1,2

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 3-connected cubic graph on 2n - 2 vertices. Such graphs are counted by A059687 (see also A204198). (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ronald M. Foster, Geometrical circuits of electrical networks, Transactions of the American Institute of Electrical Engineers, 51 (1932), 309-317.

Wikipedia, Cyclomatic number.

Wikipedia, Series-parallel graph.

For n = 1 the only circuit is a single self-loop.

For n = 2 there are two circuits: a pair of self-loops and a triple of parallel edges.

Cf. A377569, A059687, A204198.

Sequence in context: A359986 A006116 A122082 * A107948 A220840 A058673

Adjacent sequences: A002628 A002629 A002630 * A002632 A002633 A002634

nonn,more,nice

approved