Graphs
which are linked cycles
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This is joint work by Peter Adams (The University of Queensland), Roger Eggleton (Illinois State University) and James MacDougall (The University of Newcastle).
Links to the full data on which the paper is based are provided below.
The following files present information about initial links G of order g and linked cycles H of order h with 1 ≤ g < h ≤ 10. The tables are keyed to each graph G, identified by its SEAM number Sn:r. All four files specify, for each G, the numbers of isomorphism classes of linked cycles of order h generated by G for each h ≤ 10, together with the total number of isomorphism classes of all orders not exceeding 10.
In addition to this information, the first file (covering graphs G of order 1 ≤ g ≤ 6) contains other information, as follows. Column 'K' specifies the number of isomorphism classes of kernels for G. Column 'Triples" gives the number of ordered triples (K, K*, σ) for G. After the total number of isomorphism classes of linked cycles of all orders not exceeding 10, this file also explicitly lists the SEAM numbers of the rank 1 linked cycles generated (those with h = g+1).
The
following files present information about linked cycles H. The tables are keyed
to each graph H which is a linked cycle,
identified by its SEAM number Sn:r.
For each H the next entry in the
table is the fundamental generator G,
also by SEAM number; this is the earliest graph which generates H as a linked cycle. The rest of the entry specifies the
fundamental automorphism α which, when
iteratively applied to G, generates a
graph isomorphic to H.
How
is the fundamental automorphism represented? For this purpose the vertices of G are labelled with the integers 0, 1,
..., g – 1, so subtract 1 from each vertex label in the specification of G in the SEAM lists (see orders 4 to 8, order 9 and
order 10), since there the labels 1, 2, ..., g are used. All new vertices of H (that is, vertices not in G) are designated by letters a, b, c, ... (For lexicographic ordering, the numbers precede the
letters.) The fundamental automorphism is chosen, among all possible automorphisms, according to three ranked criteria: (1) it
has least period; (2) it has fewest cycles; (3) it has the lexicographically
earliest nondecreasing list of cycle orders. We simply specify the first such automorphism arising in our computations. Each fundamental automorphism
is compactly
specified by listing the vertices in its cycles without
parentheses, separating cycles by a comma, and using lexicographic order within
and between the cycles.