cp's OEIS Frontend

A361448 Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop) whose removal does not disconnect the graph, loops allowed.

%I A361448 #10 Mar 21 2023 23:08:34
%S A361448 1,2,10,66,511,4536,45519,512661,6436571,89505875,1369509795,
%T A361448 22908806774,416408493351,8178599551905,172690849144538,
%U A361448 3902128758180500,93970611848528998,2402929936231885063,65029668312580777779,1856984518220396165657,55803367549204703645086
%N A361448 Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop) whose removal does not disconnect the graph, loops allowed.
%C A361448 a(0) = 1 by convention. Loops add two to the degree of a node.
%C A361448 Instead of a rooted edge, the graph can be considered to have a pair of external legs (or half-edges). The external legs add 1 to the degree of a node, but do not contribute to the connectivity of the graph.
%C A361448 The 4-regular version of this sequence is A352174 since removing a single edge from a connected even degree regular graph cannot disconnect the graph.
%F A361448 G.f.: B(x) - x*C(x)^2 where B(x) is the g.f. of A361446 and C(x) is the g.f. of A361412.
%e A361448 a(2) = 10 = A361447(2) + 1 because there is one graph where the orientation of the rooted edge makes a difference:
%e A361448     1       __
%e A361448    /| \    |  |
%e A361448    ||  3---4  |
%e A361448    \| /    |__|
%e A361448     2
%e A361448 The nodes are labeled 1,2,3,4. There is a double edge between nodes 1 and 2 and a loop at node 4. Roots at the edges (1,3) and (3,1) are considered different because orientation is considered. Roots at (1,3) and (2,3) are considered the same because the resulting graphs are isomorphic. Roots at (3,4) or (4,3) are disallowed because the removal of that edge would disconnect the graph.
%Y A361448 Cf. A005967 (unrooted), A129427, A352174, A352175, A361412, A361446, A361447.
%K A361448 nonn
%O A361448 0,2
%A A361448 _Andrew Howroyd_, Mar 12 2023