15,029 research outputs found
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
A family of diameter-based eigenvalue bounds for quantum graphs
We establish a sharp lower bound on the first non-trivial eigenvalue of the
Laplacian on a metric graph equipped with natural (i.e., continuity and
Kirchhoff) vertex conditions in terms of the diameter and the total length of
the graph. This extends a result of, and resolves an open problem from, [J. B.
Kennedy, P. Kurasov, G. Malenov\'a and D. Mugnolo, Ann. Henri Poincar\'e 17
(2016), 2439--2473, Section 7.2], and also complements an analogous lower bound
for the corresponding eigenvalue of the combinatorial Laplacian on a discrete
graph. We also give a family of corresponding lower bounds for the higher
eigenvalues under the assumption that the total length of the graph is
sufficiently large compared with its diameter. These inequalities are sharp in
the case of trees.Comment: Substantial revision of v1. The main result, originally for the first
eigenvalue, has been generalised to the higher ones. The title has been
changed and the proofs substantially reorganised to reflect the new result,
and a section containing concluding remarks has been adde
On the bounds for the largest Laplacian eigenvalues of weighted graphs
AbstractWe consider weighted graphs, such as graphs where the edge weights are positive definite matrices. The Laplacian eigenvalues of a graph are the eigenvalues of the Laplacian matrix of a graph G. We obtain an upper bound for the largest Laplacian eigenvalue and we compare this bound with previously known bounds
A relative bound for independence
We prove an upper bound for the independence number of a graph in terms of
the largest Laplacian eigenvalue, and of a certain induced subgraph. Our bound
is a refinement of a well-known Hoffman-type bound.Comment: 10 pages; preprint, comments are welcom
An eigenvalue bound for the Laplacian of a graph
AbstractWe present a lower bound for the smallest non-zero eigenvalue of the Laplacian of an undirected graph. The bound is primarily useful for graphs with small diameter
Semidefinite programming and eigenvalue bounds for the graph partition problem
The graph partition problem is the problem of partitioning the vertex set of
a graph into a fixed number of sets of given sizes such that the sum of weights
of edges joining different sets is optimized. In this paper we simplify a known
matrix-lifting semidefinite programming relaxation of the graph partition
problem for several classes of graphs and also show how to aggregate additional
triangle and independent set constraints for graphs with symmetry. We present
an eigenvalue bound for the graph partition problem of a strongly regular
graph, extending a similar result for the equipartition problem. We also derive
a linear programming bound of the graph partition problem for certain Johnson
and Kneser graphs. Using what we call the Laplacian algebra of a graph, we
derive an eigenvalue bound for the graph partition problem that is the first
known closed form bound that is applicable to any graph, thereby extending a
well-known result in spectral graph theory. Finally, we strengthen a known
semidefinite programming relaxation of a specific quadratic assignment problem
and the above-mentioned matrix-lifting semidefinite programming relaxation by
adding two constraints that correspond to assigning two vertices of the graph
to different parts of the partition. This strengthening performs well on highly
symmetric graphs when other relaxations provide weak or trivial bounds
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