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### Distribution of Cycle Lengths in Graphs

Genghua Fan
2002 Journal of combinatorial theory. Series B (Print)
that if G is a nonbipartite 3-connected graph with minimum degree at least 3k for any positive integer k, then G contains 2k cycles of consecutive lengths m, m+1, ..., m+2k − 1 for some integer m \ k+  ...  Bondy and Vince proved that every graph with minimum degree at least three contains two cycles whose lengths differ by one or two, which answers a question raised by Erdő s.  ...  Evidently a bipartite graph cannot have cycles whose lengths differ by one. It is natural to ask what graphs contain cycles whose lengths differ by two.  ...

### A generalization of Fan's results: Distribution of cycle lengths in graphs

Jun Zhang, Jinghua Xiang
2009 Discrete Mathematics
Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser.  ...  B 84 (2002) 187 -202] proved that if G is a graph with minimum degree δ(G) ≥ 3k for any positive integer k, then G contains k + 1 cycles C 0 , C 1 , .  ...  cycles of consecutive lengths m, m + 1, m + 2, . . . , m + 2k − 1 for some integer m ≥ k + 2.  ...

### On the Tanner Graph Cycle Distribution of Random LDPC, Random Protograph-Based LDPC, and Random Quasi-Cyclic LDPC Code Ensembles [article]

Ali Dehghan, Amir H. Banihashemi
2018 arXiv   pre-print
We prove that for a random bipartite graph, with a given (irregular) degree distribution, the distributions of cycles of different length tend to independent Poisson distributions, as the size of the graph  ...  In particular, depending on the protograph and the value of c, the expected number of cycles of length c, in this case, can be either Θ(N) or Θ(1), where N is the lifting degree (code length).  ...  Related to this, it is also interesting to obtain the distribution of cycles of a given length in an ensemble of Tanner graphs (LDPC codes).  ...

### The number of edges in a maximum cycle—distributed graph

Yongbing Shi
1992 Discrete Mathematics
., The number of edges in a maximum cycle-distributed graph, Discrete Mathematics 104 (1992) 205-209.  ...  A graph G is said to be a cycle-distributed graph if no two cycles in G have the same length. In particular, a graph G containing at most one cycle is a cycle-distributed graph.  ...  Let f(n) (f*(n),f*(n)) be the maximum possible number of edges in a cycle-distributed graph (simple cycle-distributed graph, 2-connected simple cycledistributed graph) on n vertices.  ...

### Statistical Analysis of Binary Functional Graphs of the Discrete Logarithm [article]

Mitchell Orzech
2016 arXiv   pre-print
Thus, we found the theoretical asymptotic distributions of certain properties of the graph.  ...  Therefore, we focus on the statistical analysis of certain properties of the graph of the discrete logarithm.  ...  We will focus on statistics pertaining to the rho length of these graphs, as tail and cycle length have been discussed in  .  ...

### Geodesic Cycle Length Distributions in Delusional and Other Social Networks

Alex Stivala
2020 Journal of Social Structure
In this work, I re-analyze these delusional social networks using exponential random graph models (ERGMs) and investigate the distribution of the lengths of geodesic cycles.  ...  cycle (in more precise graph theory terms, this is a geodesic cycle).  ...  Geodesic cycles in random graphs were studied in Benjamini et al. (2011) , and the length of the longest geodesic cycles in random graphs in Li and Shi (2018) .  ...

### Small Cycles in Small Worlds [article]

Petra M. Gleiss, Peter F. Stadler, Andreas Wagner, David A. Fell
2000 arXiv   pre-print
We characterize the distributions of short cycles in a large metabolic network previously shown to have small world characteristics and a power law degree distribution.  ...  Short cycles reduce the length of detours when a connection is clipped, so we propose that long cycles in metabolism may have been selected against in order to shorten transition times and reduce the likelihood  ...  On the other hand, the distribution of cycle lengths is the same in all MCBs of a given graph  .  ...

### Kinetic theory of random graphs: From paths to cycles

E. Ben-Naim, P. L. Krapivsky
2005 Physical Review E
At the gelation point, the typical length of paths and cycles, l, scales with the component size k as l k^1/2.  ...  Treating linking as a dynamic aggregation process, rate equations for the distribution of node to node distances (paths) and of cycles are formulated and solved analytically.  ...  The cycle length distribution is w l = t l 2l . (31) In particular, at the gelation point, the cycle length distribution is inversely proportional to the cycle length  w l = (2l) −1 .  ...

### Page 2611 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews
Summary: “The cycle length distribution of a graph of order n is (c,€2,--*,Cn), Where c; is the number of cycles of length i. The following result is proved: Let A C E(K,), |A| <3 andn>|A|+ ; 3.  ...  Cycles of length three in bridge graphs are studied from a different point of view, namely that of the characterization of minimal elements in certain related posets: ordered bridge three-cycles (10 minimal  ...

### Kinetic Theory of Random Graphs

E. Ben-Naim
2005 AIP Conference Proceedings
Statistical properties of evolving random graphs are analyzed using kinetic theory.  ...  Treating the linking process dynamically, structural characteristics such as links, paths, cycles, and components are obtained analytically using the rate equation approach.  ...  In a similar way, the joint distribution of cycles in finite components of a given size is coupled to the joint distribution of paths of a given length in components of a given size.  ...

### Eigenvalues of random graphs with cycles [article]

Pau Vilimelis Aceituno
2020 arXiv   pre-print
In this note we offer a different perspective on this field by focusing on the cycles in a graph.  ...  We use it to explore properties of the eigenvalues of adjacency matrices of graphs with short cycles and of circulant directed graphs.  ...  We define the normalized weight of the cycles of length L ρ L = 1 N ∑ c∈C L w c (1) where C L is the set of cycles of length L, and w c = ∏ e∈c w(e), the multiplication of weights of the edges e in cycle  ...

### Graph Distance from the Topological View of Non-backtracking Cycles [article]

Leo Torres, Pablo Suarez-Serrato, Tina Eliassi-Rad
2018 arXiv   pre-print
graphs by considering their non-backtracking cycles.  ...  Here we introduce a theoretically sound and efficient new measure of graph distance, based on the "length spectrum" function from algebraic topology, which compares the structure of two undirected, unweighted  ...  Concretely, when comparing two graphs G, H, instead of comparing L G and L H directly, we compare the number of cycles in G of length 3 vs. the number of cycles in H of the same length, as well as the  ...

### On Computing the Multiplicity of Cycles in Bipartite Graphs Using the Degree Distribution and the Spectrum of the Graph [article]

Ali Dehghan, Amir H. Banihashemi
2019 arXiv   pre-print
In this paper, the result of Blake and Lin is extended to compute the number of cycles of length g+2, ..., 2g-2, for bi-regular bipartite graphs, as well as the number of 4-cycles and 6-cycles in irregular  ...  Most recently, Blake and Lin proposed a computational technique to count the number of cycles of length g in a bi-regular bipartite graph, where g is the girth of the graph.  ...  Distribution of cycles in different ensembles of bipartite graphs was studied in  .  ...

### Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons

Devdatt Dubhashi, Alessandro Mei, Alessandro Panconesi, Jaikumar Radhakrishnan, Aravind Srinivasan
2005 Journal of computer and system sciences (Print)
Motivated by routing issues in ad hoc networks, we present polylogarithmic-time distributed algorithms for two problems.  ...  We then show how to construct dominating sets that have the above properties, as well as the "low stretch" property that any two adjacent nodes in the network have their dominators at a distance of at  ...  Consider the graph G . Keep deleting edges that appear in cycles of length less than 1 + 2 log n until no such cycles remain. In the end, we will be left with at most 3n edges.  ...

### Distributed LTL Model Checking Based on Negative Cycle Detection [chapter]

Luboš Brim, Ivana Černá, Pavel Krčál, Radek Pelánek
2001 Lecture Notes in Computer Science
We come up with entirely different approach which is dependent on locating cycles with negative length in a directed graph with real number length of edges.  ...  This paper addresses the state explosion problem in automata based LTL model checking. To deal with large space requirements we turn to use a distributed approach.  ...  The problem is to find a negative length cycle in a directed graph whose edges have real number lengths.  ...
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