Labeled Enumeration

Harary, Frank

doi:10.1016/b978-0-12-324245-7.50005-8

Cited by 358 publications

(484 citation statements)

References 0 publications

Supporting

Mentioning

475

Contrasting

Unclassified

Order By: Relevance

“…There are as yet no formulas, direct or recursive, by means of which the number for a given E, or a given V or F, or a given combination of these elements, can be calculated [18], [25]. Enumeration of any of these classes must proceed by first constructing the members and then counting them one by one.…”

mentioning

confidence: 99%

The number of polyhedral (3-connected planar) graphs

Duijvestijn¹,

Pianzola²

1981

Math. Comp.

View full text Add to dashboard Cite

Abstract. Data is presented on the number of 3-connected planar graphs, isomorphic to the graphs of convex polyhedra, with up to 22 edges. The numbers of such graphs having the same number of edges, and the same number of vertices and faces, are tabulated. Conjectured asymptotic formulas by W. T. Tutte and by R. C. Mullin and P. J. Schellenberg are discussed. Additional data beyond 22 edges are given enabling the number of 10-hedra to be presented for the first time, as well as estimates of the number of 11-hedra and dodecahedra.

show abstract

mentioning

confidence: 99%

The number of polyhedral (3-connected planar) graphs

Duijvestijn¹,

Pianzola²

1981

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…To estimate the number of ways to construct a given structure S, denoted as N (S), we need to consider the automorphisms of a graph. An automorphism of a graph G is an adjacency preserving permutation of vertices of G. The collection Aut(G) of all automorphisms of G is called the automorphism group of G. In the sequel, Aut(S) of a structure S denotes Aut(G) for some labeled graph G such that G ∼ = S. In group theory, it is well known that N (S) = n!/|Aut(S)| [11]. With these definitions, we proved in [4] that…”

Section: Structural Entropymentioning

confidence: 99%

Fast Algorithm for Optimal Compression of Graphs

Choi

2010

2010 Proceedings of the Seventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

View full text Add to dashboard Cite

We consider the problem of finding optimal description for general unlabeled graphs. Given a probability distribution on labeled graphs, we introduced in [4] a structural entropy as a lower bound for the lossless compression of such graphs. Specifically, we proved that the structural entropy for the Erdős-Rényi random graph, in which edges are added with probability p, is`n 2´h (p) − n log n + O(n), where n is the number of vertices and h(p) = −p log p − (1 − p) log(1−p) is the entropy rate of a conventional memoryless binary source. In this paper, we prove the asymptotic equipartition property for such graphs. Then, we propose a faster compression algorithm that asymptotically achieves the structural entropy up to the first two leading terms with high probability. Our algorithm runs in O(n + e) time on average where e is the number of edges. To prove its asymptotic optimality, we introduce binary trees that one can classify as in-between tries and digital search trees. We use analytic techniques such as generating functions, Mellin transform, and poissonization to establish our findings. Our experimental results confirm theoretical results and show the usefulness of our algorithm for real-world graphs such as the Internet, biological networks, and social networks. IntroductionBrooks argues in [3] that there is "no theory that gives us a metric for information embodied in structure." Shannon himself alluded to it fifty years earlier in his little known 1953 paper [20]. In fact, Brooks emphasizes the importance of the quantification of information in physical structure. In computer science, however, it is more important to understand the information embodied in abstract structures that are of our particular interests in this paper. For instance, how can we quantify the amount of information in the structure of graphs such as the Internet, social networks, and biological networks? How can we understand and utilize the "structure" of non-conventional data structures such as biological data, topographical maps, medical data, and volumetric data? As the first step to understanding information in such structures, we focus on structure in graphs.

show abstract

“…The total number of inequivalent orbits for a given weight X n 1 Y n 2 can be determined using the Pólya-Redfield theorem, by reading out the coefficient of X n 1 Y n 2 from a so-called Cycle Index computed using appropriate figure inventories and the cyclestructure of the permutation representation of C L on ∆. 18 However, such approach will not help us here to determine the length of each orbit, which is the key piece of information to determine the Hückel annulene lengths, thus their energies. The problem can instead be solved by counting the orbits of weight X n 1 Y n 2 for each allowed symmetry describing the orbits of necklace configurations in F M S .…”

Section: Solution Of the Necklace Enumeration Problemmentioning

confidence: 99%