PROJECT DETAILS - Department of Mathematics (sf) - III UG

Name of the Guide with S.No Reg.No Name of the Student Department Title of the Project designation 1 15BM4423 K.N. Abiyurekha B.Sc Mathematics (SF) 2 15BM4429 S. Divya B.Sc Mathematics (SF) Dr. Mrs. A. Anis Fathima, Bellman Ford Algorithm in 3 15BM4431 K. Jaya Alamelu B.Sc Mathematics (SF) M.Sc., M.Phil., PGDCA., 4 15BM4456 J. Powmiya B.Sc Mathematics (SF) Ph.D., 5 15BM4464 R. Sumithra B.Sc Mathematics (SF) 6 15BM4445 P. Manimekalai B.Sc Mathematics (SF) 7 15BM4446 B. Muthukavitha B.Sc Mathematics (SF) Dr. Mrs. A. Anis Fathima, Ford Fulkerson Algorithm in 8 15BM4451 G. Naveena B.Sc Mathematics (SF) M.Sc., M.Phil., PGDCA., Graph Theory 9 15BM4453 S. Pavithra(15/7/98) B.Sc Mathematics (SF) Ph.D., 10 15BM4462 R. Sneha B.Sc Mathematics (SF) 11 15BM4426 D. Baby sudha B.Sc Mathematics (SF) 12 15BM4428 A. Devipriya B.Sc Mathematics (SF) Distance Regular and Distance Ms.J.Priyadharshini,M.Sc.,M. 13 15BM4435 S. Karthika B.Sc Mathematics (SF) Transitive Graphs in Graph Phil., 14 15BM4444 G. Maheshwari B.Sc Mathematics (SF) Theory 15 15BM4460 M. Santhiya B.Sc Mathematics (SF) 16 15BM4422 S. Abinaya B.Sc Mathematics (SF) 17 15BM4427 T. Chellam Rajalakshmi B.Sc Mathematics (SF) Vector Calculs in three Ms.J.Priyadharshini,M.Sc.,M. 18 15BM4436 K. Kavibharathi B.Sc Mathematics (SF) dimensional space Phil., 19 15BM4450 S. Nandhini B.Sc Mathematics (SF) 20 15BM4468 J. Vinothini B.Sc Mathematics (SF) 21 15BM4425 B. Aruthra B.Sc Mathematics (SF) 22 15BM4430 G. Ishwarya B.Sc Mathematics (SF) An application of metric space Mrs. A. Samsath, 23 15BM4434 K. Kalaivani B.Sc Mathematics (SF) and metric dimension in Graph M.Sc.,M.Phil., 24 15BM4438 R. Kaviyapriya B.Sc Mathematics (SF) theory 25 15BM4465 C. Surya Prabha B.Sc Mathematics (SF) 26 15BM4424 V. Arokiya Sundhari B.Sc Mathematics (SF) 27 15BM4439 C. Keerthana B.Sc Mathematics (SF) Distance in Graph theory and Mrs. A. Samsath, 28 15BM4440 S. Keerthana B.Sc Mathematics (SF) its application M.Sc.,M.Phil., 29 15BM4442 G. Kiruthikadevi B.Sc Mathematics (SF) 30 15BM4457 S. Powranika B.Sc Mathematics (SF) 31 15BM4437 M. Kaviya B.Sc Mathematics (SF) 32 15BM4447 M. Muthulakshmi B.Sc Mathematics (SF) 33 15BM4455 R. Pavithra B.Sc Mathematics (SF) Cordial Labelling of Degree Mrs. T. Vanjikkodi, M.Sc., 34 15BM4458 P. Rajeswari B.Sc Mathematics (SF) Splitting graphs M.Phil., PGDCA 35 15BM4463 R. Sugathakumari B.Sc Mathematics (SF) 36 15BM4466 P.Swathi B.Sc Mathematics (SF) 37 15BM4433 S. Kalaiselvi B.Sc Mathematics (SF) 38 15BM4443 M. Kousalya B.Sc Mathematics (SF) 39 15BM4449 K. Nandhini B.Sc Mathematics (SF) Path Optimization Algorithm Ms. G. Dhivya, M.Sc., 40 15BM4452 P. Nithya Priya B.Sc Mathematics (SF) and Dijkstra’s Algorithm M.Phil., 41 15BM4454 S. Pavithra(2/6) B.Sc Mathematics (SF) 42 15BM4459 G. Saidharani B.Sc Mathematics (SF) 43 15BM4432 J. Jeyapritha B.Sc Mathematics (SF) Graphical and Queuing Model 44 15BM4441 K. Kiruthika B.Sc Mathematics (SF) of Banking Operations in Ms. J.P. Thempaavai, M.Sc., 45 15BM4448 K. Nandhika B.Sc Mathematics (SF) Intercontinental Bank Plc, M.Phil., 46 15BM4461 G. Sarumathi B.Sc Mathematics (SF) Nigeria 47 15BM4469 K. Vishnu Priya B.Sc Mathematics (SF) DISTANCE TRANSITIVE AND DISTANCE REGULAR GRAPHS IN GRAPH THEORY

A Group Project Report

Submitted in partial fulfilment of the requirement for the award of the degree of

Bachelor of Science in Mathematics

Submitted by

D. BABY SUDHA (15BM4426)

A. DEVIPRIYA (15BM4428)

S. KARTHIKA (15BM4434)

G. MAHESHWARI (15BM4444)

M. SANTHIYA (15BM4459)

Under the guidance of

Ms. J. PRIYADHARSHINI M.Sc., M.Phil.,

Assistant Professor

Department of Mathematics

Sri G.V.G Visalakshi College for Women (Autonomous)

(Affiliated to Bharathiar University, Coimbatore)

Re-Accredited at ‘A’ Grade by NAAC (CGPA 3.53) An ISO 9001:2008 Certified Institution

Udumalpet – 642 128

October – 2017

CERTIFICATE

This is to certify that the Group Project work entitled “DISTANCE TRANSITIVE AND DISTANCE REGULAR GRAPHS IN GRAPH THEORY” is a bonafied record work done by Group members D.Baby sudha(15BM4426), A. Devipriya(15BM4428), S.Karthika (15BM4434), G.Maheshwari (15BM4444), M.Santhiya (15BM4459) submitted in Partial fulfilment of the requirements for the award of the degree of Bachelor of Science in Mathematics at Sri G.V.G Visalakshi College for Women (Autonomous), Udumalpet during the academic year 2015-2018.

Submitted for the Viva- voce held on ______.

Signature of the Guide Signature of the H.O.D

Internal Examiner External Examiner

DECLARATION

We hereby declare that the project entitled “DISTANCE TRANSITIVE AND DISTANCE REGULAR GRAPHS IN GRAPH THEORY”submitted in partial fulfilment of the requirements for the award of the degree of Bachelor of Science in Mathematics is a record of original project done by us during the period of study in Sri G.V.G Visalakshi College for Women (Autonomous) Udumalpet, under the supervision and guidance ofMs. J. Priyadharshini M.Sc., M.Phil., Assistant Professor, Department of Mathematics.

Place: Udumalpet Signature of the Candidates,

Date:

ACKNOWLEDGEMENT

First and foremost, we record our sincere gratitude to the Almighty for her blessings for the successful completion of the project.

We take immense pleasure in expressing our thanks to the management for their continuous support throughout the project. We are extremely grateful to the Principal Dr.(Mrs.) K. Punithavalli M.Com., M.Phil., PGDCA.,Ph.D., Sri G.V.G Visalakshi College for Women (Autonomous) for providing all the facilities to do this project.

We take privilege in expressing our sincere, heartfelt thanks and gratitude to Dr. (Mrs) A. AnisFathima M.Sc., M.Phil., PGDCA., Ph.D., Head and Assistant Professor , Department of Mathematics for her continuous support throughout the project.

We record our profound sense of indebtness to our guide Ms.J.Priyadharshini M.Sc., M.Phil., for taking a keen interest and given us valuable suggestions, guidance in all aspects of this group project.

We express our deep gratitude and humble thanks to the faculty members of the department for their valuable suggestions and lending helping hand for the successful completion of our group project.

Finally we would like to thank our parents and Friends who have contributed much to complete the group project work on time.

D. BABY SUDHA (15BM4426)

A. DEVIPRIYA (15BM4428)

S.KARTHIKA (15BM4434)

G. MAHESHWARI (15BM4444)

M.SANTHIYA (15BM459)

CONTENT

TITILE

ABSTRACT

INTRODUCTION

CHAPTER -1 [DISTANCE TRANSITIVE]

CHAPTER -2 [DISTANCE REGULAR ]

CHAPTER -3 [TYPES OF DISTANCE TRANSITIVE]

CHAPTER -4 [PRIMITIVE & IMPRIMITIVE GRAPHS]

CONCLUSION

BIBILOGRAPHY

ABSTRACT

Graph theory is a study of graphs, which are mathematical structures used to get model pairwise relation between objects. In this we will discuss about the distance transitive graphs and distance regular graphs.

In this project, we discuss about the connectivity of distance transitive to the concept of vertex transitive and distance. And we will also define the distance partition and Intersection array to show a graph is distance regular. We connect the concept of distance transitive graphs to explain the three different types of distance transitive graphs. We also study the concept of the distance transitive to know about the primitive and imprimitive graphs.

INTRODUCTION

In this paper, we will discuss about the distance and the intersection algebra for these graphs which is based upon its intersection numbers. Distance-transitive graphs were first defined by Norman L. Biggs and D.H. Smith in1971. In the Mathematical field of graph theory, a distance-transitive graph is a graph such that given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance i, then there is an automorphism of the graph that carries v to x and w to y.

Every distance-transitive graph is distance- regular but the converse does not necessarily hold, as first shown by Adel'son-Vel'skiietal.

PRELIMINARIES:

GRAPH:

A graph is a diagram consists of points called vertices, joined by lines called edges each edge join exactly two vertices.

PATH:

A path is a walk in which all the edges and all the vertices are different.

ISOMORPHISM:

Two digraphs C and D are isomorphic if the graph D can be obtain by relabeling the vertices of C that is, if there is one to one correspondence between the vertices of C and those of D, such that the arcs joining each pair of vertices in C agree both number and direction with the arcs joining the corresponding pair of vertices in D.

ADJACENT:

The vertices v and w of the diagram are adjacent vertices if they are joined (in either direction) by an arc e.

u w

REGULAR GRAPH:

A graph is regular if it vertices all have the same degree.

STRONGLY REGULAR GRAPH

If a graph G has an intersection array, we say G is distance- regular. If a distance- regular graph has diameter 2, then it is a strongly-regular graph.

CONGRUENCE

A G- congruence is an equivalence relation≡ on X which is preserved under the action of G, that is x ≡ y <=> g(x)≡g(y) for all g G.

For any action we always have the following∈ two G- congruence

The trivial G- congruence where x≡y <=>x=y;

The universal G- congruence where x≡y for all x,y X.

∈ SYMMETRIC

A graph is Vertex Transitive if every vertex can be mapped to any other vertex by some automorphism that is it is symmetric.

DIAMETER

In a graph G, the greatest path between two vertices is called as the diameter of the graph G.

CHAPTER I

TRANSITIVE

Definition 1.1: Automorphism

An automorphism of a graph G is an isomorphism between G and itself . Thus, an automorphism of G is a permutation

: V(G) V(G) such that

(v) →  uv E(G)

This condition∝ 푢 preserves ∝ ∈ adjacency 퐸퐺 and∈ non adjacency. This means that two vertices are adjacent after the map if and only if they where adjacent before the mapping.

Some important facts of automorphism,

automorphism of G then are automorphism of G.

(ii)푖 퐼푓Identity 푛푑 mapping is also an automorphism of G.

(iii) If is an automorphism of G then -1 is an automorphism of G.

Therefore the set of automorphism of G forms a group under the operation of composition and its denoted by Aut (G)

Example 1.2: a

c b

Maps the vertices : a → a

b → c

c → b We can call this permutation map a map from the vertex set of G to vertex set of G.

= V(G) V(G) ,

= = (bc) a c This abracket c btells that b gets maps to C and C gets maps to b.

We are going to see all of the automorphism of K3 identity is always automorphism

= = (a) (b) (c) here a, b, c are fixed a c 휀 a c b

Reflextion 1 = = (a) (bc) a c α a c b

2 = = (b) (ac) a c a c b

= = (c) (ab) a c a c b

r1 = = (abc) a c a c b

r2= = (acb) a c a c b

Aut (K3) = { , 1, 2, 3, r1,r2}

This group is isomorphic휀 to the symmetric group of the order three factorial which is 6.

There are six things in the group.

Definition 1.3: Transitive

The transitivity T of a graph is based on the relative numbers of triangles in the graph, compared to total number of connected triples of nodes.

A Vertex Transitive graph is a graph G such that, given ant two vertices v1 and v2 of G, there is some automorphism.

Definition 1.4: Vertex Transitive

In the mathematical field of graph theory, a vertex –transitive graph is a graph G such that ,given any two vertices v1 and v2 of G , there is some automorphism

f : V(G)

→ 푉퐺 such that f(v1) =v2.

In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical .

Every symmetry graph without isolated vertices is vertex-transitive and every vertex-transitive graph is regular. However , not all vertex-transitive graph are symmetric, and not all regular graph are vertex transitive.

Example 1.5:

Finite vertex-transitive graphs include the symmetric graphs ( such as peterson graph). The finite are also vertex transitive. Although every cayley graph is vertex transitive, there exist other vertex transitive graphs that are not cayley graphs.

Definition 1.6: Edge Transitive

In mathematical field of graph theory, an edge-transitive graph is a graph G such that given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2. In other words a graph is edge transitive if its automorphism group acts transitively upon its edge.

Examples And Properties 1.7:

Edge transitive graphs include any km,n and any symmetric graph , such as the vertices and edges of the . Symmetric graphs are also vertex transitive graph (if they are connected) but in general edge transitive graphs need not to be vertex transitive the gray graph is an example of a graph which is edge transitive but not vertex transitive. An edge transitive graph that is also regular but not vertex transitive is called semi-symmetric.

Defintion 1.8: Distance

For Vertices U, V belongs to VT the distance from u→ v is defined to be the least number of edges in a path from u→ v and is denoted by d (u, v).

Example 1.9

u

v w

d(u,v)=1 u w

x

y v

d(u,v)=3, d (u,v)=2 d(u,v)=1

Definition 1.10: Distance Transitive Graph

In the mathematical field of graph theory, a distance transitive graph is a graph such that given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance ,there is an automorphism of the graph that carries v to x and w to y. A distance transitive graph is interesting partly because it has a large automorphism group. Some interesting finite groups are the automorphism group of distance transitive graph especially of those whose diameter is 2.

Example 1.11:

Johnsons graph

Hamming graph

Grassman graph

CHAPTER 2

DISTANCE REGULAR GRAPH

Definition 2.1: Intersection Array

If a graph G is of diameter d (d>=2) is distance regular if and only if there is an array of integers

{b0,b1,b2,.....bd-1; c1,c2,.....cd} such that for all 1 j , bj gives the number of neighbours of u at distance j+1 from v and cj, gives the neighbours of u at distance j 1 from v for any pair of vertices u ≤ ≤ and v at distance j on G. And we denote the intersection array by, −

c1 ...... cd-1 cd

G= a∗o a1 ...... ad-1 ad

bo b1 ...... bd-1

Where c0, bd are undefined (which is what the *∗ represents). The easiest way to understand these intersection numbers to think of a distance partition of G. If we fix v and choose u Gj(v), then cj denotes the number of neighbours of u that are closer to v, aj the umber of neighbours at ∈ the same distance, and bj that are further away.

Example 2.2:

* 1 2 3

Τ(Q3) = 0 0 0 0 3 2 1 * Some simple observations are that a0=0(as there is only one vertex in G0(v)), c1=1 (for the same reason),b0=k, where k is the valency of the graph,and that all columns sum to k.A less trivial observation 9which requires proof) is that the top row is weakly increasing (i.e. 1=c1

(i.e., k>b1>...... >bd-1).there are also a number of other numerical constraints which can be derived.

Definition 2.3: ith Distance Matrix

th In connected graph G of diameter d, for 0 ≤ i ≤ d, the i distance matrix Ai given by

1 if d(vr, vs)=i

(Ai)rs = (or)

0 otherwise

For all vr,vs VT

∈ These are generalisation of the adjacency matrix, as clearly A1 =A(G). Note that A0= I (as d(vr, vs)=0 if and only if vr=vs), as that the sum of all the distance matrices is the all-once matrix J. From now on. We shall only consider distance - regular graphs.

Definition 2.4: Partition

A partition is a way of writing an integers n as a sum of positive integers where the order of the addends is not sufficient possibly subject to one or more additional constants.

Definition 2.5: Distance Partition

For a graph G, Vertex v VG, and for 0 i d = diam(G), define

Gi(v) = {w∈ V: d(u,v)=i}≤ ≤

∈ These cells G0(V) ,G1(V) ...... Gd(V) form a distance partition (or a level decomposition) of G.

Example 2.6:

V

G0(V) G1(V) G2(V) G3(V) We can also use this distance partition to provide an alternative characterisation of distance transitive graphs.

ALGEBRAIC CONSTRAINTS ON INTERSSECTION ARRAY

Definition 2.7: Adjacency Matrices

The adjacency matrix of a graph G with n vertices is the n by n matrix A (G) with entries

1 if vi v

~ Aij= (or)

0 otherwise

Example 2.8:

The K4

V2 V1

V3 V4 v4

0 1 1 1

Aij= 1 0 1 1

1 1 0 1

1 1 1 0

The eigenvalues of G are defined to be those of A (G). The adjacency algebra, denoted of

G is the set of all linear combinations of powers of Ą (G) (with coefficients in C). This is a vectoĄ Gr space were the elements (which are n by n matrices) can be multiplied together, so forms an algebra. i i A , this algebra is clearly commutativeג∑ Because each element of Ą (G) has the form

Theorem 2.9:

th l The number of paths in G from vertex vi to vertex vj of length l is given i,j entry in A .

Proof:

This is done by induction on l.

If G is connected, the condition tells us a property of the adjacency algebra.

Corollary

If G is connected, and has diameter d, then the dimension of Ą (G) is atleast d+1.

Proof:

By considering vertices along a path between to vertices at distance d, it follows that

{A0, A1, A2,...... ,Ad}

Is a linearly independent set of size d+1 in Ą

Theorem 2.10:

For a distance regular graph G of valency k and diameter d, has basis {A0,A1,……Ad} and its diameter id d+1. 휜푮

Proof:

Clearly {A0,A1,……Ad} is a linearly independent set, as in a vgiven position there is a non-zero in only on Ai. Now, the minimum polynomial of A is

Min(A)= (A-kI)J = (A-kJ)(A0+A1+…+Ad),

A polynomial in A of degree d+1. Hence the dimension of , is d+1.

Corollary 휜푮

{A0, A1,…...... , Ad} is another basis for .

휜푮 Now we consider the linear transformation of given by multiplying an element X= i Ai 풅 by A. then 휜푮 ∑풊=ퟎ 풔

XA= i (bi-1Ai-1+aiAi+ci+1Ai+1) 풅 풊=ퟎ In other ∑words,풔 we can represent a linear transformation by B(G), in the intersection matrix: 0 1

K a1 c2

b1 a2 .

b2 . . = B(G)

. . cd-1

. ad-1 cd

bd-1 ad we notice that this look very like the intersection array tilted slightly and with lots of zeros thrown in the empty spaces. Now,as this matrix represents the same linear transformation as A (with respect to different bases), they must have the same eigenvalues. Now , there are d+1 distinct eigenvalues , which we call λ0

λ1 … λd. As eigenvalues of B, each have ,multiplicity 1. As eigenvalues of A, they will (in > general> ) >have multiplicities greater than 1. However, it transpires quite surprisingly that these numbers can be obtained directly from B.

Let ui denote the left eigenvector of B corresponding to λI , and vi the corresponding right eigenvector.

CHAPTER 3

DIFFERENT TYPES OF DISTANCE TRANSITIVE GRAPHS There are three different types of Distance transitive graphs. They are 1) Johnsons graph 2) Hamming graph 3) Grassmanns graph

Definition 3.1: Johnson Graphs

Again, we consider x={1,…..n}. Let Xk denote the family of all k-subsets of X clearly

│Xk│= . The Johnson graph J(n,k) is defined as follows. It has vertex set Xk; two vertices 푛 labeled by u= {i1,…,ik}and v={j1,…..,jk} are adjacent if and only if they differ in one element, i.e, if 푘 │u∩v│=k-1

The symmetric group Sn acts transitively on Ẋk. it follows that d(u,v)=m if and only if│u∩v│=k- m, and that if │u∩v│=k-m, then for g Sn, │g(u)∩g(v)│=k-m. That J(n,k) is distance transitive follows from this. ∈

JOHNSON GRAPH

Named after SELMER M. JOHNSON

Vertices 푛 푘 Edges 푘푛−푘 푛 2 푘 Diameter min (k, n-k)

Properties (k (n-k)) regular ,vertex transitive,

Distance Transitive,

Hamilton connected

Example 3.2:

The johnson graph J(4,2) is .

* 1 4

0 2 0

( J , ) =4 1 *

The octahedron J(4,2)

HAMMING GRAPH The Hamming graphs are defined in a similar way to the Johnson graphs. Again, we consider a set of size n (typically, we think of Zn, the set of integers modulo n, or the finite field with q elements Fq), but this time we consider d, the Cartesian product of d copies of . Definition 3.3: The hamming graph, denoted H(d; n), has vertex set d, with two vertices being adjacent if, when regarded as ordered d-tuples, they differ in exactly one coordinate. It is clear that two vertices are at distance k if and only if they differ in exactly k co-ordinates. (Recall the definition of the Hamming distance of two vectors being the number of co-ordinates in which they differ.) Let us consider the following example:

Example 3.4:

The hamming graphs H(d,2) are the d-dimensional hypercubes, Qd, Q4 Is shown below.

4 3 2 1 ٭

0 0 0 0 0 = (Q4) ٭ 1 2 3 4

Their automorphism group S2 Wr Sd is known as the hyperoctahedral group. GRASSMANN GRAPH The Grassmann graphs are another family of graphs defined similarly to the Johnson graphs. This time, however, we are concerned with subspaces of a vector space, rather than with subsets of a set. DEFINITION 3.3: Let V be an n-dimensional vector space over Fnq, the finite field with q elements, where q is a power of a prime. Then let Vk be the set of all k-dimensional subspaces of V. We define the Grassmann graph, denoted by G (q;n;k), as having vertex set Vk, and two vertices (i.e. subspaces) U;W being adjacent if and only if dim(U \W) = k1. There are numerous similarities between the Grassmann and Johnson graphs (more so than with the Hamming graphs).

CHAPTER 4

PRIMITIVE AND IMPRIMITIVE GRAPHS

Definition 5.1: Block A block of a graph G is a block of the automorphism group Aut(G) acting on the vertex set VG. As always, we have the trivial blocks Ø, fvg and VG. Definition 5.2: Primitive If a distance-transitive graph has only trivial blocks, we say Gis a primitive graph. Otherwise, Ghas a non-trivial block system and is called an imprimitive graph. Definition 5.3: Imprimitive If there are no other G- congruence’s, we say the action of G on X is primitive. Otherwise, it is called Imprimitive. The equivalence classes are known as blocks. These form a partition of X which is unaffected by the action of G. A distance transitive graph G is (im)primitive if and only if aut(G) is (im)primitive on VG.

Example 5.3:

The complete graphs, Kn are all primitive distance-transitive graphs. (K5 is illustrated overleaf.)

This is because Aut(Kn) = Sn, so every possible permutation of the vertices is an automorphism.

Thus there is no non-trivial subset of VKn that is fixed under the action of Sn. Hence Kn has no non- trivial blocks, so is a primitive graph

Example 5.4:

The complete bipartite graphs Kn;n are imprimitive distance-transitive graphs. (K3;3 is illustrated). w V1

u v V2 K3;3, an imprimitive graph

The relation ≡ where u ≡ v if and only if u, v V1 or u, v V2 is clearly an equivalence relation (G) , on VKn, n that is invariant under the action of Aut∈ , with the∈ bipartition V1 V2 forming the block system. For example, as labeled in the graph.

We have u ≡ v but v ≡ w. As Kn,n are clearly distance-transitive and have a nontrivial block system, they are imprimitive. The complete characterization of imprimitive graphs was one of the first major results obtained in the study of distance-transitive graphs.

Examples 5.5:

(i) The complete graphs Kn are primitives, as there is no partition of the vertices which is

invariant under Sn.

(ii) The complete bipartite graphs Kn,n are imprimitive; the blocks of imprimitivity are

precisely the two halves of the bipartition. K4,4 is shown below.

Imprimitive distance-transitive graphs are characterized in a nice way. This result is due to D.H. Smith [3] in (1971).

Suppose that G is a distance-transitive graph, with diam(G) >2. Then:

 A block of G containing a vertex v must be a union pf cells Gi (v).  The only possibilities for blocks are

(a)G 0(v) UG 2 (v) UG 4 (v) U……G n (v) (where ἠ is the maximum even distance inG, so is either d or (d-1), or

(b) G0 (v)U Gd(v).  Case (a) occurs if and only if G is bipartite.  Case (b) occurs if and only if G is antipodal.

Note that G is antipodal if, for u,w 0(v)U Gd(v), u w, we have d(u,w)=d. Clearly, if

Gd(v) consists only of a single vertex, ∈then G must be antipodal,≠ so in this case we call G trivially antipodal.  Hence a distance-transitive graph is imprimitive if and only if it is bipartite or antipodal. Observe that is an inclusive ‘or’: it is possible for both situations to occur simultaneously. For instance, the cube is both bipartite and antipodal. If a graph is trivially antipodal, it is easy to spot (we just have to count the number of vertices in the outermost cell). Bipartite distance-transitive (or, indeed, distance-regular) graphs are also easy to spot: this can be determined immediately from the intersection array. PROPOSITION 5.6: Suppose G is distance-regular, with intersection array

* c1 …. cd-1 cd

G= a0 a1 ad-1 ad

b0 b1 bd-1 *

then G is bipartite if and only if a0 =a1 =………..ad=0 PROOF We prove the contrapositive in both directions.

First, suppose aj=0 for some j, so there adjacent vertices v, w Gi (u) also, there are paths π,ρ of the length j from u to v and from u to w respectively. Let x be the∈ last vertex where π and ρ meet and suppose d(x, v) =d(x, w)-=m. then we can from a circuit of length 2m+1, an odd number. So G is not bipartite. Conversely, suppose G is not bipartite. Then G contains some odd circuit , of length 2δ+1 say. Choose some vertex u in . Then there exist two adjacent vertices v, w in such that d(u,v)=d(u,w)= δ, so v, w G δ (u) as shown.

………..∈

u

……… w

G0(u) G1(u) G δ(u)

Consequently we have that a δ is non zero. We now return to some of the infinite families of distance-transitive graps we considered and determine if they are primitive. PRIMITIVE HAMMING GRAPHS We shall check if these can be bipartite graph and/or antipodal. Now, the intersection array of H(d,n) is

* 1 …. j …. d-1 d 0 …. …. …. …. …. d(n-2) d(n-1) (d-1)(n-1)…. (d-j)(n-1) …. (n-1) 1 *

As mentioned above, this corresponds to a bipartite graph if and only if the middle row is al zeros. Now, the columns must sum to the valency, which is d (n─1) so we want

d(n-1) – j-(d-j)(n-1)=0  nd-d-j-nd+d+nj-j=0  nj-2j=0  n=2. Thus the only bipartite Hamming graphs are H (d,2) the hypercubes. We note that the hypercubes are trivially antipodal. For any distance regular graph G and a fixed vertex v, by counting the number of edges between cells G i(v) and G i+1(v) it follows that kj bj

=kj+1 cj+1 (where kj=Gj (v)). Thus we can obtain a formula for the number of vertices in G a(v),

Kd ……푑− = ……푑 In the case of H(d,n) this gives,

Kd ……푑− = ……푑

푑푛 − 푑 − 푛 − … .푛 − = × × … .× d

푑! 푛 − = d = (n-1)푑! . Clearly this equals 1 iff n=2. If n>2, however, we have vertices u=000…0, V=111…1 and w=211….1, where d(u, v)=d, but (d, w)=1. Hence the Hamming graphs H(d, n) for n>2 are not antipodal. Thus the only imprimitive hamming graphs are the hypercubes, so all others must be primitive.

PRIMITIVE JOHNSON GRAPHS

We use the same method again, first checking if J(n,k) is bipartite. The intersection array of J(n,k) is given by,

* 12 …. i2 …. (k-1)2 k2 0 …. …. …. …. …. nk-2k k(n-k) (k-1)(n-k-1) …. (k-i)(n-k-i) …. n-2k+1 *

If the middle row is all zeroes, then for each i we have k(n-1) – i2-(k-i)(n-k-i)=0 and (from the last column) nk-2k2=0. This last condition implies n=2k, so substituting this gives us, for all i, k(2k-k) – i2-(k-i)(2k-k-i)=0  k2-i2-(k-i)2=0  2ki-2i2=0  i=k, Which is absurd. So J(n,k) can never be bipartite.

We now observe that if n= 2k, J(n,k) is trivally antipodal. If vertices u={u1,u2,...,uk} and v={v1,v2,…,vk} are at maximum distance (which clearly is k), we have ui ≠vj for all I,j. Therefore these from a partition of the underlying set, so there can be no other vertex at distance k from u. However, if n>2k (we can assume this, as there is an isomorphism from J(n,k) to J(n,n-k), by taking the complement of each k- subset) we have vertices u=={u1,u2,...,uk}, v={v1,v2,…,vk} and w={ v1,v2,…,vk-1, x }, where x≠ ui ,vj for all i,j. Hence d(u,v)= d(u,w)=k, but d(u,w)=1. So the only imprimitive Johnson graphs are J(2k,k). We conclude these notes by mentioning that there are only four primitive distance transitive graphs of valency 3. These are the complete graph K4, the Peterson graph, the and the Biggs described them as “Three Remarkable Graphs” in his paper [2], where full descriptions can be found. We have already seen the Peterson graph. The Coxeter graph and Biggs-Smith graph are shown below.

The Coxeter graph