A371396 -id:A371396 - cp's OEIS frontend

A369859 Minimum chord of a chordal ring graph with 2*n vertices and minimal diameter.

1, 1, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 15, 7, 7, 9, 7, 19, 7, 9, 9, 9, 9, 9, 9, 11, 19, 19, 9, 9, 21, 11, 29, 23, 23, 11, 11, 25, 11, 11, 11, 11, 11, 13, 13, 23, 23, 41, 11, 25, 25, 13, 15, 27, 27, 27, 13, 13, 29, 29, 13, 13, 13, 31, 13, 15, 15, 15, 17, 27, 13, 43, 29, 29, 29, 45, 17, 17, 31, 31, 31, 39

Offset: 1

Miquel A. Fiol, Apr 30 2024

nonn

Given integers n and c (odd), the chordal ring graph CR(2*n,c) is a bipartite graph with vertex set Z_{2*n}, and edges {i,i+1}, {i,i-1}, {i,i+c} if i is odd, and {i,i-c} if i is even.

Cf. A371396.

A373503 The minimum diameter of a chordal ring graph with 2*n vertices.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 10, 10, 9, 10, 11, 11, 10, 10, 11, 11, 11, 10, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 12, 12, 11, 11, 12, 12

Offset: 1

Miquel A. Fiol, Jul 10 2024

nonn

P. Morillo, F. Comellas, and M. A. Fiol, The optimization of Chordal Ring Networks, Commun. Technol., Eds. Q. Yasheng and W. Xiuying. World Scientific, (1987), 295-299.

Cf. A369859, A371396.

A374622 Maximum number of vertices of a chordal ring mixed graph CRM(N,c) with diameter n.

Original entry on oeis.org

8, 10, 18, 16, 32, 34, 50, 44, 72, 74, 98, 88, 128, 130, 162, 148, 200, 202, 242, 224, 288, 290, 338, 316, 392, 394, 450, 424, 512, 514, 578, 548, 648, 650, 722, 688, 800, 802, 882, 844, 968, 970, 1058, 1016, 1152, 1154

Offset: 3

Miquel A. Fiol, Jul 14 2024

nonn

			For n = 9, the maximum number of vertices a(9) = 50 is attained by the chordal ring mixed graph CRM(50,9).

C. Dalfó, G. Erskine, G. Exoo, M. A. Fiol, and J. Tuite, On bipartite (1, 1, k)-mixed graphs, (2024).

Cf. A371396.

If n is odd, a(n) = (n+1)^2/2.
Conjecture: If n is even, n=0 mod 4, a(n) = n^2/2+2;
            If n (> 2) is even, n=2 mod 4, a(n) = n*(n/2 - 1) + 4.
Conjectured g.f.: 2*(1 + x + 2*x^2 + x^3 + 2*x^4 - 3*x^5 + 4*x^6 - x^7 + x^8)/((1 - x)^3*(1 + x + x^2 + x^3)^2). - Stefano Spezia, Jul 14 2024

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A369859 Minimum chord of a chordal ring graph with 2*n vertices and minimal diameter.

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A373503 The minimum diameter of a chordal ring graph with 2*n vertices.

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A374622 Maximum number of vertices of a chordal ring mixed graph CRM(N,c) with diameter n.

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