Tomaz Pisanski - cp's OEIS frontend

A153846 Number of non-isomorphic I-graphs I(n,j,k) on 2n vertices (1<=j,k<=Floor[(n-1)/2]).

1, 1, 2, 3, 2, 4, 4, 6, 3, 11, 4, 7, 10, 10, 5, 14, 5, 17, 12, 11, 6, 28, 10, 14, 13, 21, 8, 35, 8, 22, 17, 18, 17, 41, 10, 19, 20, 40, 11, 44, 11, 31, 32, 23, 12, 60, 16, 36, 25, 37, 14, 49, 24, 50, 27, 30, 15, 93, 16, 31, 40, 46, 29, 64, 17, 47, 32, 63, 18, 96, 19, 38, 49, 51, 30

Offset: 3

Tomaz Pisanski, Jan 08 2009

nonn

The I-graph I(n,j,k) is a graph with vertex set V(I(n,j,k)) = {u_0,u_1,...,u_{n-1},v_0,v_1,...,v_{n-1}} and edge set E(I(n,j,k)) = {u_i u_{i+j}, u_i v_i, v_i v_{i+k} : i=0,...,n-1}, where the subscripts are to be read modulo n. The I-graphs generalize the family of generalized Petersen graphs.

I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star, The Foster Census (Charles Babbage Research Centre, 1988), ISBN 0-919611-19-2.

Marko Boben, Tomaz Pisanski, Arjana Zitnik, I-graphs and the corresponding configurations J. Combin. Des. 13 (2005), no. 6, 406--424.
B. Horvat, T. Pisanski; A. Zitnik. Isomorphism checking of I-graphs, Graphs Comb. 28, No. 6, 823-830 (2012).
M. Watkins, A theorem on Tait colorings with an application to the generalized Petersen graphs, J. Combin. Theory 6 (1969), 152-164.
Eric Weisstein's World of Mathematics, Graph Expansion

Cf. A077105, A153847.

A153847 Number of non-isomorphic connected I-graphs I(n,j,k) on 2n vertices (1<=j,k<=Floor[(n-1)/2]).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 4, 3, 7, 4, 5, 7, 6, 5, 8, 5, 10, 9, 8, 6, 14, 8, 10, 9, 13, 8, 19, 8, 12, 13, 13, 13, 19, 10, 14, 15, 20, 11, 25, 11, 19, 19, 17, 12, 26, 14, 22, 19, 22, 14, 26, 19, 26, 21, 22, 15, 40, 16, 23, 25, 24, 23, 37, 17, 28, 25, 37, 18, 38, 19, 28, 31, 31, 25, 43, 20

Offset: 3

Tomaz Pisanski, Jan 08 2009

nonn

The I-graph I(n,j,k) is a graph with vertex set V(I(n,j,k)) = {u_0,u_1,...,u_{n-1},v_0,v_1,...,v_{n-1}} and edge set E(I(n,j,k)) = {u_i u_{i+j}, u_i v_i, v_i v_{i+k} : i=0,...,n-1}, where the subscripts are to be read modulo n. The I-graphs generalize the family of generalized Petersen graphs.

I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star, The Foster Census (Charles Babbage Research Centre, 1988), ISBN 0-919611-19-2.

Marko Boben, Tomaz Pisanski, Arjana Zitnik, I-graphs and the corresponding configurations J. Combin. Des. 13 (2005), no. 6, 406--424.
M. Watkins, A theorem on Tait colorings with an application to the generalized Petersen graphs, J. Combin. Theory 6 (1969), 152-164.
Eric Weisstein's World of Mathematics, Graph Expansion

Cf. A077105, A153846.

A112921 Number of nonisomorphic Y-graphs Y(n:i,j,k) on 4n vertices (or nodes) for 1<=i,j,k

Original entry on oeis.org

1, 1, 2, 4, 4, 6, 8, 10, 7, 24, 10, 20, 26, 26, 15, 44, 19, 54, 44, 44, 26, 102, 38, 62, 57, 96, 40, 164, 46, 104, 91, 102, 91, 213, 64, 128, 124, 222, 77, 290, 85, 212, 200, 184, 100, 388, 128, 268, 199, 292, 126

Offset: 3

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), Oct 06 2005

A Y-graph Y(n:i,j,k) has 4n vertices arranged in four segments of n vertices. Let the vertices be v_{x,y} for x=0,1,2,3 and y in the integers modulo n. The edges are v_{1,y}v_{1,y+i}, v_{2,y}v_{2,y+j}, v_{2,y}v_{2,y+k} and v_{0,y}v_{x,y}, where y=0,1,...,n-1 and x=1,2,3 and the subscript addition is performed modulo n.

			Y(7:1,2,3) is the Coxeter graph, the only (connected) symmetric (vertex- and edge-transitive) Y-graph of girth 7 or less.

I. Z. Bouwer, W. W. Chernoff, B. Monson, and Z. Starr (Editors), "Foster's Census", Charles Babbage Research Centre, Winnipeg, 1988.

J. D. Horton and I. Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory B 53 (1991) 114-129.

Cf. A112922, A112923, A112924, A112921, A107452.

A112917 Number of nonisomorphic H-graphs H(n:i,j;k,m) on 6n vertices (or nodes) for 1<=i,j,k,m

Original entry on oeis.org

1, 1, 4, 6, 7, 13, 19, 31, 24, 76, 41, 77, 116, 116, 87, 226, 115, 307, 276, 308, 201, 671, 317, 523, 478, 786, 403, 1495

Offset: 3

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), Oct 06 2005

An H-graph H(n:i,j;k,m) has 6n vertices arranged in six segments of n vertices. Let the vertices be v_{x,y} for x=0,1,2,3,4,5 and y in the integers modulo n. The edges are v_{0,y}v_{1,y}, v_{0,y}v_{2,y}, v_{0,y}v_{3,y}, v_{1,y}v_{4,y}, v_{1,y}v_{5,y} (inner edges) and v_{2,y}v_{2,y+i}, v_{3,y}v_{3,y+j}, v_{4,y}v_{3,y+k}, v_{5,y}v_{5,y+m} (outer edges) where y=0,1,...,n-1 and subscript addition is performed modulo n.

			The only connected symmetric H-graphs are H(17:1,4;2,8) and H(34:1,13;9,15) which are also listed in Foster's Census.

I. Z. Bouwer, W. W. Chernoff, B. Monson, and Z. Starr (Editors), "Foster's Census", Charles Babbage Research Centre, Winnipeg, 1988.

J. D. Horton and I. Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory B 53 (1991) 114-129.

Cf. A112918, A112919, A112920, A112921, A107452.

A112918 Number of nonisomorphic connected H-graphs H(n:i,j;k,m) on 6n vertices (or nodes) for 1<=i,j,k,m

Original entry on oeis.org

1, 1, 4, 5, 7, 12, 18, 27, 24, 69, 41, 70, 111, 103, 87, 202, 115, 275, 268, 284, 201, 583, 313, 482, 459, 708, 403, 1347

Offset: 3

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), Oct 06 2005

An H-graph H(n:i,j;k,m) has 6n vertices arranged in six segments of n vertices. Let the vertices be v_{x,y} for x=0,1,2,3,4,5 and y in the integers modulo n. The edges are v_{0,y}v_{1,y}, v_{0,y}v_{2,y}, v_{0,y}v_{3,y}, v_{1,y}v_{4,y}, v_{1,y}v_{5,y} (inner edges) and v_{2,y}v_{2,y+i}, v_{3,y}v_{3,y+j}, v_{4,y}v_{3,y+k}, v_{5,y}v_{5,y+m} (outer edges) where y=0,1,...,n-1 and subscript addition is performed modulo n. H-graph H(n:i,j;k,m) is connected if and only if gcd(n,i,j,k,m) = 1.

			The only connected symmetric H-graphs are H(17:1,4;2,8) and H(34:1,13;9,15) which are also listed in Foster's Census.

I. Z. Bouwer, W. W. Chernoff, B. Monson, and Z. Starr (Editors), "Foster's Census", Charles Babbage Research Centre, Winnipeg, 1988.

J. D. Horton and I. Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory B 53 (1991) 114-129.

Cf. A112917, A112919, A112920.

A112919 Number of nonisomorphic connected bipartite H-graphs H(n:i,j;k,m) on 6n vertices (or nodes) for 1<=i,j,k,m

Original entry on oeis.org

0, 1, 0, 1, 0, 4, 0, 4, 0, 12, 0, 7, 0, 16, 0, 18, 0, 33, 0, 24, 0, 67, 0, 41, 0, 71, 0, 111

Offset: 3

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), Oct 06 2005

An H-graph H(n:i,j;k,m) has 6n vertices arranged in six segments of n vertices. Let the vertices be v_{x,y} for x=0,1,2,3,4,5 and y in the integers modulo n. The edges are v_{0,y}v_{1,y}, v_{0,y}v_{2,y}, v_{0,y}v_{3,y}, v_{1,y}v_{4,y}, v_{1,y}v_{5,y} (inner edges) and v_{2,y}v_{2,y+i}, v_{3,y}v_{3,y+j}, v_{4,y}v_{3,y+k}, v_{5,y}v_{5,y+m} (outer edges) where y=0,1,...,n-1 and subscript addition is performed modulo n.

			The only connected symmetric bipartite H-graph is H(34:1,13;9,15) which is also listed in Foster's Census.

I. Z. Bouwer, W. W. Chernoff, B. Monson, and Z. Starr (Editors), "Foster's Census", Charles Babbage Research Centre, Winnipeg, 1988.

J. D. Horton and I. Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory B 53 (1991) 114-129.

Cf. A112917, A112918, A112920.

A112920 Number of nonisomorphic connected bipartite H-graphs H(n:i,j;k,m) with girth 6 on 6n vertices (or nodes) for 1<=i,j,k,m

Original entry on oeis.org

0, 0, 0, 1, 5, 3, 5, 3, 13, 8, 19, 27, 9, 19, 33, 74, 41, 19, 61, 75, 61, 137, 51, 108, 95, 111, 99, 217

Offset: 3

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), Oct 06 2005

An H-graph H(n:i,j;k,m) has 6n vertices arranged in six segments of n vertices. Let the vertices be v_{x,y} for x=0,1,2,3,4,5 and y in the integers modulo n. The edges are v_{0,y}v_{1,y}, v_{0,y}v_{2,y}, v_{0,y}v_{3,y}, v_{1,y}v_{4,y}, v_{1,y}v_{5,y} (inner edges) and v_{2,y}v_{2,y+i}, v_{3,y}v_{3,y+j}, v_{4,y}v_{3,y+k}, v_{5,y}v_{5,y+m} (outer edges) where y=0,1,...,n-1 and subscript addition is performed modulo n.

			The smallest H-graph with girth 6 is H(6:1,1;1,1).

I. Z. Bouwer, W. W. Chernoff, B. Monson, and Z. Starr (Editors), "Foster's Census", Charles Babbage Research Centre, Winnipeg, 1988.

J. D. Horton and I. Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory B 53 (1991) 114-129.

Cf. A112917, A112919, A112920.

A112922 Number of nonisomorphic connected Y-graphs Y(n:i,j,k) on 4n vertices (or nodes) for 1<=i,j,k

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 7, 19, 10, 16, 23, 20, 15, 33, 19, 43, 39, 37, 26, 73, 36, 52, 49, 75, 40, 127, 46, 78, 83, 87, 85, 149, 64, 109, 113, 163, 77, 227, 85, 167, 167, 158, 100, 266, 124, 222, 183, 229, 126

Offset: 3

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), Oct 06 2005

A Y-graph Y(n:i,j,k) has 4n vertices arranged in four segments of n vertices. Let the vertices be v_{x,y} for x=0,1,2,3 and y in the integers modulo n. The edges are v_{1,y}v_{1,y+i}, v_{2,y}v_{2,y+j}, v_{2,y}v_{2,y+k} and v_{0,y}v_{x,y}, where y=0,1,...,n-1 and x=1,2,3 and the subscript addition is performed modulo n. It is connected if and only if gcd(n,i,j,k) = 1.

			Y(7:1,2,3) is the Coxeter graph, the only symmetric (vertex- and edge-transitive) Y-graph of girth 7 or less.

I. Z. Bouwer, W. W. Chernoff, B. Monson, and Z. Starr (Editors), "Foster's Census", Charles Babbage Research Centre, Winnipeg, 1988.

J. D. Horton and I. Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory B 53 (1991) 114-129.

Cf. A112921, A112923, A112924.

A112923 Number of nonisomorphic connected bipartite Y-graphs Y(n:i,j,k) on 8n vertices (or nodes) for 1<=i,j,k<=n.

Original entry on oeis.org

1, 1, 2, 2, 5, 4, 5, 7, 9, 7, 14, 10, 15, 23, 15, 15, 27, 19, 28, 39, 29, 26, 45, 36, 39

Offset: 2

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), Oct 06 2005

A Y-graph Y(n:i,j,k) has 4n vertices arranged in four segments of n vertices. Let the vertices be v_{x,y} for x=0,1,2,3 and y in the integers modulo n. The edges are v_{1,y}v_{1,y+i}, v_{2,y}v_{2,y+j}, v_{2,y}v_{2,y+k} and v_{0,y}v_{x,y}, where y=0,1,...,n-1 and x=1,2,3 and the subscript addition is performed modulo n.

			Y(4:1,1,1) is the smallest bipartite Y-graph.
Y(14:1,3,5) is the smallest bipartite symmetric (vertex- and edge-transitive) Y-graph.

I. Z. Bouwer, W. W. Chernoff, B. Monson, and Z. Starr (Editors), "Foster's Census", Charles Babbage Research Centre, Winnipeg, 1988.

J. D. Horton and I. Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory B 53 (1991) 114-129.

Cf. A112921, A112922, A112924.

A112924 Number of nonisomorphic connected Y-graphs Y(n:i,j,k) with girth 6 on 4n vertices (or nodes) for 1<=i,j,k<=n.

Original entry on oeis.org

0, 0, 0, 1, 3, 2, 3, 2, 5, 3, 6, 6, 4, 4, 8, 12, 9, 4, 12, 10, 11, 19, 10, 12, 15, 12, 14, 22, 15, 12, 20, 16, 18, 31, 18, 18, 24, 16, 20, 50, 21, 20, 28, 22, 23, 50, 27, 24, 32, 24, 26

Offset: 3

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), Oct 06 2005

A Y-graph Y(n:i,j,k) has 4n vertices arranged in four segments of n vertices. Let the vertices be v_{x,y} for x=0,1,2,3 and y in the integers modulo n. The edges are v_{1,y}v_{1,y+i}, v_{2,y}v_{2,y+j}, v_{2,y}v_{2,y+k} and v_{0,y}v_{x,y}, where y=0,1,...,n-1 and x=1,2,3 and the subscript addition is performed modulo n.

			Y(6:1,1,1) is the smallest Y-graph with girth 6.

I. Z. Bouwer, W. W. Chernoff, B. Monson, and Z. Starr (Editors), "Foster's Census", Charles Babbage Research Centre, Winnipeg, 1988.

J. D. Horton and I. Z. Bouwer, Symmetric Y-graphs and H-graphs, J. Comb. Theory B 53 (1991) 114-129.

Cf. A112921, A112922, A112923.

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User: Tomaz Pisanski

A153846 Number of non-isomorphic I-graphs I(n,j,k) on 2n vertices (1<=j,k<=Floor[(n-1)/2]).

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A153847 Number of non-isomorphic connected I-graphs I(n,j,k) on 2n vertices (1<=j,k<=Floor[(n-1)/2]).

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A112921 Number of nonisomorphic Y-graphs Y(n:i,j,k) on 4n vertices (or nodes) for 1<=i,j,k

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A112917 Number of nonisomorphic H-graphs H(n:i,j;k,m) on 6n vertices (or nodes) for 1<=i,j,k,m

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A112918 Number of nonisomorphic connected H-graphs H(n:i,j;k,m) on 6n vertices (or nodes) for 1<=i,j,k,m

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A112919 Number of nonisomorphic connected bipartite H-graphs H(n:i,j;k,m) on 6n vertices (or nodes) for 1<=i,j,k,m

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A112920 Number of nonisomorphic connected bipartite H-graphs H(n:i,j;k,m) with girth 6 on 6n vertices (or nodes) for 1<=i,j,k,m

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A112922 Number of nonisomorphic connected Y-graphs Y(n:i,j,k) on 4n vertices (or nodes) for 1<=i,j,k

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A112923 Number of nonisomorphic connected bipartite Y-graphs Y(n:i,j,k) on 8n vertices (or nodes) for 1<=i,j,k<=n.

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