16 research outputs found
Cubic graphs and the golden mean
The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We investigate the validity of the inequality for infinite, transitive, simple, cubic graphs, where is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds for are presented for transitive cubic graphs with girth either or , and for certain quasi-transitive cubic graphs.This work was supported in part by the Engineering and Physical Sciences Research Council under grant EP/I03372X/1. ZL acknowledges support from the Simons Foundation under grant #351813 and the National Science Foundation under grant DMS-1608896
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
Some Optimally Adaptive Parallel Graph Algorithms on EREW PRAM Model
The study of graph algorithms is an important area of research in computer science, since graphs offer useful tools to model many real-world situations. The commercial availability of parallel computers have led to the development of efficient parallel graph algorithms.
Using an exclusive-read and exclusive-write (EREW) parallel random access machine (PRAM) as the computation model with a fixed number of processors, we design and analyze parallel algorithms for seven undirected graph problems, such as, connected components, spanning forest, fundamental cycle set, bridges, bipartiteness, assignment problems, and approximate vertex coloring. For all but the last two problems, the input data structure is an unordered list of edges, and divide-and-conquer is the paradigm for designing algorithms. One of the algorithms to solve the assignment problem makes use of an appropriate variant of dynamic programming strategy. An elegant data structure, called the adjacency list matrix, used in a vertex-coloring algorithm avoids the sequential nature of linked adjacency lists.
Each of the proposed algorithms achieves optimal speedup, choosing an optimal granularity (thus exploiting maximum parallelism) which depends on the density or the number of vertices of the given graph. The processor-(time)2 product has been identified as a useful parameter to measure the cost-effectiveness of a parallel algorithm. We derive a lower bound on this measure for each of our algorithms
Locality of Random Digraphs on Expanders
We study random digraphs on sequences of expanders with bounded average
degree and weak local limit. The threshold for the existence of a giant
strongly connected component, as well as the asymptotic fraction of nodes with
giant fan-in or giant fan-out are local, in the sense that they are the same
for two sequences with the same weak local limit. The digraph has a bow-tie
structure, with all but a vanishing fraction of nodes lying either in the
unique strongly connected giant and its fan-in and fan-out, or in sets with
small fan-in and small fan-out. All local quantities are expressed in terms of
percolation on the limiting rooted graph, without any structural assumptions on
the limit, allowing, in particular, for non tree-like limits.
In the course of proving these results, we prove that for unoriented
percolation, there is a unique giant above criticality, whose size and critical
threshold are again local. An application of our methods shows that the
critical threshold for bond percolation and random digraphs on preferential
attachment graphs is , with an infinite order phase transition at .Comment: Added a proof on infinite order phase transition of PA graphs.
Revised introduction and moved the proof on applications to appendi
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Laplacians Of Cellular Sheaves: Theory And Applications
Cellular sheaves are a discrete model for the theory of sheaves on cell complexes. They carry a canonical cochain complex computing their cohomology. This thesis develops the theory of the Hodge Laplacians of this complex, as well as avenues for their application to concrete engineering and data analysis problems. The sheaf Laplacians so developed are a vast generalization of the graph Laplacians studied in spectral graph theory. As such, they admit generalizations of many results from spectral graph theory and the spectral theory of discrete Hodge Laplacians. A theory of approximation of cellular sheaves is developed, and algorithms for producing spectrally good
approximations are given, as well as a generalization of the notion of expander graphs. Sheaf Laplacians allow development of various dynamical systems associated with sheaves, and their behavior is studied. Finally, applications to opinion dynamics, extracting network structure from data, linear control systems, and distributed optimization are outlined