OFFSET
1,4
COMMENTS
Appears to be the same as number of 3-connected cubic graphs on 2, 4, 6, 8, 10, 12 and 14 vertices. - Gordon Royle, Jun 02 2003
From Elijah Beregovsky, Feb 18 2026: (Start)
Geometrical circuits, introduced by Foster as an abstraction for electrical circuits, are multigraphs without leaf nodes and with additional equivalence relations:
1. A circuit consisting of two disconnected components C1 and C2 is equivalent "by separation" to the circuit resulting from identifying any one vertex from C1 with a single vertex of C2, thus connecting the components at the chosen vertices.
2. A circuit consisting of several subcircuits connected in series (i.e. multigraphs of the form u-v-w-u where hyphens stand for any subcircuit, and subcircuits have no edges between them) is equivalent "by series interchange" to the circuit resulting from exchanging any pair of subcircuits or substituting any subcircuit with its mirror image.
3. A circuit C is equivalent to any of the circuits resulting from subdivision of any edge in C.
Nullity of a multigraph, better known as cyclomatic number, is the minimum number r of edges that must be removed from the graph to break all its cycles, making it into a tree or forest. It can be found as r = |E| - |V| + c, where c is the number of connected components. Note that nullity is preserved by the circuit equivalence relations.
Any circuit of nullity n > 1 can be constructed by contracting edges of a "basic circuit" -- a cubic multigraph on 2n - 2 vertices that has no elements in series and that cannot be "separated" as in the first equivalence relation. This means a(n) coincides with 3-connected cubic simple graphs A204198(n-1) for all n > 2. (End)
REFERENCES
R. M. Foster, Topologic and algebraic considerations in network synthesis, pp. 8-18 in Proceedings of the Symposium on Modern Network Synthesis, New York, 1952, pp. 8-18. Polytechnic Institute of Brooklyn, New York, N. Y., 1952.
LINKS
Ronald M. Foster, Geometrical circuits of electrical networks, Transactions of the American Institute of Electrical Engineers, 51 (1932), 309-317.
Aleksandar Krapež, Slobodan K. Simić, and Dejan V. Tošić, Parastrophically uncancellable quasigroup equations, Aequat. Math. 79 (3) (2010) 261-280.
Wikipedia, Cyclomatic number.
Wikipedia, Series-parallel graph.
FORMULA
a(n) = A204198(n-1) for n > 2. - Elijah Beregovsky, Feb 18 2026
EXAMPLE
n=1: single self-loop; n=2: 3 edges in parallel between 2 nodes; n=3: K_4; n=4: K_{3,3} and the trivalent graph with 6 nodes and 9 edges formed by the 1-skeleton of a triangular prism.
CROSSREFS
KEYWORD
nonn,nice,more
AUTHOR
N. J. A. Sloane, Feb 06 2001
EXTENSIONS
a(8)-a(15) added from A204198 by Elijah Beregovsky, Feb 18 2026
STATUS
approved