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
a graph whose automorphism group contains a full cycle = A circulant graph = a circulant (for brevity) = a Cayley graph of a cyclic group. #(non-isomorphic circulants); enumeration by order only, or by order and degree (generating polynomial). Analytic: exact enumeration represented by (closed) formulae (opposed to: constructive, numerical, algorithmic, approximate). 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, p 3 and perspectives 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, p 3 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, or by order and degree (generating polynomial). Analytic: exact enumeration represented by (closed) formulae (opposed to: constructive, numerical, algorithmic, approximate). 1-a
via three P´ olya-type subproblems (pairs of multipliers from a P´ olya-type formula with respect to b) Directed, oriented, self-complementary, tournament, etc. “closed-like” formula (due to the one-multiplier equivalence) numerous interconnections and formal identities olya-type scheme without an explicit formula yet . . . oschel, 1996]: reductive P´ The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: circulants of prime orders p [. . . ]: similar formulae; c) Circulants of prime-squared orders p 2 [Klin–L–P¨ d) Circulants of odd prime-power orders p k [L–P, 2000]: e) Circulants of square-free orders pqr : : : , and 2 pqr : : : ( 2 � p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: 2
oschel, 1996]: via three P´ “closed-like” formula (due to the one-multiplier equivalence) olya-type scheme without an explicit formula yet . . . reductive P´ b) Directed, oriented, self-complementary, tournament, etc. olya-type subproblems (pairs of multipliers from numerous interconnections and formal identities The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: olya-type formula with respect to Z ∗ a P´ p (multipliers m | p − 1 ) circulants of prime orders p [. . . ]: similar formulae; c) Circulants of prime-squared orders p 2 [Klin–L–P¨ d) Circulants of odd prime-power orders p k [L–P, 2000]: e) Circulants of square-free orders pqr : : : , and 2 pqr : : : ( 2 � p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: 2-a
oschel, 1996]: via three P´ “closed-like” formula (due to the one-multiplier equivalence) olya-type scheme without an explicit formula yet . . . reductive P´ olya-type subproblems (pairs of multipliers from The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: olya-type formula with respect to Z ∗ a P´ 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 p 2 [Klin–L–P¨ d) Circulants of odd prime-power orders p k [L–P, 2000]: e) Circulants of square-free orders pqr : : : , and 2 pqr : : : ( 2 � p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: 2-b
“closed-like” formula (due to the one-multiplier equivalence) reductive P´ olya-type scheme without an explicit formula yet . . . The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: olya-type formula with respect to Z ∗ a P´ 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 p 2 [Klin–L–P¨ oschel, 1996]: olya-type subproblems (pairs of multipliers from Z ∗ via three P´ p 2 ) d) Circulants of odd prime-power orders p k [L–P, 2000]: e) Circulants of square-free orders pqr : : : , and 2 pqr : : : ( 2 � p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: 2-c
“closed-like” formula (due to the one-multiplier equivalence) The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: olya-type formula with respect to Z ∗ a P´ 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 p 2 [Klin–L–P¨ oschel, 1996]: olya-type subproblems (pairs of multipliers from Z ∗ via three P´ p 2 ) d) Circulants of odd prime-power orders p k [L–P, 2000]: reductive P´ olya-type scheme without an explicit formula yet . . . e) Circulants of square-free orders pqr : : : , and 2 pqr : : : ( 2 � p < q < : : : ) [Alspach–Mishna, 2002], [K–L–P, 2003]: 2-d
The state of art (until recently) a) Undirected circulants of prime orders p [Turner, 1967]: olya-type formula with respect to Z ∗ a P´ 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 p 2 [Klin–L–P¨ oschel, 1996]: olya-type subproblems (pairs of multipliers from Z ∗ via three P´ p 2 ) d) Circulants of odd prime-power orders p k [L–P, 2000]: reductive P´ olya-type scheme without an explicit formula yet . . . e) Circulants of square-free orders pqr . . . , and 2 pqr . . . ( 2 ≤ p < q < . . . ) [Alspach–Mishna, 2002], [K–L–P, 2003]: “closed-like” formula (due to the one-multiplier equivalence) 2-e
together with a natural generalization of it. O ! if and only if 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] = f 1 ; 2 ; : : : ; n � 1 g . ( ( x; y ) 2 E (�) ( ) x � y 2 X: Symmetric: x 2 X ( ) n � x 2 X ). Conjecture ( * ) [L, 1998]. For any even n = 2 m and X; Y � [ n � 1] , � �(2 m; X ) = �(2 m; Y ) � �(2 m; X 4 f m g ) = �(2 m; Y 4 f m g ) : O ! * 4 denotes the set-theoretic symmetric difference. Theorem [Muzychuk, 2012]. Conjecture ( * ) is valid 3
together with a natural generalization of it. O ! if and only if 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 = 2 m and X; Y � [ n � 1] , � �(2 m; X ) = �(2 m; Y ) � �(2 m; X 4 f m g ) = �(2 m; Y 4 f m g ) : O ! * 4 denotes the set-theoretic symmetric difference. Theorem [Muzychuk, 2012]. Conjecture ( * ) is valid 3-a
Theorem [Muzychuk, 2012]. Conjecture ( * ) is valid ! together with a natural generalization of it. 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 = 2 m and X, Y ⊆ [ n − 1] , ∼ Γ(2 m, X ) Γ(2 m, Y ) = if and only if ∼ Γ(2 m, X △ { m } ) Γ(2 m, Y △ { m } ) . = → * O ← O △ denotes the set-theoretic symmetric difference. 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 = 2 m and X, Y ⊆ [ n − 1] , ∼ Γ(2 m, X ) Γ(2 m, Y ) = if and only if ∼ Γ(2 m, X △ { m } ) Γ(2 m, Y △ { m } ) . = → * O ← O △ denotes the set-theoretic symmetric difference. Theorem [Muzychuk, 2012]. Conjecture ( * ) is valid together with a natural generalization of it. ! 3-c
This identity was discovered empirically and verified Rather unexpected: its analog does not hold for Cayley graphs of Abelian groups in general. Enumerative corollary c ( n, r ) denotes the number of undirected circulants of order n and degree r . Corollary . c (2 m, 2 s + 1) = c (2 m, 2 s ) , 0 ≤ s < m. Immediate (bijection). Reduces odd degrees to even. by exhaustive search for m � 25 [B. McKay, 1995]. 4
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