http://nova.newcastle.edu.au/vital/access/services/Feed ${session.getAttribute("locale")} 5 http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:7966 A graph H of order h is an n-linked cycle if it has an induced subgraph G of order g < h and an automorphism α: H → H of order n ≥ 2 such that H = ⋃{αr(G) : 0 ≤ r < n} and G has an induced subgraph K of order k < g such that αr(G) ⋂ αr⁺¹(G) ≅ K for 0 ≤ r < n. Then G is the initial link of this linked cycle, K is its kernel, α|G is the link isomorphism, and any pair (G, α) allowing H to be expressed as a linked cycle yields a generalized factorization of H. For a given standard ordering of all finite graphs, the "earliest" pair (G, α) is a fundamental representation of H. There are 2 593 574 linked cycles among all graphs of order h ≤ 10. The paper gives an overview, and a fundamental representation of each of them is provided on a supporting website. 2011-10-11T02:00:03.120Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:7962 A linked pair is a graph H = G₀ ⋃ G₁ formed from (1) a given finite graph G,(2) isomorphic induced proper subgraphs K and K*, not necessarily distinct, and (3) a graph isomorphism σ: K* → K. The graph isomorphism τ: G₀ → G₁ is the extension of a to G₀ = G so that G₀ ⋂ G₁ = K. The ingredients of the linked pair construction are the initial link G, the kernel K, the prekernel K*, and the shift σ; the extended map τ is the link isomorphism. Iteration of this construction for any n ≥ 2 yields an n-linked chain ⋃{Gr = τr(G) : 0 ≤ r < n}, with Gr ⋂ Gr₊₁ ≅ K for 0 ≤ r < n, and eventually leads to the free linked chain ⋃{Gr = τr(G) : r ∈ Z}. For suitable n, imposing a periodic equivalence relation on the vertices of the free linked chain yields an n-linked cycle, corresponding to requiring Gn = G₀ in the n-linked chain. The resultant finite graph is the union of a sequence of n induced subgraphs isomorphic to G, consecutive pairs having intersection isomorphic to K; the collapsed isomorphism, τ is an automorphism of order n, All graphs with an automorphism of order n > 2, and many with an automorphism of order n = 2, are in fact n-linked cycles, and this viewpoint leads to a "generalized factorization" of such graphs. 2011-10-11T02:00:01.979Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:6508 Extending earlier data summaries for graphs of order n ≤ 9, this paper describes structural characteristics and relationships for the 12005168 graphs of order 10. It summarises data for their degree sequences, their component structure, their cycle structure, and their poset structure under the subgraph partial order. A standardized listing of the order 10 graphs, along with related data, is provided on the website www.maths.uq.edu.au/~pa/research/poset10.html 2010-06-16T05:50:02.118Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:6509 The set G(n) of unlabelled simple graphs of order n is a poset with partial ordering G ≤ H whenever G is a spanning subgraph of H. On the website www.maths.uq.edu.au/~pa/research/poset9.html we have made available a tabulation of the Hasse diagram for G(9), a digraph of order 274668 and size 4147388, extending our recent tabulations for G(n) with 4 ≤ n ≤ 8. The present paper is a descriptive summary of features of G(9) derived from the tabulation, including: the maximum number of graphs in G(9) with the same degree sequence is 3020, corresponding to 2¹3²4³5²6¹; there are 36 self-complementary graphs in G(9), but 10794 graphs with self-complementary degree sequences; there are 49 graphs in G(9) that are edge-transitive, and 134996 that have no edge-symmetry; the maximum number of immediate successors of a graph in G(9) is 28, and 12 graphs attain this maximum; the number of immediate successors of a graph in G(9) is distributed unimodally, with peak at 16 attained by 25010 graphs. All underlying data are available on the website. 2010-06-16T05:50:01.344Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:6510 The level set G(n,m) comprises all unlabelled simple graphs of order n and size m, and is partitioned into similarity classes, comprising all graphs with the same degree sequence. When graphs are ordered lexicographically by their signature, a unique numerical list of structural descriptors, the similarity classes of G(n,m) occur in contiguous blocks; the first graph in each similarity class is its sentinel. The sentinel of the first similarity class in each G(n,m) is determined, and shown to be the unique realization of its degree sequence. The degree sequence of the last similarity class in each G(n,m) is also determined, as are the exact size range for which it has more than one realization and the exact size range for which its sentinel has more than one component. 2010-06-16T05:50:01.339Z ]]> http://nova.newcastle.edu.au/vital/access/manager/Repository/uon:1699 The poset G(n) comprises the unlabelled simple graphs of order n, with partial ordering G ≤ H whenever G is a spanning subgraph of H. We define a modified Steinbach numbering of the graphs in G(n), apply this numbering to each G(n) with n ≤ 8, and use it to tabulate the Hasse diagram structure of the posets with 4 ≤ n ≤ 8 together with key aspects of the independence structure of these posets. In particular, the Hasse diagram of G(8) is a directed graph of order 12346 and size 125066; the poset G(8) has 51952895 independent pairs of graphs, and 96775426396 independent triples. We present 14 tables of descriptive data for G(n) with 4 ≤ n ≤ 8. All of the underlying data can be found on our webpage www.maths.uq.edu.au/pa/research/posets4to8.html 2010-06-16T04:08:46.472Z ]]>