Some Recent Advances in the Analytic Enumeration of Circulant Graphs - - PowerPoint PPT Presentation

Some Recent Advances in the Analytic Enumeration of Circulant Graphs

Valery Liskovets

Institute of Mathematics National Academy of Sciences Minsk, Belarus liskov@im.bas-net.by

International Conference “Modern Trends in Algebraic Graph Theory” Villanova, PA, USA, June 2 – 5, 2014

Contents A brief survey

• The state of art
• Enumeration of undirected circulants of even order

and odd degree

• Enumeration of circulants of odd prime-power order:

the general case, p3 and perspectives A circulant graph = a circulant (for brevity) = a graph whose automorphism group contains a full cycle = a Cayley graph of a cyclic group. #(non-isomorphic circulants); enumeration by order only,

• r by order and degree (generating polynomial).

Analytic: exact enumeration represented by (closed) formulae (opposed to: constructive, numerical, algorithmic, approximate).

1

Contents A brief survey

• The state of art
• Enumeration of undirected circulants of even order

and odd degree

• Enumeration of circulants of odd prime-power order:

the general case, p3 and perspectives A circulant graph = a circulant (for brevity) = a graph whose automorphism group contains a full cycle = a Cayley graph of a cyclic group. #(non-isomorphic circulants); enumeration by order only,

• r by order and degree (generating polynomial).

Analytic: exact enumeration represented by (closed) formulae (opposed to: constructive, numerical, algorithmic, approximate).

1-a

The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: a P´

• lya-type formula with respect to

b) Directed, oriented, self-complementary, tournament, etc. circulants of prime orders p [. . . ]: similar formulae; numerous interconnections and formal identities c) Circulants of prime-squared orders p2 [Klin–L–P¨

• schel, 1996]:

via three P´

• lya-type subproblems (pairs of multipliers from

d) Circulants of odd prime-power orders pk [L–P, 2000]: reductive P´

• lya-type scheme without an explicit formula yet . . .

e) Circulants of square-free orders pqr : : : , and 2pqr : : : (2 p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: “closed-like” formula (due to the one-multiplier equivalence)

2

The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: a P´

• lya-type formula with respect to Z∗

p (multipliers m|p − 1)

b) Directed, oriented, self-complementary, tournament, etc. circulants of prime orders p [. . . ]: similar formulae; numerous interconnections and formal identities c) Circulants of prime-squared orders p2 [Klin–L–P¨

• schel, 1996]:

via three P´

• lya-type subproblems (pairs of multipliers from

d) Circulants of odd prime-power orders pk [L–P, 2000]: reductive P´

• lya-type scheme without an explicit formula yet . . .

e) Circulants of square-free orders pqr : : : , and 2pqr : : : (2 p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: “closed-like” formula (due to the one-multiplier equivalence)

2-a

The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: a P´

• lya-type formula with respect to Z∗

p (multipliers m|p − 1)

b) Directed, oriented, self-complementary, tournament, etc. circulants of prime orders p [. . . ]: similar formulae; numerous interconnections and formal identities c) Circulants of prime-squared orders p2 [Klin–L–P¨

• schel, 1996]:

via three P´

• lya-type subproblems (pairs of multipliers from

d) Circulants of odd prime-power orders pk [L–P, 2000]: reductive P´

• lya-type scheme without an explicit formula yet . . .

e) Circulants of square-free orders pqr : : : , and 2pqr : : : (2 p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: “closed-like” formula (due to the one-multiplier equivalence)

2-b

The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: a P´

• lya-type formula with respect to Z∗

p (multipliers m|p − 1)

b) Directed, oriented, self-complementary, tournament, etc. circulants of prime orders p [. . . ]: similar formulae; numerous interconnections and formal identities c) Circulants of prime-squared orders p2 [Klin–L–P¨

• schel, 1996]:

via three P´

• lya-type subproblems (pairs of multipliers from Z∗

p2)

d) Circulants of odd prime-power orders pk [L–P, 2000]: reductive P´

• lya-type scheme without an explicit formula yet . . .

e) Circulants of square-free orders pqr : : : , and 2pqr : : : (2 p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: “closed-like” formula (due to the one-multiplier equivalence)

2-c

The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: a P´

• lya-type formula with respect to Z∗

p (multipliers m|p − 1)

b) Directed, oriented, self-complementary, tournament, etc. circulants of prime orders p [. . . ]: similar formulae; numerous interconnections and formal identities c) Circulants of prime-squared orders p2 [Klin–L–P¨

• schel, 1996]:

via three P´

• lya-type subproblems (pairs of multipliers from Z∗

p2)

d) Circulants of odd prime-power orders pk [L–P, 2000]: reductive P´

• lya-type scheme without an explicit formula yet . . .

e) Circulants of square-free orders pqr : : : , and 2pqr : : : (2 p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: “closed-like” formula (due to the one-multiplier equivalence)

2-d

The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: a P´

• lya-type formula with respect to Z∗

p (multipliers m|p − 1)

b) Directed, oriented, self-complementary, tournament, etc. circulants of prime orders p [. . . ]: similar formulae; numerous interconnections and formal identities c) Circulants of prime-squared orders p2 [Klin–L–P¨

• schel, 1996]:

via three P´

• lya-type subproblems (pairs of multipliers from Z∗

p2)

d) Circulants of odd prime-power orders pk [L–P, 2000]: reductive P´

• lya-type scheme without an explicit formula yet . . .

e) Circulants of square-free orders pqr . . . , and 2pqr . . . (2 ≤ p < q < . . . ) [Alspach–Mishna, 2002], [K–L–P, 2003]: “closed-like” formula (due to the one-multiplier equivalence)

2-e

Little further progress. Two cases.

• 1. Undirected circulants of odd degree

Odd: due to “spokes” * in circulants of even order: *

O.

= (n; X) denotes the undirected circulant of order n with a symmetric connection set X [n 1] = f1; 2; : : : ; n 1g. ((x; y) 2 E() ( ) x y 2 X: Symmetric: x 2 X ( ) n x 2 X). Conjecture (*) [L, 1998]. For any even n = 2m and X; Y [n 1], (2m; X)

• =

(2m; Y ) if and only if (2m; X 4 fmg)

• =

(2m; Y 4 fmg): 4 denotes the set-theoretic symmetric difference.

O

! *

O

Theorem [Muzychuk, 2012]. Conjecture (*) is valid together with a natural generalization of it. !

3

Little further progress. Two cases.

• 1. Undirected circulants of odd degree

Odd: due to “spokes” * in circulants of even order: *

O.

Γ = Γ(n, X) denotes the undirected circulant of order n with a symmetric connection set X ⊆ [n − 1] = {1, 2, . . . , n − 1}. ((x, y) ∈ E(Γ) ⇐ ⇒ x − y ∈ X. Symmetric: x ∈ X ⇐ ⇒ n − x ∈ X). Conjecture (*) [L, 1998]. For any even n = 2m and X; Y [n 1], (2m; X)

• =

(2m; Y ) if and only if (2m; X 4 fmg)

• =

(2m; Y 4 fmg): 4 denotes the set-theoretic symmetric difference.

O

! *

O

Theorem [Muzychuk, 2012]. Conjecture (*) is valid together with a natural generalization of it. !

3-a

Little further progress. Two cases.

• 1. Undirected circulants of odd degree

Odd: due to “spokes” * in circulants of even order: *

O.

Γ = Γ(n, X) denotes the undirected circulant of order n with a symmetric connection set X ⊆ [n − 1] = {1, 2, . . . , n − 1}. ((x, y) ∈ E(Γ) ⇐ ⇒ x − y ∈ X. Symmetric: x ∈ X ⇐ ⇒ n − x ∈ X). Conjecture (*) [L, 1998]. For any even n = 2m and X, Y ⊆ [n − 1], Γ(2m, X) ∼ = Γ(2m, Y ) if and only if Γ(2m, X △ {m}) ∼ = Γ(2m, Y △ {m}). △ denotes the set-theoretic symmetric difference.

O ←

→ *

O

Theorem [Muzychuk, 2012]. Conjecture (*) is valid together with a natural generalization of it. !

3-b

Little further progress. Two cases.

• 1. Undirected circulants of odd degree

Odd: due to “spokes” * in circulants of even order: *

O.

Γ = Γ(n, X) denotes the undirected circulant of order n with a symmetric connection set X ⊆ [n − 1] = {1, 2, . . . , n − 1}. ((x, y) ∈ E(Γ) ⇐ ⇒ x − y ∈ X. Symmetric: x ∈ X ⇐ ⇒ n − x ∈ X). Conjecture (*) [L, 1998]. For any even n = 2m and X, Y ⊆ [n − 1], Γ(2m, X) ∼ = Γ(2m, Y ) if and only if Γ(2m, X △ {m}) ∼ = Γ(2m, Y △ {m}). △ denotes the set-theoretic symmetric difference.

O ←

→ *

O

Theorem [Muzychuk, 2012]. Conjecture (*) is valid together with a natural generalization of it. !

3-c

Enumerative corollary c(n, r) denotes the number of undirected circulants

• f order n and degree r.

Corollary. c(2m, 2s + 1) = c(2m, 2s), 0 ≤ s < m. Immediate (bijection). Reduces odd degrees to even. This identity was discovered empirically and verified by exhaustive search for m 25 [B. McKay, 1995]. Rather unexpected: its analog does not hold for Cayley graphs

• f Abelian groups in general.

4

Enumerative corollary c(n, r) denotes the number of undirected circulants

• f order n and degree r.

Corollary. c(2m, 2s + 1) = c(2m, 2s), 0 ≤ s < m. Immediate (bijection). Reduces odd degrees to even. This identity was discovered empirically and verified by exhaustive search for m ≤ 25 [B. McKay, 1995]. Rather unexpected: its analog does not hold for Cayley graphs

• f Abelian groups in general.

4-a

• 2. pk-Circulants, p odd prime

n = pk, Γ(n, X). The connection set X is partitioned into k “layers” in accordance with the divisibility by powers of p: X = X(0) ˙ ∪ X(1) ˙ ∪ · · · ˙ ∪ X(k−1) Fundamental isomorphism theorem [Klin–P¨

• schel, 1980].

Two circulants Γ = Γ(pk, X) and Γ′ = Γ(pk, X′) are isomorphic if and only if their respective layers are multiplicatively equivalent, i.e. X′

(i) = miX(i),

i = 0, 1, . . . , k − 1, for a set of multipliers m0, m1, . . . , mk−1 that satisfy the following constraints: whenever the layer X(i) satisfies the non-invariance condition (1 + pk−i−j−1)X(i) ̸= X(i) (Rij) for some i ∈ {0, 1, . . . , k − 2} and j ∈ {0, 1, . . . , k − 2 − i}, the successive multipliers mi, . . . , mk−j−1 meet the congruences mi+1 ≡ mi (mod pk−i−j−1) mi+2 ≡ mi+1 (mod pk−i−j−2) · · · mk−j−1 ≡ mk−j−2 (mod p)

      

(Eij)

5

Initial enumeration (sketchily) [L–P¨

• schel, 2000]

By the isomorphism theorem, the count of pk-circulants splits additively into k! “primary” P´

• lya-type subproblems Qf, where

Qf is parameterized by a function f : [0, k − 2] → [0, k − 2] such that i + f(i) ≤ k − 1 for all i. Qf is subtly specified by the set of the non-invariance conditions (1 + pk−i−j−1)X(i) ̸= X(i) for the pairs (i, j), j = f(i), and the set of the corresponding systems of congruences. Qf: to count orbits of the Abelian group Gf = (Z∗

pk ⊕ Z∗ pk−1 ⊕ · · · ⊕ Z∗ p)\Ef

acting multiplicatively component-wise on a set Cf

• f connection sets satisfying the mentioned restrictions.

\Ef denotes taking the subgroup of all multiplier tuples that meet all the congruences specified by f.

6

Further reduction k! orbit enumeration subproblem have been reduced to Ck basic problems PB for certain “basic” sets B of non-invariance pairs ij. Ck =

1 k+1

(2k

k

)

: the k-th Catalan number. 1, 2, 5, 14, 42, 132,. . . To specify a problem PB, it is convenient to represent its basic set B as a decorated South–East k-walk on the integer lattice . . . . . . Example: 1 2 3 4 5

7

Further reduction k! orbit enumeration subproblem have been reduced to Ck basic problems PB for certain “basic” sets B of non-invariance pairs ij. Ck =

1 k+1

(2k

k

)

: the k-th Catalan number. 1, 2, 5, 14, 42, 132,. . . To specify a problem PB, it is convenient to represent its basic set

B as a decorated South–East k-walk on the integer lattice . . . . . .

Example:

The 4-walk with B = {01, 10}: ⇓ 4

• |

3

• |

· 2

• |
• ·

1

• —o

|

• ·

∗

• —o—o—o

j/i 1 2 3 4

7-a

Table of basic orbit enumeration subproblems for pk-circulants, k ≤ 4 [L–P, 2000]

k No

B = {ij}∗

Equalities ∗∗ Congruences 1 1 ∅

• 2

1 ∅

• 2

00 m1 = m0

• 3

1 ∅

• 2

00 m2 = m1 = m0

• 3

01 m1 = m0

• 4

10 m2 = m1

• 5

01,10 m2 = m1 m1 ≡ m0 (mod p) 4 1 ∅

• 2

00 m3 = m2 = m1 = m0

• 3

01 m2 = m1 = m0

• 4

02 m1 = m0

• 5

10 m3 = m2 = m1

• 6

11 m2 = m1

• 7

20 m3 = m2

• 8

01,10 m3 = m2 = m1 m1 ≡ m0 (mod p2) 9 01,20 m3 = m2, m1 = m0 m2 ≡ m1 (mod p) 10 02,10 m3 = m2 = m1 m1 ≡ m0 (mod p) 11 02,11 m2 = m1 m1 ≡ m0 (mod p) 12 02,20 m3 = m2, m1 = m0

• 13

11,20 m3 = m2 m2 ≡ m1 (mod p) 14 02,11,20 m3 = m2 m2 ≡ m1 (mod p), m1 ≡ m0 (mod p)

∗ Basic pairs of indices for inequalities (1 + pk−i−j−1)X(i) ̸= X(i). ∗∗ Instead of congruences, for convenience.

8

Circulants of prime-cubed orders In 2013, Josef Lauri and his student Victoria Gatt succeeded in counting undirected and directed circulants of orders p3 for small p. Namely, for p = 3 and 5. In the analytic enumeration (only a part of their efforts) they followed the above-mentioned general scheme. Hard work. Seems almost sufficient to be generalized directly to arbitrary prime p. In particular, d(27) = 3728891 – the first previously unknown value for odd orders (d stands for #(directed circulants)). Verified constructively. Several promising intermediate formulae and identities. . . .

9

Basic subproblems for counting p3-circulants

k No

B = {ij}

Equalities Congruences 3 1 ∅

• 2

00 m2 = m1 = m0

• 3

01 m1 = m0

• 4

10 m2 = m1

• 5

01,10 m2 = m1 m1 ≡ m0 (mod p)

Five rather sophisticated P´

• lya-type subproblems:

C = A1 + A2 + A3 + A4 + A5. C: generic designation for the p3-circulant enumerator of a class. It makes sense to refine these terms by inclusion-exclusion in order to replace the specifying inequalities (B) by equalities: C = A1+A21A22+A31A32+A41A42+A51A521A522+A523 (implicit at G–L). 11 terms instead of 5. Just as we used 3 terms instead of 2 for counting p2-circulants.

10

Basic subproblems for counting p3-circulants

k No

B = {ij}

Equalities Congruences 3 1 ∅

• 2

00 m2 = m1 = m0

• 3

01 m1 = m0

• 4

10 m2 = m1

• 5

01,10 m2 = m1 m1 ≡ m0 (mod p)

Five rather sophisticated P´

• lya-type subproblems:

C = A1 + A2 + A3 + A4 + A5. C: generic designation for the p3-circulant enumerator of a class. It makes sense to refine these terms by inclusion-exclusion in order to replace the specifying inequalities (B) by equalities: C = A1+A21−A22+A31−A32+A41−A42+A51−A521−A522+A523 (implicit at G–L). 11 terms instead of 5. Just as we used 3 terms instead of 2 for counting p2-circulants.

10-a

The general prime-power case revisited Still challenging. Forthcoming tasks. First of all, to finish the case p3 for arbitrary p and diverse classes of circulants such as self-complementary. To simplify the resulting formula as much as possible. To analyze identities and other interconnections between intermediate quantities. What further? k = 4 with 14 tedious subproblems (moreover, after the refinement by inclusion-exclusion there will be 45 terms to be specified and calculated)?

• No. The next should (and can, I believe) be

the general case pk for an arbitrary k and all prime p > 2 !

11

The general prime-power case revisited Still challenging. Forthcoming tasks. First of all, to finish the case p3 for arbitrary p and diverse classes of circulants such as self-complementary. To simplify the resulting formula as much as possible. To analyze identities and other interconnections between intermediate quantities. What further? k = 4 with 14 tedious subproblems (moreover, after the refinement by inclusion-exclusion there will be 45 terms to be specified and calculated)?

• No. The next should (and can, I believe) be

the general case pk for an arbitrary k and all prime p > 2 !

11-a

The general prime-power case revisited Still challenging. Forthcoming tasks. First of all, to finish the case p3 for arbitrary p and diverse classes of circulants such as self-complementary. To simplify the resulting formula as much as possible. To analyze identities and other interconnections between intermediate quantities. What further? k = 4 with 14 tedious subproblems (moreover, after the refinement by inclusion-exclusion there will be 45 terms to be specified and calculated)?

• No. The next should (and can, I believe) be

the general case pk for an arbitrary k and all prime p > 2 !

11-b

On a general formula for pk-circulants A presumable “quasi-closed” algorithmized formula with the computer-calculated terms as the solutions of the corresponding computer-generated basic P´

• lya-type orbit enumeration problems.

With the help of a program package such as GAP or Maple. Some further simplifications and hidden interconnections (quite probable, particularly for small k, and most intriguing). One beautiful feature: after the refinement by inclusion-exclusion, the expected general formula will contain sk terms, where sk stand for the famous (small) Schr¨

• der numbers:

1, 3, 11, 45, 197, 903, . . . Are also known as super-Catalan numbers. sk grow exponentially but faster than the Catalan numbers: sk (3 + 2 2)k (instead of Ck 4k).

Digression in conclusion: I know of 160+ distinct combinatorial interpretations of the Schr¨

• der numbers!

12

On a general formula for pk-circulants A presumable “quasi-closed” algorithmized formula with the computer-calculated terms as the solutions of the corresponding computer-generated basic P´

• lya-type orbit enumeration problems.

With the help of a program package such as GAP or Maple. Some further simplifications and hidden interconnections (quite probable, particularly for small k, and most intriguing). One beautiful feature: after the refinement by inclusion-exclusion, the expected general formula will contain sk terms, where sk stand for the famous (small) Schr¨

• der numbers:

1, 3, 11, 45, 197, 903, . . . Are also known as super-Catalan numbers. sk grow exponentially but faster than the Catalan numbers: sk (3 + 2 2)k (instead of Ck 4k).

Digression in conclusion: I know of 160+ distinct combinatorial interpretations of the Schr¨

• der numbers!

12-a

On a general formula for pk-circulants A presumable “quasi-closed” algorithmized formula with the computer-calculated terms as the solutions of the corresponding computer-generated basic P´

• lya-type orbit enumeration problems.

With the help of a program package such as GAP or Maple. Some further simplifications and hidden interconnections (quite probable, particularly for small k, and most intriguing). One beautiful feature: after the refinement by inclusion-exclusion, the expected general formula will contain sk terms, where sk stand for the famous (small) Schr¨

• der numbers:

1, 3, 11, 45, 197, 903, . . . Are also known as super-Catalan numbers. sk grow exponentially but faster than the Catalan numbers: sk ≈ (3 + 2 √ 2)k (instead of Ck ≈ 4k).

Digression in conclusion: I know of 160+ distinct combinatorial interpretations of the Schr¨

• der numbers!

12-b

Thank you! My special gratitude goes to Misha KLIN, who attracted me to these problems 20 years ago.

13