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About 1,870 papers for “graph enumeration”

Graph Enumeration

Vivek S. Borkar, Vladimir Ejov, Jerzy A. Filar et al. · International series in management science/operations research/International series in operations research & management science · 2012

Cited by 0

Graph Enumeration

Vivek S. Borkar, Vladimir Ejov, Jerzy A. Filar et al. · RePEc: Research Papers in Economics

Abstract Graph enumeration is a study in graph theory that deals with counting nonisomorphic graphs that have a particular property. Harary and Palmer [58] provide an excellent introduction to the topic of graph enumeration. On counting labeled cubic graphs, there has been a series of results, most notably Read [86], Read [87], Wormald [101], and Wormald [102], which collectively present various approaches for counting labeled cubic graphs, and labeled cubic graphs with a given connectivity. In comparison to labeled cubic graphs, the numeration of unlabeled cubic graphs is a significantly more challenging problem [102]. In 1977, Robinson [88] presented a method to count unlabeled cubic graphs.

Cited by 0

Graph enumeration

Michael A. Henning, Jan H. van Vuuren · RePEc: Research Papers in Economics

Cited by 0

ENUMERATION OF GRAPH COVERINGS

Lee Jaeun · Yokohama National University Repository (Yokohama National University) · 1999

Enumerative research is presently a major center of interest in topological graph theory, as in the work of Archdeacon et al [2], Gross and Furst [4], Hofmeister [9]-[13] , Kwak and Lee [20]-[24], Mizuno and Sato [25]-[28] and [32], Mohar [29], Mull, Rieper and White [30], Negami [31], etc. In this paper, we survey some results on the enumeration of graph coverings and introduce some unsolved problems.

Cited by 0Open access

Combinatorial species and graph enumeration

Andy Hardt, Pete McNeely, Tung Phan et al. · arXiv (Cornell University) · 2013

In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the distinction between labeled and unlabeled structures. The cycle index series of a species encodes the labeled and unlabeled enumerative data of that species. Moreover, by using species operations, we are able to solve for the cycle index series of one species in terms of other, known cycle indices of other species. Section 3 is an exposition of species theory and Section 4 is an enumeration of point-determining bipartite graphs using this toolset. In Section 5, we extend a result about point-determining graphs to a similar result for point-determining Φ-graphs, where Φ is a class of graphs with certain properties. Finally, Appendix A is an expository on species computation using the software Sage [9] and Appendix B uses Sage to calculate the cycle index series of point-determining bipartite graphs.

Cited by 2Open access

Enumeration of connected graph coverings

Kwak, JH, Lee, J · Open Access System for Information Sharing (Pohang University of Science and Technology) · 2016

The number of the isomorphism classes of n-fold coverings of a graph G is enumerated by the authors (Canad. J. Math. XLII (1990), 747-761) and Hofmeister (Discrete Math. 98 (1991), 275-183). But the enumeration of the isomorphism classes of connected n-fold coverings of a graph has not been studied except for n = 2. In this paper, we enumerate the isomorphism classes of connected n-fold coverings of a graph G for any n. As a consequence, we obtain a formula for finding the total number of conjugacy classes of subgroups of a given index of a finitely generated free group. (C) 1996 John Wiley & Sons, Inc.

Cited by 0

Graph enumeration and random graphs

Lander Ramos Garrido · 2017

In this thesis we use analytic combinatorics to deal with two related problems: graph enumeration and random graphs from constrained classes of graphs. We are interested in drawing a general picture of some graph families by determining, first, how many elements are there of a given possible size (graph enumeration), and secondly, what is the typical behaviour of an element of fixed size chosen uniformly at random, when the size tends to infinity (random graphs). The problems concern graphs subject to global conditions, such as being planar and/or with restrictions on the degrees of the vertices. In Chapter 2 we analyse random planar graphs with minimum degree two and three. Using techniques from analytic combinatorics and the concepts of core and kernel of a graph, we obtain precise asymptotic estimates and analyse relevant parameters for random graphs, such as the number of edges or the size of the core, where we obtain Gaussian limit laws. More challenging is the extremal parameter equal to the size of the largest tree attached to the core. In this case we obtain a logarithmic estimate for the expected value together with a concentration result. In Chapter 3 we study the number of subgraphs isomorphic to a fixed graph in subcritical classes of graphs. We obtain Gaussian limit laws with linear expectation and variance when the fixed graph is 2-connected. The main tool is the analysis of infinite systems of equations by Drmota, Gittenberger and Morgenbesser, using the theory of compact operators. Computing the exact constants for the first estimates of the moments is in general out of reach. For the class of series-parallel graphs we are able to compute them in some particular interesting cases. In Chapter 4 we enumerate (arbitrary) graphs where the degree of every vertex belongs to a fixed subset of the natural numbers. In this case the associated generating functions are divergent and our analysis uses instead the so-called configuration model. We obtain precise asymptotic estimates for the number of graphs with given number of vertices and edges and subject to the degree restriction. Our results generalize widely previous special cases, such as d-regular graphs or graphs with minimum degree at least d. En aquesta tesi utilitzem l'analítica combinatòria per treballar amb dos problemes relacionats: enumeració de grafs i grafs aleatoris de classes de grafs amb restriccions. En particular ens interessa esbossar un dibuix general de determinades famílies de grafs determinant, en primer lloc, quants grafs hi ha de cada mida possible (enumeració de grafs), i, en segon lloc, quin és el comportament típic d'un element de mida fixa triat a l'atzar uniformement, quan aquesta mida tendeix a infinit (grafs aleatoris). Els problemes en què treballem tracten amb grafs que satisfan condicions globals, com ara ésser planars, o bé tenir restriccions en el grau dels vèrtexs. En el Capítol 2 analitzem grafs planar aleatoris amb grau mínim dos i tres. Mitjançant tècniques de combinatòria analítica i els conceptes de nucli i kernel d'un graf, obtenim estimacions asimptòtiques precises i analitzem paràmetres rellevants de grafs aleatoris, com ara el nombre d'arestes o la mida del nucli, on obtenim lleis límit gaussianes. També treballem amb un paràmetre que suposa un repte més important: el paràmetre extremal que es correspon amb la mida de l'arbre més gran que penja del nucli. En aquest cas obtenim una estimació logarítmica per al seu valor esperat, juntament amb un resultat sobre la seva concentració. En el Capítol 3 estudiem el nombre de subgrafs isomorfs a un graf fix en classes de grafs subcrítiques. Quan el graf fix és biconnex, obtenim lleis límit gaussianes amb esperança i variància lineals. L'eina principal és l'anàlisi de sistemes infinits d'equacions donada per Drmota, Gittenberger i Morgenbesser, que utilitza la teoria d'operadors compactes. El càlcul de les constants exactes de la primera estimació dels moments en general es troba fora del nostre abast. Per a la classe de grafs sèrie-paral·lels podem calcular les constants en alguns casos particulars interessants. En el Capítol 4 enumerem grafs (arbitraris) el grau de cada vèrtex dels quals pertany a un subconjunt fix dels nombres naturals. En aquest cas les funcions generatrius associades són divergents i la nostra anàlisi utilitza l'anomenat model de configuració. El nostre resultat consisteix a obtenir estimacions asimptòtiques precises per al nombre de grafs amb un nombre de vèrtexs i arestes donat, amb la restricció dels graus. Aquest resultat generalitza àmpliament casos particulars existents, com ara grafs d-regulars, o grafs amb grau mínim com a mínim d.

Cited by 0Open access

Graph Compositions I: Basic Enumeration

Knopfmacher, A., Mays, M.E. · Zenodo (CERN European Organization for Nuclear Research) · 2001

The idea of graph compositions generalizes both ordinary compositions of positive integers and partitions of finite sets. In this paper we develop formulas, generating functions, and recurrence relations for composition counting functions for several families of graphs.

Cited by 0Open access

Enumeration of forests in a graph

C. J. Liu, Yutze Chow · Proceedings of the American Mathematical Society · 1981

The enumeration of forests of different orders in a graph is carried out by a procedure that involves formal sums and certain annihilation operators on the space of such sums.The results here extend the well-known matrix-tree theorem to the general case of forests.

Cited by 24Open access

Symbolic method and directed graph enumeration

Élie de Panafieu, Sergey Dovgal · arXiv (Cornell University) · 2019

We introduce the arrow product, a systematic generating function technique for directed graph enumeration. It provides short proofs for previous results of Gessel on the number of directed acyclic graphs and of Liskovets, Robinson and Wright on the number of strongly connected directed graphs. We also recover Robinson's enumerative results on directed graphs where all strongly connected components belong to a given family.

Cited by 7Open access

Extremal graph theory: supersaturation and enumeration

Hong Liu · IDEALS (University of Illinois Urbana-Champaign) · 2015

In this thesis, we study supersaturation and enumeration problems in extremal combinatorics. In Chapter 2, with Balogh, we disprove a conjecture of Erdos and Tuza concerning the number of different ways one can create a copy of K_4, a complete graph on 4 vertices, in a K_4-free graph.
\nIn Chapter 3, we extend a classical result of Kolaitis, Promel and Rothschild on the typical structure of graphs forbidding a clique of fixed order as a subgraph, showing that the order of the forbidden clique can be as large as some polylogarithmic function of the order of the host graph. This is based on joint work with Balogh, Bushaw, Collares Neto, Morris and Sharifzadeh.
\nIn Chapter 4 and Chapter 5, we study the number of maximal sum-free subsets of the set [n] := {1, 2, . . . , n}. Together with Balogh, Sharifzadeh and Treglown, we show that, for each 1 ≤ i ≤ 4, there are constants Ci such that the number of maximal sum-free subsets in [n] is (C_i + o(1))2^{n/4}, where i ≡ n mod 4. This resolves a conjecture of Cameron and Erdos.
\nIn Chapter 6, with Balogh and Sharifzadeh, we study the number of subsets of [n] which does not contain an arithmetic progression of a fixed length. This addresses another question of Cameron and Erdos and provides an optimal bound for infinitely many n. As corollaries, we improve the known transference results on arithmetic progressions.

Cited by 0Open access

Γ-species, quotients, and graph enumeration

Andrew Gainer-Dewar · arXiv (Cornell University) · 2012

The theory of Γ-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion. This new approach is then applied to two graph-enumeration problems which were previously unsolved in the unlabeled case-bipartite blocks and general k-trees.

Cited by 0Open access

Circuit enumeration in an undirected graph

Pentti A. Honkanen · 1978

In his paper J. T. Welch [7] presents an algorithm for determining the circuits of a graph in a mechanical manner. Unfortunately, when an implementation of this procedure was attempted by N. E. Gibbs a very large class of counter examples was discovered. Gibbs rectified this situation in his algorithm [1]. However, a close study of Gibbs' algorithm shows that it is essentially an enumerative case considering all possible combinations. Hsu and Honkanen [3] developed a set of algorithms which used Welch's Concept where it applied and then examined the remaining circuits essentially in a manner like Gibbs' algorithm but did not require examining all possible combinations. R. Tarjan [6] pointed out several flaws in these algorithms which are corrected in this paper.

Cited by 3Open access

Enumeration of Forests in a Graph

C. J. Liu, Yutze Chow · Proceedings of the American Mathematical Society · 1981

The enumeration of forests of different orders in a graph is carried out by a procedure that involves formal sums and certain annihilation operators on the space of such sums.The results here extend the well-known matrix-tree theorem to the general case of forests.

Cited by 10

Accelerating Maximal Clique Enumeration via Graph Reduction

Wen Deng, Weiguo Zheng, Hong Cheng · arXiv (Cornell University) · 2023

As a fundamental task in graph data management, maximal clique enumeration (MCE) has attracted extensive attention from both academic and industrial communities due to its wide range of applications. However, MCE is very challenging as the number of maximal cliques may grow exponentially with the number of vertices. The state-of-the-art methods adopt a recursive paradigm to enumerate maximal cliques exhaustively, suffering from a large amount of redundant computation. In this paper, we propose a novel reduction-based framework for MCE, namely RMCE, that aims to reduce the search space and minimize unnecessary computations. The proposed framework RMCE incorporates three kinds of powerful reduction techniques including global reduction, dynamic reduction, and maximality check reduction. Global and dynamic reduction techniques effectively reduce the size of the input graph and dynamically construct subgraphs during the recursive subtasks, respectively. The maximality check reduction minimizes the computation for ensuring maximality by utilizing neighborhood dominance between visited vertices. Extensive experiments on 18 real graphs demonstrate the effectiveness of our proposed method. It achieves remarkable speedups up to 44.7x compared to existing approaches.

Cited by 1Open access