995 research outputs found
On Vertex Bisection Width of Random -Regular Graphs
Vertex bisection is a graph partitioning problem in which the aim is to find
a partition into two equal parts that minimizes the number of vertices in one
partition set that have a neighbor in the other set. We are interested in
giving upper bounds on the vertex bisection width of random -regular graphs
for constant values of . Our approach is based on analyzing a greedy
algorithm by using the Differential Equations Method. In this way, we obtain
the first known upper bounds for the vertex bisection width in random regular
graphs. The results are compared with experimental ones and with lower bounds
obtained by Kolesnik and Wormald, (Lower Bounds for the Isoperimetric Numbers
of Random Regular Graphs, SIAM J. on Disc. Math. 28(1), 553-575, 2014).Comment: 31 pages, 2 figure
Absorption Time of the Moran Process
The Moran process models the spread of mutations in populations on graphs. We
investigate the absorption time of the process, which is the time taken for a
mutation introduced at a randomly chosen vertex to either spread to the whole
population, or to become extinct. It is known that the expected absorption time
for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which
allows the behaviour of the process on undirected graphs to be analysed using
the Markov chain Monte Carlo method. We show that this does not extend to
directed graphs by exhibiting an infinite family of directed graphs for which
the expected absorption time is exponential in the number of vertices. However,
for regular directed graphs, we show that the expected absorption time is
Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds
and give improved bounds for other families of graphs, based on isoperimetric
number. Our results are obtained via stochastic dominations which we
demonstrate by establishing a coupling in a related continuous-time model. The
coupling also implies several natural domination results regarding the fixation
probability of the original (discrete-time) process, resolving a conjecture of
Shakarian, Roos and Johnson.Comment: minor change
On the Smallest Eigenvalue of Grounded Laplacian Matrices
We provide upper and lower bounds on the smallest eigenvalue of grounded
Laplacian matrices (which are matrices obtained by removing certain rows and
columns of the Laplacian matrix of a given graph). The gap between the upper
and lower bounds depends on the ratio of the smallest and largest components of
the eigenvector corresponding to the smallest eigenvalue of the grounded
Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently
obtain a tight characterization of the smallest eigenvalue for certain classes
of graphs. Specifically, for Erdos-Renyi random graphs, we show that when a
(sufficiently small) set of rows and columns is removed from the Laplacian,
and the probability of adding an edge is sufficiently large, the smallest
eigenvalue of the grounded Laplacian asymptotically almost surely approaches
. We also show that for random -regular graphs with a single row and
column removed, the smallest eigenvalue is . Our bounds
have applications to the study of the convergence rate in continuous-time and
discrete-time consensus dynamics with stubborn or leader nodes
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