109 research outputs found
The icosahedron is clique divergent
AbstractA clique of a graph G is a maximal complete subgraph. The clique graph k(G) is the intersection graph of the set of all cliques of G. The iterated clique graphs are defined recursively by k0(G)=G and kn+1(G)=k(kn(G)). A graph G is said to be clique divergent (or k-divergent) if limn→∞|V(kn(G))|=∞. The problem of deciding whether the icosahedron is clique divergent or not was (implicitly) stated Neumann-Lara in 1981 and then cited by Neumann-Lara in 1991 and Larrión and Neumann-Lara in 2000. This paper proves the clique divergence of the icosahedron among other results of general interest in clique divergence theory
The clique graphs of the hexagonal lattice -- an explicit construction and a short proof of divergence
We present a new, explicit and very geometric construction for the iterated
clique graphs of the hexagonal lattice which makes apparent its
clique-divergence and sheds light on some previous observations, such as the
boundedness of the degrees and clique sizes of as
The fundamental group of the clique graph
AbstractGiven a finite connected bipartite graph B=(X,Y) we consider the simplicial complexes of complete subgraphs of the square B2 of B and of its induced subgraphs B2[X] and B2[Y]. We prove that these three complexes have isomorphic fundamental groups. Among other applications, we conclude that the fundamental group of the complex of complete subgraphs of a graph G is isomorphic to that of the clique graph K(G), the line graph L(G) and the total graph T(G)
Critical phenomena in complex networks
The combination of the compactness of networks, featuring small diameters,
and their complex architectures results in a variety of critical effects
dramatically different from those in cooperative systems on lattices. In the
last few years, researchers have made important steps toward understanding the
qualitatively new critical phenomena in complex networks. We review the
results, concepts, and methods of this rapidly developing field. Here we mostly
consider two closely related classes of these critical phenomena, namely
structural phase transitions in the network architectures and transitions in
cooperative models on networks as substrates. We also discuss systems where a
network and interacting agents on it influence each other. We overview a wide
range of critical phenomena in equilibrium and growing networks including the
birth of the giant connected component, percolation, k-core percolation,
phenomena near epidemic thresholds, condensation transitions, critical
phenomena in spin models placed on networks, synchronization, and
self-organized criticality effects in interacting systems on networks. We also
discuss strong finite size effects in these systems and highlight open problems
and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references,
extende
Locally grid graphs: classification and Tutte uniqueness
We define a locally grid graph as a graph in which the structure around each vertex is a 3×3 grid ⊞, the canonical examples being the toroidal grids Cp×Cq. The paper contains two main results. First, we give a complete classification of locally grid graphs, showing that each of them has a natural embedding in the torus or in the Klein bottle. Secondly, as a continuation of the research initiated in (On graphs determined by their Tutte polynomials, Graphs Combin., to appear), we prove that Cp×Cq is uniquely determined by its Tutte polynomial, for p,q⩾6
Conservative Sparsification for Efficient Approximate Estimation
Linear Gaussian systems often exhibit sparse structures. For systems which grow as a function of time, marginalisation of past states will eventually introduce extra non-zero elements into the information matrix of the Gaussian distribution. These extra non-zeros can lead to dense problems as these systems progress through time. This thesis proposes a method that can delete elements of the information matrix while maintaining guarantees about the conservativeness of the resulting estimate with a computational complexity that is a function of the connectivity of the graph rather than the problem dimension. This sparsification can be performed iteratively and minimises the Kullback Leibler Divergence (KLD) between the original and approximate distributions. This new technique is called Conservative Sparsification (CS). For large sparse graphs employing a Junction Tree (JT) for estimation, efficiency is related to the size of the largest clique. Conservative Sparsification can be applied to clique splitting in JTs, enabling approximate and efficient estimation in JTs with the same conservative guarantees as CS for information matrices. In distributed estimation scenarios which use JTs, CS can be performed in parallel and asynchronously on JT cliques. This approach usually results in a larger KLD compared with the optimal CS approach, but an upper bound on this increased divergence can be calculated with information locally available to each clique. This work has applications in large scale distributed linear estimation problems where the size of the problem or communication overheads make optimal linear estimation difficult
Feynman integrals and hyperlogarithms
We study Feynman integrals in the representation with Schwinger parameters
and derive recursive integral formulas for massless 3- and 4-point functions.
Properties of analytic (including dimensional) regularization are summarized
and we prove that in the Euclidean region, each Feynman integral can be written
as a linear combination of convergent Feynman integrals. This means that one
can choose a basis of convergent master integrals and need not evaluate any
divergent Feynman graph directly.
Secondly we give a self-contained account of hyperlogarithms and explain in
detail the algorithms needed for their application to the evaluation of
multivariate integrals. We define a new method to track singularities of such
integrals and present a computer program that implements the integration
method.
As our main result, we prove the existence of infinite families of massless
3- and 4-point graphs (including the ladder box graphs with arbitrary loop
number and their minors) whose Feynman integrals can be expressed in terms of
multiple polylogarithms, to all orders in the epsilon-expansion. These
integrals can be computed effectively with the presented program.
We include interesting examples of explicit results for Feynman integrals
with up to 6 loops. In particular we present the first exactly computed
counterterm in massless phi^4 theory which is not a multiple zeta value, but a
linear combination of multiple polylogarithms at primitive sixth roots of unity
(and divided by ). To this end we derive a parity result on the
reducibility of the real- and imaginary parts of such numbers into products and
terms of lower depth.Comment: PhD thesis, 220 pages, many figure
- …