition 3.
We can use the vertex-transitivity property to characterize all the simple eigenvalues of T. Proof Let x be a real eigenvector corresponding to the simple eigenvalue A, and let P be a permutation matrix representing an automorphism n of T.
Since Y is vertex-transitive, we deduce t h a t all the entries of x have the same absolute value. This is impossible for an odd number of summands of equal absolute value, and so our first statement is proved. This statement is false if we assume merely that the graph is regular of valency 3.
If we strengthen our assumptions by postulating t h a t V is symmetric, then the simple eigenvalues are restricted still further. Proof We continue with the notation of the previous proof. A vertex-transitive graph with a prime number of vertices must be a circulant graph. The graph F G, H, O is connected and vertex-transitive. Symmetric graphs The condition of vertex-transitivity is not a very powerful one, as is demonstrated by the fact that we can construct at least one vertex-transitive graph from each finite group, by means of the Cayley graph construction.
The condition t h a t a graph be sym- metric is apparently only slightly stronger, for a vertex-transitive graph is symmetric if and only if each vertex-stabilizer Gv is transitive on the set of vertices adjacent to v. Both these graphs are vertex-transitive, but only K3 3 is symmetric. I t is strange that symmetric graphs do have distinctive properties which are not shared by all vertex-transitive graphs. This was first demonstrated by Tutte in the case of trivalent graphs.
I t is sometimes convenient to regard a single vertex v as a 0-arc [v]. The automorphism group G F is transitive on 1-arcs if and only if r is symmetric since a 1-arc is just a pair of adjacent vertices.
The only connected graph of valency one is K2, and this graph is 1-transitive. From now on, we shall usually assume that the graphs under consideration have valency not less than three. For such graphs we have the following elementary inequality. Proof r contains a circuit of length g and it is easy to see a non-closed path of length g. I t is helpful to think of the operation of taking a successor of [a] in terms of' shunting' [a] through one step in F.
Suppose we ask whether repeated shunting will transform one s-arc into any other. If there are vertices of valency one in F then our shunting might be halted in a 'siding', while if all vertices have valency two we cannot reverse the direction of our 'train'. However, if each vertex of F has valency not less than three, and F is con- nected, then our intuition is correct and the shunting procedure is universally adequate.
Proof The proof of this is not deep, but it does require careful examination of several different cases. We refer the reader to the book by Tutte , pp. That is,  is also in the orbit of [a] under H. Now Lemma The girth of 93 is five, so t h a t Proposition Conversely, both Yi and F0 are contained in Fi9 so we have the result. The following proposition shows that the answer is 'yes', with an important exception.
The exception is in the case of bipartite graphs. We say that this subgroup preserves the bipartition. U contains all vertices whose distance from oc0 is two, and consequently all vertices whose distance from oc0 is even. Thus the girth constraint in Proposition I n the next chapter we shall specialize the results of Proposi- tions Then an automorphism of T which fixes [a] must be the identity.
Consequently we know the orders of all the groups in the sequence: The structure of these groups can be elucidated by investigating certain sets of generators for them. These generators are derived from the sets Yi denned for the general case in Chapter We note, as a consequence of Proposition We shall need the following notation: For the rest of this chapter, X will denote the subgroup of 6? Proof We shall use the notation and results of Propositions Another simple example is the 2-transitive graph Q3 depicted in Fig.
The proof of this very important result is due to Tutte , with later improvements by Sims and Djokovic The result is a purely algebraic consequence of the presentation of the stabilizer sequence given in Proposition Let A denote the largest natural number such that Fx is Abelian. LEMMA Then by part 3 of Lemma Hence we may argue as follows: Proof If t is at least four, then Proposition However, the results of Lemma I t remains to exclude the possibility t — 1, which is done by means of the following special argument.
Also, by part 3 of Lemma That is, x0 fixes x5 as. Consequently x0 fixes the whole 7-arc , and this contradicts Proposition Join two vertices by an edge if and only if the corre- sponding permutations have different shape and they commute.
For instance, ab is joined to ab cd ef , ab ce df and ab cf de , while ab cd ef is joined to ab , cd and ef. Then the generators for the stabilizer sequence may be chosen as follows: The vertices correspond to the triangles in PG 2, 3 and two vertices are adjacent whenever the corresponding triangles have one common point and their remaining four points are distinct and collinear.
I n each case the group acts primitively on the vertices Wong The abstract groups are Wong The covering-graph construction I n this short chapter we shall study a technique which enables us to construct a new symmetric graph from a given one.
A particu- lar form of this construction is sufficient to imply the existence of infinitely many connected trivalent 5-transitive graphs.
For any group K, a K-chain on T is a function j: I t is easy to check that the definition of adjacency depends only on the unordered pair of vertices. The covering graph T Z2, j is depicted in Fig. This is a group whose elements are the ordered pairs K, g in K x 6? A straightforward calculation, using the definition of com- patibility, shows that this permutation of VT is an auto- morphism of T.
Then [K] G is transitive on the t-arcs of Y. Proof JLet K0,V0 , I n fact, for a given graph Y and group K, it is very likely that the only covering graph is the trivial one con- sisting of if components each isomorphic with Y. However, it is possible to choose K depending on Y in such a way that a non- trivial covering graph always exists. Proof Pick a vertex v of Y, and let r o denote the component of r which contains the vertex l,v.
Conversely, the vertex K, v is in r o only if K represents the edges of a circuit in Y. Proof We know t h a t there is at least one connected trivalent 5-transitive graph O it was constructed at the end of the previous chapter.
Applying the construction of Theorem We may repeat this process as often as we please, obtaining an infinite sequence of graphs with the required properties. This graph represents the configuration of Desargues's theorem in projective geometry Coxeter This is the most restrictive of our hierarchy of sym- metry conditions, and it has the most interesting consequences.
I t is helpful to recast the definition. For any connected graph r , and each v in VF we define r. Small graphs may be depicted in a manner which emphasizes this partition by arranging their vertices in columns, according to distance from an arbitrary vertex v.
For example, K3 3 is displayed in this way in Fig. Proof Let us suppose t h a t F is distance-transitive, and use the notation of the first sentence of this chapter. The converse is proved by reversing the argument.
Fortunately, there are many identities involving the intersection numbers, and we shall find that just 2d of them are sufficient to determine the rest. These numbers have the following simple interpretation in terms of the diagrammatic representation of T introduced at the beginning of this chapter.
U0 bx For example, the cube Qz is a distance-transitive graph with diameter 3; from the representation in Fig. Further, since each column of the intersection array sums to k, if we are given the first and third rows we can calculate the middle row.
However, the original notation of Definition Then we have the following equations and inequalities: Pick a path v, , We shall obtain much more restrictive conditions in the next chapter. In fact, this situation is encountered in important applications, and we are justified in making the following definition. We shall now construct a simple basis for the adjacency algebra of a distance-regular graph.
Thus AAJ r s is equal to the r, s -entry of the matrix on the right-hand side. The form of the recursion shows that the degree of pi is at most i, and since A 0 , A1? However, Proposition 2. These eigenvalues, and the strange story of their multiplicities, form the subject of the next chapter. Further, the coefficient tMj is just the intersection number shij.
Concurrently, the work of Schur and Wielandt on the commuting algebra, or centralizer ring, of a permu- tation group, culminated in the paper of Higman which employs graph-theoretic ideas very closely related to those of this chapter.
The connection between the theory of the com- muting algebra and distance-transitive graphs is also treated in [B, pp. We end this chapter by remarking that several well-known families of graphs have the distance-transitivity property. For instance, the complete graphs Kn and the complete bipartite graphs Kkk have this property. Their diameters are 1 and 2 respectively, and the intersection arrays are: A survey of distance- transitive graphs whose diameter is two, and their connection with sporadic simple groups, can be found in [B, pp.
But Y is not distance-transitive; in fact there is no auto- morphism taking a vertex ai to a vertex bj Adel'son-Velskii The feasibility of intersection arrays I n this chapter we shall study the question of when an arbitrary array is the intersection array of a distance-regular graph. The theoretical material is an improved formulation of that given in [B, pp. The results of Proposition I n addition, the inequalities of parts 2 and 3 of Proposition There are also some elementary parity condi- tions: These conditions are not very restrictive, and they are satisfied by many arrays which are not realized by any graph.
I n this case, simple but special arguments are sufficient to prove t h a t there is no graph. Our aim now is to obtain a strong condition which rules out a multitude of examples of this kind. Thus, we have proved: What is more, the matrix B x alone is sufficient! We shall now give a practical demonstration of this fact. Since B is the image of the adjacency matrix A under a faithful representation, the mini- mum polynomials of A and B coincide, and so the eigenvalues of A are those of B.
However, the multi- plicity m A of A as an eigenvalue of A will usually be greater than one; we claim t h a t m A can be calculated from B alone. Let us regard A as an indeterminate, and define a sequence of polynomials in A, with rational coefficients, by the recursion: The polynomial vt A has degree i in A, and comparing the defini- tion with Lemma The last row of B is not used; it gives rise to an equa- tion representing the condition t h a t v A is an eigenvector of B corresponding to the eigenvalue A.
The roots of this equation in A are the eigenvalues A0, A 1? Proof Each eigenvalue of B is simple, and so there is a one- dimensional space of corresponding eigenvectors. Let K denote the diagonal matrix with diagonal entries k0,kv This turns out to be a very powerful condition.
I t is strange that, although these conditions are not sufficient for the existence of a graph with the given array, they are neverthe- less so restrictive that most known feasible arrays are in fact realizable.
We shall now give an example of the practical application of the feasibility conditions in a case where the conditions are satisfied. An important set of examples where the conditions are not satisfied will be encountered in Chapter The eigenvalues and multiplicities of this graph were found in a different way in Chapter 3.
Their diameters d and numbers of vertices n are Biggs and Smith Friendship is interpreted as a symmetric, irreflexive relation. This result may be proved as follows: This array is not feasible, so the first possibility must hold G.
Higman, unpublished. This can be deduced from the results of Wielandt The automorphism group of this graph is Janko's simple group of order Conway Primitivity and imprimitivity I n this chapter we shall investigate the permutation-group - theoretic notion of primitivity in the context of symmetric and distance-transitive graphs.
In the latter case we shall prove that the automorphism group is imprimitive if and only if one of two simple graph-theoretic conditions is satisfied. We begin by summarizing some terminology which is ex- plained at greater length in [B, Section 1. A block B, in the action of a group ffona set X, is a subset of X such that B and g B are either disjoint or identical, for each g in G. If G is transi- tive on X, then we say t h a t the permutation group X,0 is primitive if the only blocks are the trivial blocks, that is, those with cardinality 0,1 or X.
I n the case of an imprimitive permu- tation group X, G , the set X is partitioned into a disjoint union of non-trivial blocks, which are permuted by G. We shall refer to this partition as a block system. A graph Y is said to be primitive or imprimitive according as the group 6r F acting on VY has the corresponding property. Suppose C contains two adjacent vertices u and v. Clearly, this block system is not a colour-partition of Tz.
The rest of this chapter is devoted to an investigation of the consequences of the stronger hypothesis of distance-transitivity. We shall show that, in the imprimitive case, the vertex-colouring induced by a block system is either a 2-colouring, or a colouring of another quite specific kind.
Proof Let w be any vertex in Tj u. Further repetitions of the argument show that Trj u c: Then a non-trivial block in the action of G T on VT which contains the vertex u is one of the two sets i Mu r2 u u r 4 M u From parts 2 and 3 of Proposi- tion Thus B contains the set 1 above, and if it contained any other vertices, it would contain two adjacent vertices and would be trivial. We deduce that B is the set 1.
I t is quite common for an imprimitive group to be imprimitive in more ways than one, and our example demonstrates this behaviour in the two ways allowed by Proposition We now turn to an investigation of the graph-theoretical con- sequences of the two kinds of imprimitivity in distance-transitive graphs. Proof Since T contains triangles and is distance-transitive, every ordered pair of adjacent vertices belongs to a triangle.
If z e T2 u , then X contains two adjacent vertices, contrary to Proposition Proof Suppose F is bipartite. If X were not a block, then we should have an automorphism g of F such that X and g X intersect but are not identical.
From this contradiction we conclude that X is a block. Conversely, suppose X is a block. From this contradiction we deduce t h a t r has no odd circuits and its bipartite. The complete tripartite graphs Kr r r exemplify the case dealt with in Lemma The cubes Qk are trivially antipodal, since every vertex has a unique vertex at maximum distance from it; these graphs are at the same time bipartite.
The dodecahedron is also trivially anti- podal and not bipartite. Examples of graphs which are non- trivially antipodal and not bipartite are the complete tripartite graphs Kr r r which have diameter 2, and the line graph of Petersen's graph, which has diameter 3.
Proof Suppose T is antipodal. Both possibilities can occur in the same graph. Proof A non-trivial block is of the type 1 or 2 described in Proposition I n the case of a block of type 1 , Proposition If the diameter is 1, then the graph is complete, and consequently primitive.
If the diameter is 2, a block of type 1 is simultaneously of type 2. Consequently, if the graph is not bipartite, it must be antipodal. Automorphic graphs are apparently very rare. For instance, there are exactly three automorphic graphs whose valency is three Biggs and Smith and just one whose valency is four. The three trivalent graphs can be neatly described by the following diagrams Fig. I n the same way, the first diagram represents Petersen's graph, and the last diagram represents a graph with vertices.
From the intersection arrays we may calculate the spectra, and hence the complexity: Primitivity and imprimitivity 22 7 Antipodal graphs covering Kkjk Let T be a distance- transitive antipodal graph, whose derived graph is Kk k, and suppose each block in F contains r vertices.
Minimal regular graphs with given girth The results of Chapter 21, on the feasibility of intersection arrays, can be applied to a wide range of combinatorial problems.
Using the notation of Definition We have now proved t h a t the diameter of T is at least d, and t h a t T has at least n0 k, g vertices. If V has exactly n0 k,g vertices, then its diameter is d, and it has the stated intersection array.
The form of this array shows that V has no odd circuits, and so it is bipartite. I t is important to remark that, for a given k and g, there will always be some graph such t h a t there are no smaller graphs with the same valency and girth.
However, this graph need not be a k, gr -graph, for it will not necessarily attain the lower bound, n0 k, g , for the number of vertices. Consequently, there is no 3, 7 -graph. Proposition The methods of Damerell are most closely related to those of our tract, and we shall apply the techniques of his paper to both odd and even values of g. We shall study the feasibility of the following intersection matrix: I t is helpful to view these equations in the following way. Proof 1 The existence of the eigenvalues k and — k follows from the fact t h a t T is fc-valent and bipartite.
Now the eigen- values of r are by Proposition The equation of Lemma Suppose that A is an eigenvalue of our matrix B. From this equation we have 7 — qsinot. But it is well known [I.
Niven, Irrational numbers Wiley, , p. Thus we have the case 1. But there are three eigenvalues in all: Discounting the first possibility, we can check explicitly that the three others do lead to feasible intersection arrays. If the roots A1? We saw in the proof of Proposition Now the formula for m A is the quotient of two quadratic expressions in A, and so m A is integral only if A is at worst a quadratic irrational.
Suppose A is a quadratic irrational. But this must be a multiple of the minimal equation for A, which is monic with integer coefficients. For instance, a k, 6 -graph exists if and only if there is a projective plane with k points on each line Singleton The 57,5 -graph is rather enigmatic; many claims of its non-existence have been made, but none published.
However, the results of Aschbacher show that there is no distance-transitive 57, 5 -graph, and so the con- struction of the graph if there is one is certain to be highly complicated. A sequence whose terms are alternately points and lines, each term being incident with its successor, is called a chain; it is a proper chain if there are no repeated terms, except possibly when the first and last terms are identical when we speak of a closed chain.
A non-degenerate generalized m-gon is an incidence system with the properties: The auto- morphism group of this graph is the group of order obtained from PSU 3,52 by adjoining the field automorphism of GJF 5 2 Hoffman and Singleton There are 63 points of the plane which do not lie on their polar lines, and they form 63 self-polar triangles Edge The graph is not vertex-transitive, since there is no auto- morphism taking a 'point' vertex to a 'triangle' vertex.
I t has vertices, and its automorphism group has order 8 Bibliography Adel'son-Velskii, G. Example of a graph without a transitive auto- morphism group, Soviet Math. Aschbacher, M. The non-existence of rank three permutation groups of degree and subdegree 57, J. Algebra 19, Austin, T. The enumeration of point labelled chromatic graphs and trees, Canad. Baker, G.
Linked-cluster expansion for the graph-vertex coloration problem, J. Theory B 10, Bannai, E. Ito On Moore graphs, J. Tokyo Sec. Benson, C.
Minimal regular graphs of girth eight and twelve, Canad. Biggs, N. Expansions of the chromatic polynomial, Discrete Math. Damerell and D. Sands Recursive families of graphs, J.
Theory B 12, Smith On trivalent graphs, Bull. London Math. Birkhoff, G. A determinant formula for the number of ways of coloring a map, Ann. Bose, R. Mesner Linear associative algebras, Ann. Bouwer, I. A no-nonsense, crystal Will make a good introduction Review Download PDF.
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