23,695 research outputs found
Average hitting times in some -equitable graphs
It is known that the average hitting times of simple random walks from any
vertex to any other vertex in distance-regular graphs are determined by their
intersection array. In this paper, we introduce a new graph classification
called -equitable, utilizing both the equitable partition and the function
, which represents a generalization of distance-regular graphs. We determine
the average hitting times from any vertex to any other vertex in -equitable
graphs by using their parameter referred to as the quotient matrix.
Furthermore, we prove that there is some function such that the Cartesian
product of two strongly regular graphs is -equitable. We then calculate the
quotient matrix for these graphs and determine the average hitting times from
any vertex to any other vertex in these graphs. In the same manner, we
determine the average hitting times on some generalized Paley graphs
Spectral partitioning in equitable graphs
Graph partitioning problems emerge in a wide variety of complex systems, ranging from biology to finance, but can be rigorously analyzed and solved only for a few graph ensembles. Here, an ensemble of equitable graphs, i.e., random graphs with a block-regular structure, is studied, for which analytical results can be obtained. In particular, the spectral density of this ensemble is computed exactly for a modular and bipartite structure. Kesten-McKay's law for random regular graphs is found analytically to apply also for modular and bipartite structures when blocks are homogeneous. An exact solution to graph partitioning for two equal-sized communities is proposed and verified numerically, and a conjecture on the absence of an efficient recovery detectability transition in equitable graphs is suggested. A final discussion summarizes results and outlines their relevance for the solution of graph partitioning problems in other graph ensembles, in particular for the study of detectability thresholds and resolution limits in stochastic block models
Spectral properties of random graphs with fixed equitable partition
We define a graph to be -regular if it contains an equitable partition
given by a matrix . These graphs are generalizations of both regular and
bipartite, biregular graphs. An -regular matrix is defined then as a matrix
on an -regular graph consistent with the graph's equitable partition. In
this paper we derive the limiting spectral density for large, random
-regular matrices as well as limiting functions of certain statistics for
their eigenvector coordinates as a function of eigenvalue. These limiting
functions are defined in terms of spectral measures on -regular trees. In
general, these spectral measures do not have a closed-form expression; however,
we provide a defining system of polynomials for them. Finally, we explore
eigenvalue bounds of -regular graph, proving an expander mixing lemma,
Alon-Bopana bound, and other eigenvalue inequalities in terms of the
eigenvalues of the matrix .Comment: 24 pages, 3 figure
Graph removal lemmas
The graph removal lemma states that any graph on n vertices with o(n^{v(H)})
copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite
its innocent appearance, this lemma and its extensions have several important
consequences in number theory, discrete geometry, graph theory and computer
science. In this survey we discuss these lemmas, focusing in particular on
recent improvements to their quantitative aspects.Comment: 35 page
A characterization of testable hypergraph properties
We provide a combinatorial characterization of all testable properties of
-graphs (i.e. -uniform hypergraphs). Here, a -graph property
is testable if there is a randomized algorithm which makes a
bounded number of edge queries and distinguishes with probability between
-graphs that satisfy and those that are far from satisfying
. For the -graph case, such a combinatorial characterization was
obtained by Alon, Fischer, Newman and Shapira. Our results for the -graph
setting are in contrast to those of Austin and Tao, who showed that for the
somewhat stronger concept of local repairability, the testability results for
graphs do not extend to the -graph setting.Comment: 82 pages; extended abstract of this paper appears in FOCS 201
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