Constructive and analytic enumeration of circulant graphs with p 3 vertices; p = 3 , 5 Joint work with Victoria Gatt, Mikhail Klin, Valery Liskovets (with help from Matan Ziv-Av) October 11, 2016
Background We work within the framework of two papers: Klin, Liskovets and R. P¨ oschel, Analytic enumeration of circulant graphs with prime-squared number of vertices , S´ em. Lotharing. Combin., 1996, Volume B36d. Contructive (structural) approach : based on the known classification of circulant graphs in terms of S -rings. This not only counts the nonisomorphic circulant graphs but enables us, in principle at least, actually to list them. Analytic (multiplier) approach : based on isomorphism theorems for circulant graphs of prime-power orders (a correction of the false conjecture of ´ Ad´ am) as given in Liskovets and P¨ oschel, Counting circulant graphs of prime-power order by decomposing into orbit enumeration problems . Discrete Math., 2000, Volume 214, 173–191. This allows the use of Redfield–P´ olya type enumeration.
Motivation 1. There are almost no computational results for the number of isomorphism classes of circulant graphs of prime-cubed orders. (We restrict ourselves to orders 27 and 125 because the number of circulant graphs obtained for order 125 is already very large.) 2. The generating functions which we obtain reveal some surprising relations between their intermediate terms and which lead us to make some general conjectures. 3. An important aspect of counting circulant graphs stems from the relationship with the falsity of ´ Ad´ am’s Conjecture. This conjecture is, in turn, related to an important more general thread of research in algebraic graph theory: Cayley-graph Isomorphism. 4. To illustrate the use of Schur rings in AGT.
Definitions A circulant graph is a Cayley graph Cay( G = Z n , S ) of a cyclic group G = Z n where S ⊆ G (called the connecting set of the Cayley graph) such that 0 �∈ S and S generates G = Z n . Cay( G = Z n , S ) has G = Z n as vertex-set and there is an arc from vertex h to vertex g in Cay( G = Z n , S ) if g = h + s for some s ∈ S . When − S = S (that is, s ∈ S if and only if − s ∈ S ), the circulant graph is also referred to as an undirected circulant graph otherwise it is a directed circulant graph . The valency of a vertex in an undirected graph is equal to the number of edges containing that vertex. The valency of a vertex v in a directed graph is equal to the number of arcs of the form ( v , x ). An edge { a , b } is considered to be the union of the two arcs ( a , b ) and ( b , a ), and this is consistent with our definitions of valency for directed and undirected graphs. Undirected graphs contain only arcs paired this way but directed graphs could also be “mixed”, in the sense that they could contain both arcs and edges.
´ Ad´ am’s Conjecture If Γ 1 = Cay( Z n , S ) and Γ 2 = Cay( Z n , T ) such that there exists an m ∈ Z ∗ n with mS = { ms : s ∈ S } = T , then Γ 1 and Γ 2 are isomorphic. In this case we say that the connecting sets are equivalent . In 1967 ´ Ad´ am conjectured that the converse is also true, that is, two isomorphic circulant graphs have equivalent connecting sets. This conjecture turned out to be false. The following is the smallest counterexample, found by Elspas and Turner in 1970. It is a pair of directed circulants: Consider Z 8 and let S = { 1 , 2 , 5 } and T = { 1 , 5 , 6 } . Then the sets S , T are not equivalent but Cay( Z 8 , S ) and Cay( Z 8 , T ), are isomorphic via the map � i + 1 � i �→ 4 + i . 2
b b b b b b b b b b b b b b b b b b Graphic illustration of failure of ´ Ad´ am’s Conjecture Γ 1 = Cay( Z 9 , { 1 , 3 , 4 , 7 } ) Γ 2 = Cay( Z 9 , { 1 , 6 , 4 , 7 } ) 0 3 0 3 6 4 1 4 1 2 5 5 2 7 7 8 8 Γ 2 Γ 1
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Graphic illustration of failure of ´ Ad´ am’s Conjecture Γ 1 = Cay( Z 25 , {± 1 , ± 4 , ± 5 , ± 6 , ± 9 , ± 11 } ) Γ 2 = Cay( Z 25 , {± 1 , ± 4 , ± 6 , ± 9 , ± 10 , ± 11 } ) 0 0 5 5 20 20 15 10 15 10 4 1 4 1 24 9 24 9 6 6 21 21 14 11 14 19 16 16 11 19 2 2 3 3 22 22 23 8 7 8 7 23 12 17 17 12 18 13 18 13 Γ 1 Γ 2
Muzychuk’s Theorem The principal theorem which gives the most correct version (with respect to order of the circulant) of ´ Ad´ am’s Conjecture is the following due to Muzychuk. Theorem Let Γ 1 and Γ 2 be two circulant graphs on n vertices, and suppose that n is square-free or twice a square-free number (and a few other small orders for undirected cases). Then Γ 1 , Γ 2 are isomorphic if and only if their connecting sets are equivalent.
The constructive (structural) approach: Schur rings A subring S of a group ring Z [ Z n ] is called a Schur ring S or S -ring over Z n if the following conditions hold: 1. S is closed under addition and multiplication and also pre-multiplication with elements from Z (i.e. S is a Z -module); 2. Simple quantities T 0 , T 1 , ..., T r − 1 exist in S such that every element σ ∈ S has a unique representation; r − 1 � σ = σ i T i i =0 3. T 0 = 0, � r − 1 i =0 T i = Z n , that is, { T 0 , T 1 , . . . , T r − 1 } is a partition of Z n ; 4. For every i ∈ { 0 , 1 , 2 , ..., r − 1 } there exists a j ∈ { 0 , 1 , 2 , . . ., r − 1 } such that T j = − T i (= { n − x : x ∈ T i } ); 5. For i , j ∈ { 1 , ..., r } , there exist non-negative integers p k ij called structure constants, such that r � p k T i · T j = ij T k k =1 The simple quantities T 0 , T 1 , ..., T r − 1 form a standard basis for S and their corresponding sets T i are basic sets of the S -ring.
Constructive (structural) approach: basic circulants of Schur rings Let S be a Schur ring over Z n generated by simple quantities T 0 , T 1 , ..., T r − 1 whose corresponding sets T i are called basic sets of the S -ring S . The circulant graphs Γ i = Cay( Z n , T i ), where 0 ≤ i ≤ r − 1, are called basic circulant graphs of S . This situation is denoted by S = � T 0 , T 1 , ... T r − 1 � . A permutation g : Z n → Z n is called an automorphism of an S -ring S , if it is an automorphism of every circulant Γ i . Therefore the intersection of the automorphism groups of the basic circulants of S , gives the automorphism group of the S -ring: r − 1 � Aut S := Aut Γ i i =0
Constructive (structural) approach: the lattice of S -rings Let L ( n ) be the lattice of all Schur rings over Z n given as a sequence L ( n ) = ( S 1 , S 2 , ... S s ) such that S j ⊆ S i implies j ≥ i .
The lattice of S -rings: counting labelled circulants We first use the lattice of Schur rings to count the number of labelled circulant graphs, as follows. 1. For directed circulants, let ˜ d ir be the number of r -element basis sets of the S -ring S i , different from the basis set T 0 = { 0 } , that is, ˜ d ir := |{ T ( x ) ∈ S i | x � = 0 and | T ( x ) | = r }| 2. For undirected circulants, let d ir be the number of r -element symmetrized (that is closed under taking of inverses) basis sets of S i , different from T 0 . That is, d ir := |{ T sym ( x ) | x � = 0 and | T sym ( x ) | = r }| 3. The generating functions of all labelled directed and undirected circulant graphs which belong to the Schur ring S i are given by ˜ f i ( t ) and f i ( t ) respectively, given by: n − 1 n − 1 f ir t r := ˜ ˜ � ˜ � (1 + t r ) d ir f i ( t ) := r =0 r =1 n − 1 n − 1 f ir t r := � � (1 + t r ) d ir f i ( t ) := r =0 r =1
The lattice of S -rings: counting unlabelled circulants The link between the number of labelled and unlabelled circulant graphs is given by this result. Lemma Let G i = Aut ( S i ) , let N ( G i ) = N S n ( G i ) be the normalizer of the group G i in S n , and let Γ be a circulant graph belonging to S i . Then (a) Aut (Γ) = G i ⇐ ⇒ Γ generates S i . (b) If Aut (Γ) = G i then there are exactly [ N ( G i ) : G i ] (that is, equal to the number of cosets of G i in N ( G i ) ) distinct circulant graphs which are isomorphic to Γ . Provided all S -rings in consideration are “Schurian”. “All S -rings of a cyclic group of prime-power order are Schurian” — Evdokimov and Ponomarenko.
The lattice of S -rings: counting unlabelled circulants So, let the generating function for the number of non-isomorphic undirected circulant graphs with automorphism group G i be given by n − 1 � g ir t r g i ( t ) = r =0 and let the generating function for the number of non-isomorphic directed circulant graphs with automorphism group G i be given by n − 1 � g ir t r g i ( t ) = ˜ ˜ r =0 And let g ( t ) = g ( n , t ) and ˜ g ( t ) = ˜ g ( n , t ) denote the generating functions for the number of non-isomorphic undirected and directed circulant graphs, respectively, with n vertices.
The generating functions for unlabelled circulants These generating functions are then given by the following theorem whose proof is based on the inclusion-exclusion principle. Theorem | G i | | N ( G j ) | � , g i ( t ) = f i ( t ) − g j ( t ) | N ( G i ) | | G j | S j ⊂ S i | G i | | N ( G j ) | (1) ˜ � , g i ( t ) = ˜ f i ( t ) − g j ( t ) ˜ | N ( G i ) | | G j | S j ⊂ S i s s � � g ( t ) = g i ( t ) , g ( t ) = ˜ ˜ g i ( t ) . i =1 i =1
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