Mixed Moore Graphs. GT2015, Nyborg

Leif K. Jørgensen Aalborg University Denmark Moore Graphs

A Moore graph of diameter D is a (regular) graph (undirected/directed/mixed) with the property that for any two vertices x, y, there is a unique path of length at most D from x to y.

In this talk the degree of a Moore graph is denoted by k. An undirected Moore graph with degree k and diameter D has order

M(k, D) := 1 + k + k(k − 1) + k(k − 1)2 + ... + k(k − 1)D−1.

The Moore bound M(k, D) is

• an upper bound on the order of a undirected graph with max. degree k and diameter D.

• an lower bound on the order of an undirected graph with min. degree k and girth 2D + 1. Theorem Hoffman + Singleton 1960 If a Moore graph with diameter D = 2 then the degree is either

• k = 2, unique Moore graph: the cycle of length 5,

• k = 3, unique Moore graph: the Petersen graph,

• k = 7, unique Moore graph: the Hoffman-Singleton graph, or

• k = 57, existence of Moore is unknown.

Theorem Damerell 1973, Bannai + Ito 1973 A Moore graph with k ≥ 3 and D ≥ 3 does not exist. Petersen graph

G = (V,E) : graph

An automorphism of G is a is a bijection φ : V → V such that

xy ∈ E ⇐⇒ φ(x)φ(y) ∈ E. Aut(G) denotes the group of all automorphisms of G.

Does there exist a Moore graph with D = 2 and k = 57. Number of vertices is 3250.

Theorem Macajˇ + Sirˇ a´nˇ 2010 A Moore graph of degree 57 has at most 375 automorphisms. G = (V,E) : graph

An automorphism of G is a is a bijection φ : V → V such that

xy ∈ E ⇐⇒ φ(x)φ(y) ∈ E. Aut(G) denotes the group of all automorphisms of G.

Does there exist a Moore graph with D = 2 and k = 57. Number of vertices is 3250.

Theorem Macajˇ + Sirˇ a´nˇ 2010 A Moore graph of degree 57 has at most 375 automorphisms. A Cayley graph Cay(X,S) of a group X with connection set S ⊂ X is the graph with vertex set X and edges xy where yx−1 ∈ S, i.e., there is an edge from x to sx, where s ∈ S.

Example: C5 = Cay(Z5, {1, −1}) 0 ZZ w Z Z Z Z Z Z Z Z Z 4 A ¡ 1 A ¡ ww A ¡ A ¡ A ¡ A ¡ A ¡ A ¡ A ¡ AA ¡¡ 23 ww Theorem (Sabidussi) A graph G is a Cayley graph of a group X if and only if X is a sharply transitive (regular) subgroup of Aut(G). Sharply transitive means that for every u, v ∈ V (G) there is a unique φ ∈ X such that φ(u) = v.

Theorem An undirected Moore other than C5 is not a Cayley graph.

Proof: An automorphism φ with φ ◦ φ = Id and φ(v) 6= v for all vertices v. Theorem (Sabidussi) A graph G is a Cayley graph of a group X if and only if X is a sharply transitive (regular) subgroup of Aut(G). Sharply transitive means that for every u, v ∈ V (G) there is a unique φ ∈ X such that φ(u) = v.

Theorem An undirected Moore other than C5 is not a Cayley graph.

Proof: An automorphism φ with φ ◦ φ = Id and φ(v) 6= v for all vertices v. Directed Moore Graphs

The number of vertices in a directed Moore graph with degree k and diameter D is −→ M(k, D) = 1 + k + k2 + ... + kD.

Theorem Plesn´ık and Znam´ 1974, Bridges and Toueg 1980. Directed Moore graphs do not exist except in the trivial cases:

• D = 1, complete directed graphs Kk+1, and

• k = 1, directed cycles CD+1. −→ M(k, D) is an upper bound on the order of a directed graph with max. out-degree k and diameter D.

Lower bound (k ≥ 3): kD−1 + kD from Kautz digraphs.

If G is directed k-regular graph with diameter D then the line-digraph L(G) is k-regular and has diameter (at most) D + 1.

D−1 The Kautz digraph L (Kk+1) is a k-regular digraph with diameter D. Mixed Moore Graphs

Mixed graph: directed and undirected edges.

Theorem (Bosak,´ 1979) A mixed graph with diameter D with the property that for every pair (x, y) of vertices there is a unique path of length at most D from x to y is either an undirected tree or else there exist numbers t and z such

• every vertex has undirected degree t

• every vertex has directed out-degree z and directed in-degree z.

In the regular case the graph is called a mixed Moore graph. Theorem (Nguyen, Miller, Gimbert 2007) Mixed Moore graphs of diameter at least 3 do not exist. Consider mixed Moore graph of diameter 2.

Theorem (Bosak,´ 1979) If a mixed Moore graph is not a directed triangle or a 5-cycle then the eigenvalues are k = t + z with multiplicity 1 and (integers) −1±c where √ 2 c = 4t − 3.

c2+3 c is an odd positive integer t = 4 and c | (4z − 3)(4z + 5).

Possible values of t: 1, 3, 7, 13, 21,.... For each value of t: inﬁnitely many values of z. n = (t + z)2 + z + 1 t = 1 z = 1, 2, 3, 4,...

The Kautz digraph on (z + 2)(z + 1) vertices is the line-digraph of a the complete directed graph on z + 2 vertices, L(Kz+2).

This is a mixed Moore graph for every z.

Gimbert 2001: Kautz digraphs are unique mixed Moore graph with t = 1. Every permutation of the vertices of the Kz+2 induces an automorphism of L(Kz+2).

Theorem L(Kz+2) is a Cayley graph if and only if z + 2 is a prime-power.

Proof: Suppose that L(Kz+2) is a Cayley graph of a group X.

Then (Sabidussi) X has a sharply transitive action on vertices of L(Kz+2). X has sharply transitive action on edges (ordered pairs of distinct vertices) of Kz+2. X has sharply 2-transitive action on vertices of L(Kz+2).

Zassenhaus (1935): z + 2 is a prime-power and X is an afﬁne group AGL(1, z + 2) (or one of a few exceptional groups). Cay(X,S) is a mixed Moore graph if and only if

• for g ∈ S there is no pair (s1, s2) ∈ S × S such that g = s1s2

• for g∈ / S there is a unique pair (s1, s2) ∈ S × S such that g = s1s2

And then k = t + z = |S| and t = |S ∩ S(−1)|, where S(−1) = {s−1 | s ∈ S}. " # α a X = { | α, a ∈ GF [z + 2], α 6= 0}. 0 1

One possible choice for the connection set " # α 1 S = { | α ∈ GF [z + 2], α 6= 0}. 0 1

" #" # " # α 1 β 1 αβ α + 1 = 0 1 0 1 0 1 Feasible cases with t > 1 and n ≤ 200:

n t z 18 3 1 40 3 3 54 3 4 84 7 2 88 3 6 108 3 7 150 7 5 154 3 9 180 3 10 For t = 3, z = 1, Bosak´ constructed a graph. It has 18 vertices.

Bosak´ graph = Pappus graph + six directed triangles is mixed Moore graph.

Nguyen, Miller, Gimbert 2007: Bosak´ graph is unique.

The Bosak´ graph is a Cayley graph Cay(X,S), where

α 0 a   X = {0 α b | α, a, b ∈ GF [3], α 6= 0} 0 0 1

2 0 0 2 0 0 2 0 1 1 0 1         S = {0 2 0 , 0 2 1 , 0 2 0 , 0 1 1} 0 0 1 0 0 1 0 0 1 0 0 1 Pappus graph and Bosa´k graph Theorem J. 2015 There exist at least two mixed Moore graphs with n = 108, t = 3, z = 7.

T We denote these graph by G108 and G108.

One is transposed of the other.

They are both Cayley graphs of groups number 15 and 17 is the GAP catalogue of groups of order 108.

Group number 17 is isomorphic to automorphism group of the Bosak´ graph, and it is also isomorphic to

α a b   X = {0 β c | α, β, a, b, c ∈ GF [3], α, β 6= 0} 0 0 1 Theorem J. 2015 Suppose that G is mixed Moore graph with n = 40, t = z = 3. Then G does not contain an undirected 5-cycle. Theorem J. A mixed Moore with t = 7, z = 2, n = 84 does not exist.

Proof: Computer search.

Feasible cases with t > 1 and n ≤ 200:

n t z Exist ? 18 3 1 Bosak´ 40 3 3 54 3 4 84 7 2 NO 88 3 6 108 3 7 Cayley graph 150 7 5 154 3 9 180 3 10 Theorem J. A mixed Moore with t = 7, z = 2, n = 84 does not exist.

Proof: Computer search.

Feasible cases with t > 1 and n ≤ 200:

n t z Exist ? 18 3 1 Bosak´ 40 3 3 54 3 4 84 7 2 NO 88 3 6 108 3 7 Cayley graph 150 7 5 154 3 9 180 3 10