Most of last month's notes grew out of my reading of Algebraic Graph Theory by Godsil and Royle. I am still exploring and learning this subject. This month, however, I dove into another book with the same title but this book is written by Norman Biggs. As a result, many of the notes that follow are drawn from Biggs's treatment of the topic.

Since I already had a good understanding of the subject from the earlier book, I decided to skip the first fourteen chapters of the new book. I began with Chapter 15, which discusses automorphisms of graphs and then moved on to the following chapters on graph symmetries. My main reason for picking up Biggs's book was to understand Tutte's well known result that any \( s \)-arc-transitive finite cubic graph must satisfy \( s \le 5. \) While I did not reach that chapter this month, I made substantial progress with the book. I hope to work through the proof of Tutte's theorem next month.

Contents

1. Degree of Vertices in an Orbit
2. Regular Non-Vertex-Transitive Graphs
3. Vertex-Transitive But Not Edge-Transitive
4. Edge-Transitivex But Not Vertex-Transitive
5. Bipartiteness as a Necessary Condition
6. Graph with an Automorphism Group
7. Permutation Groups Need Not Be Automorphism Groups
8. Symmetric Graphs

Degree of Vertices in an Orbit

If two vertices of a graph belong to the same orbit, then they have the same degree. In other words, for a graph \( X, \) if \( x, y \in V(X) \) and there is an automorphism \( \alpha \) such that \( \alpha(x) = y, \) then \( \deg(x) = \deg(y). \)

The proof is quite straightforward. Let \begin{align*} N(x) &= \{ v_1, \dots, v_r \}, \\ N(y) &= \{ w_1, \dots, w_s \} \end{align*} represent the neighbours of \( x \) and \( y \) respectively. Therefore we have \[ x \sim v_1, \; \dots, \; x \sim v_r. \] Since an automorphism preserves adjacency, we get \[ \alpha(x) \sim \alpha(v_1), \; \dots, \; \alpha(x) \sim \alpha(v_r). \] Substituting \( \alpha(x) = y, \) we get \[ y \sim \alpha(v_1), \; \dots, \; y \sim \alpha(v_r). \] Thus \[ \alpha(N(x)) = \{ \alpha(v_1), \; \dots, \; \alpha(v_r) \} \subseteq N(y). \] A similar argument works in reverse as well. By the definition of automorphism, if \( \alpha \) is an automorphism, so is \( \alpha^{-1}. \) From the definition of \( N(y) \) above, we have \[ y \sim w_1, \; \dots, \; y \sim w_s. \] Therefore \[ \alpha^{-1}(y) \sim \alpha^{-1}(w_1), \; \dots, \; \alpha^{-1}(y) \sim \alpha^{-1}(w_s). \] This is equivalent to \[ x \sim \alpha^{-1}(w_1), \; \dots, \; x \sim \alpha^{-1}(w_s). \] Thus \[ \alpha^{-1}(N(y)) = \{ \alpha^{-1}(w_1), \; \dots, \; \alpha^{-1}(w_s) \} \subseteq N(x) \] This can be rewritten as \[ \{ \alpha^{-1}(w_1), \; \dots, \; \alpha^{-1}(w_s) \} \subseteq \{ v_1, \dots, v_r \}. \] Therefore \[ N(y) = \{ w_1, \dots, w_s \} \subseteq \{ \alpha(v_1), \dots, \alpha(v_r) \} = \alpha(N(x)). \] We have shown that \( \alpha(N(x)) \subseteq N(y) \) and \( N(y) \subseteq \alpha(N(x)). \) Thus \[ \alpha(N(x)) = N(y). \] Thus \[ \lvert N(y) \rvert = \lvert \alpha(N(x)) \rvert = r. \] Therefore both \( x \) and \( y \) have \( r \) neighbours each. Hence \( \deg(x) = \deg(y). \)

Regular Non-Vertex-Transitive Graphs

The Frucht graph and the Folkman graph are examples of graphs that are \( k \)-regular but not vertex-transitive. In fact, the Folkman graph is a semi-symmetric graph, i.e. it is regular and edge-transitive but not vertex-transitive.

Vertex-Transitive But Not Edge-Transitive

The circular ladder graph \( CL_3, \) i.e. the triangular prism graph, is vertex-transitive but not edge-transitive.

Every vertex has the same local structure. Every vertex has degree \( 3 \) and it lies on exactly one of the two triangles and it has exactly one 'vertical' edge connecting it to the corresponding edge on the other triangle. Any vertex can be sent to any other by an automorphism.

Since triangle edges are in a triangle and vertical edges are in no triangle, no automorphism can send a triangle edge to a vertical edge or vice versa. Therefore the graph is not edge-transitive.

Edge-Transitivex But Not Vertex-Transitive

The complete bipartite graphs \( K_{m,n} \) with \( m \ne n \) are edge-transitive but not vertex-transitive.

Every edge connects one vertex from the \( m \)-part to one vertex from the \( n \)-part. Any permutation of vertices inside the \( m \)-part preserves adjacency. Similarly, any permutation of vertices inside the \( n \)-part preserves adjacency.

Take two arbitrary edges \[ uv, \; u'v' \in E(K_{m,n}) \] where \( u, u' \) are vertices that lie in the \( m \)-part and \( v, v' \) are vertices that lie in the \( n \)-part. Permute vertices within the \( m \)-part to send \( u \) to \( u'. \) Similarly, permute vertices within the \( n \)-part to send \( v \) to \( v'. \) This gives an automorphism that sends the edge \( uv \) to \( u'v'. \) In this manner we can find an automorphism that sends any edge to any other. Therefore, \( K_{m,n} \) is edge-transitive.

However, \( K_{m,n} \) is not vertex-transitive since no automorphism can send a vertex in the \( m \)-part to a vertex in the \( n \)-part since the vertices in the \( m \)-part have degree \( n \) and the vertices in the \( n \)-part have degree \( m. \)

Bipartiteness as a Necessary Condition

If a connected graph is edge-transitive but not vertex-transitive, then it must be bipartite.

Graph with an Automorphism Group

In 1938, Frucht proved that for every finite abstract group \( G, \) there exists a graph whose automorphism group is isomorphic to \( G . \)

Remarkably, this result remains valid even when we restrict our attention to cubic graphs. That is, for every finite abstract group \( G, \) there exists a cubic graph whose automorphism group is isomorphic to \( G. \) Moreover, the result has been extended to graphs satisfying various additional graph-theoretical properties, such as \( k \)-connectivity, \( k \)-regularity and prescribed chromatic number.

Permutation Groups Need Not Be Automorphism Groups

Consider the following specialised version of the problem discussed in the previous section: Given a permutation group on a set \( X, \) must there exist a graph with vertex set \( X \) whose automorphism group is precisely that permutation group?

The answer is no. Consider the cyclic group \( C_3 \) acting on \( X = \{ a, b, c \}. \) There is no graph \( \Gamma \) with \( V(\Gamma) = X \) and \( \operatorname{Aut}(\Gamma) \cong C_3. \) If we take \( \Gamma = K_3, \) then \( C_3 \subset S_3 = \operatorname{Aut}(K_3) \) but \( C_3 \ne \operatorname{Aut}(K_3) . \)

Symmetric Graphs

It is interesting that while we study graph symmetry through concepts such as graph automorphisms, vertex-transitivity, edge-transitivity, etc. the name symmetric graph is reserved for graphs that are \( 1 \)-arc-transitive. A vertex-transitive graph or an edge-transitive graph need not be \(1\)-arc-transitive and therefore need not be symmetric.

However, every \( s \)-arc-transitive graph is \(1 \)-arc-transitive for \( s \ge 1. \) Consequently, every \( s \)-arc-transitive graph is symmetric. Moreover, every distance-transitive graph is also \( 1 \)-arc-transitive and hence symmetric.

Formally, we say that a graph \( \Gamma \) is \( 1 \)-arc-transitive (or equivalently, symmetric) if for all \( 1 \)-arcs \( uv \) and \( u'v' \) of \( \Gamma, \) there is an automorphism \( \alpha \in \operatorname{Aut}(\Gamma) \) such that \( \alpha(uv) = u'v'. \)

Stated in more basic terms, we can say that \( \Gamma \) is symmetric if for all \( u, v, u', v' \in V(\Gamma) \) satisfying \( u \sim v \) and \( u' \sim v', \) there exists \( \alpha \in \operatorname{Aut}(\Gamma) \) such that \( \alpha(u) = u' \) and \( \alpha(v) = v'. \)

Switching gears now, we say that \( \Gamma \) is distance-transitive if for all \( u, v, u', v' \in V(\Gamma) \) satisfying \( d(u, v) = d(u', v'), \) there exists \( \alpha \in \operatorname{Aut}(\Gamma) \) such that \( \alpha(u) = u' \) and \( \alpha(v) = v'. \) Since all \( 1 \)-arcs \( uv \) and \( u'v' \) satisfy \( d(u, v) = d(u', v') = 1, \) distance-transitivity implies that there is an automorphism that sends \( uv \) to \( u'v'. \) Therefore a distance-transitive graph is also \( 1 \)-arc-transitive.

To summarise, a graph must possess a certain degree of symmetry in order to be called symmetric. It turns out that merely having a non-trivial automorphism group is not sufficient. Even being vertex-transitive or edge-transitive is not enough for a graph to be called symmetric. The graph needs to be at least \( 1 \)-arc-transitive to be called symmetric.

Another interesting aspect of this terminology is that the property of being asymmetric is not the exact opposite of being symmetric. For example, a vertex-transitive graph need not be symmetric. However, that does not make it asymmetric. A graph is called asymmetric if it has no non-trivial automorphisms, i.e. its automorphism group contains only the identity permutation. Thus, if a graph has at least two vertices and is vertex-transitive, it must admit a non-trivial automorphism that maps one vertex to another. So while such a vertex-transitive may not be symmetric, it isn't asymmetric either.

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Contents

1. Cayley Graphs
2. Vertex-Transitive Graphs
3. Arc-Transitive Graphs
4. Bipartite Graphs and Cycle Parity
5. Tutte's Theorem
6. Tutte's 8-Cage
7. Linear Congruential Generator
8. Numbering Lines

Cayley Graphs

Let \( G \) be a group and let \( C \subseteq G \) such that \( C \) is closed under taking inverses and does not contain the identity, i.e. \[ \forall x \in C, \; x^{-1} \in C, \qquad e \notin C. \] Then the Cayley graph \( X(G, C) \) is the graph with the vertex set \( V(X(G, C)) \) and edge set \( E(X(G, C)) \) defined by \begin{align*} V(X(G, C)) &= G, \\ E(X(G, C)) &= \{ gh : hg^{-1} \in C \}. \end{align*} The set \( C \) is known as the connection set.

Vertex-Transitive Graphs

A graph \( X \) is vertex-transitive if its automorphism group acts transitively on its set of vertices \( V(X). \) Intuitively, this means that no vertex has a special role. We can 'move' the graph around so that any chosen vertex becomes any other vertex. In other words, all vertices are indistinguishable. The graph looks the same from each vertex.

The \( k \)-cube \( Q_k \) is vertex-transitive. So are the Cayley graphs \( X(G, C). \) However the path graph \( P_3 \) is not vertex-transitive since no automorphism can send the middle vertex of valency \( 2 \) to an end vertex of valency \( 1. \)

Arc-Transitive Graphs

The cube \( Q_3 \) is \( 2 \)-arc-transitive but not \( 3 \)-arc-transitive. In \( Q_3, \) a \( 3 \)-arc belonging to a \( 4 \)-cycle cannot be sent to a \( 3 \)-arc that does not belong to a \( 4 \)-cycle. This is easy to explain. The end vertices of a \( 3 \)-arc belonging to a \( 4 \)-cycle are adjacent but the end vertices of a \( 3 \)-arc not belonging to a \( 4 \)-cycle are not adjacent. Therefore, no automorphism can map the end vertices of the first \( 3 \)-arc to those of the second \( 3 \)-arc.

For intuition, imagine that a traveller stands on a vertex and chooses an edge to move along. They do this \( s \) times thereby walking along an arc of length \( s, \) also known as an \( s \)-arc. By the definition of \( s \)-arcs, the traveller is not allowed to backtrack from one vertex to the previous one immediately. In an \( s \)-arc-transitive graph, these arcs look the same no matter which vertex they start from or which edges they choose. In the cube, this is indeed true for \( s = 2. \) All arcs of length \( 2 \) are indistinguishable. No matter which arc of length \( 2 \) the traveller has walked along, the graph would look the same from their perspective at each vertex along the arc. However, this no longer holds good for arcs of length \( 3 \) since there are two distinct kinds of arcs of length \( 3. \) The first kind ends at a distance of \( 1 \) from the starting vertex of the arc (when the arc belongs to a \( 4 \)-cycle). The second kind ends at a distance \( 3 \) from the starting vertex of the arc (when the arc does not belong to a \( 4 \)-cycle). Therefore the cube is not \( 3 \)-arc-transitive.

Bipartite Graphs and Cycle Parity

A graph is bipartite if and only if it contains no cycles of odd length. Equivalently, every cycle in a bipartite graph has even length. Conversely, if every cycle in a graph has even length, then the graph is bipartite.

Tutte's Theorem

For any \( s \)-arc-transitive cubic graph, \( s \le 5. \) This was demonstrated by W. T. Tutte in 1947. A proof can be found in Chapter 18 of Algebraic Graph Theory by Norman Biggs.

In 1973, Richward Weiss established a more general theorem that proves that for any \( s \)-arc-transitive graph, \( s \le 7. \) The bound is weaker but it applies to all graphs rather than only to cubic ones.

Tutte's 8-Cage

The book Algebraic Graph Theory by Godsil and Royle offers the following two descriptions of Tutte's 8-cage on 30 vertices:

Take the cube and an additional vertex \( \infty. \) In each set of four parallel edges, join the midpoint of each pair of opposite edges by an edge, then join the midpoint of the two new edges by an edge, and finally join the midpoint of this edge to \( \infty. \)

Construct a bipartite graph \( T \) with the fifteen edges as one colour class and the fifteen \( 1 \)-factors as the other, where each edge is adjacent to the three \( 1 \)-factors that contain it.

It can be shown that both descriptions construct a cubic bipartite graph on \( 30 \) vertices of girth \( 8. \) It can be further shown that there is a unique cubic bipartite graph on \( 30 \) vertices with girth \( 8. \) As a result both descriptions above construct the same graph.

Linear Congruential Generator

Here is a simple linear congruential generator (LCG) implementation in JavaScript:

function srand (seed) {
  let x = seed
  return function () {
    x = (1664525 * x + 1013904223) % 4294967296
    return x
  }
}

Here is an example usage:

> const rand = srand(0)
undefined
> rand()
1013904223
> rand()
1196435762
> rand()
3519870697

Numbering Lines

Both BSD and GNU cat can number output lines with the -n option. For example:

$ printf 'foo\nbar\nbaz\n' | cat -n
     1  foo
     2  bar
     3  baz

However I have always used nl for this. For example:

$ printf 'foo\nbar\nbaz\n' | nl
     1  foo
     2  bar
     3  baz

While nl is specified in POSIX, the cat -n option is not.

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