10 research outputs found
The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph
In this paper, we show that the largest Laplacian H-eigenvalue of a
-uniform nontrivial hypergraph is strictly larger than the maximum degree
when is even. A tight lower bound for this eigenvalue is given. For a
connected even-uniform hypergraph, this lower bound is achieved if and only if
it is a hyperstar. However, when is odd, it happens that the largest
Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower
bound. On the other hand, tight upper and lower bounds for the largest signless
Laplacian H-eigenvalue of a -uniform connected hypergraph are given. For a
connected -uniform hypergraph, the upper (respectively lower) bound of the
largest signless Laplacian H-eigenvalue is achieved if and only if it is a
complete hypergraph (respectively a hyperstar). The largest Laplacian
H-eigenvalue is always less than or equal to the largest signless Laplacian
H-eigenvalue. When the hypergraph is connected, the equality holds here if and
only if is even and the hypergraph is odd-bipartite.Comment: 26 pages, 3 figure
Mysteries around the graph Laplacian eigenvalue 4
We describe our current understanding on the phase transition phenomenon of
the graph Laplacian eigenvectors constructed on a certain type of unweighted
trees, which we previously observed through our numerical experiments. The
eigenvalue distribution for such a tree is a smooth bell-shaped curve starting
from the eigenvalue 0 up to 4. Then, at the eigenvalue 4, there is a sudden
jump. Interestingly, the eigenvectors corresponding to the eigenvalues below 4
are semi-global oscillations (like Fourier modes) over the entire tree or one
of the branches; on the other hand, those corresponding to the eigenvalues
above 4 are much more localized and concentrated (like wavelets) around
junctions/branching vertices. For a special class of trees called starlike
trees, we obtain a complete understanding of such phase transition phenomenon.
For a general graph, we prove the number of the eigenvalues larger than 4 is
bounded from above by the number of vertices whose degrees is strictly higher
than 2. Moreover, we also prove that if a graph contains a branching path, then
the magnitudes of the components of any eigenvector corresponding to the
eigenvalue greater than 4 decay exponentially from the branching vertex toward
the leaf of that branch.Comment: 22 page
Bounds for the Laplacian Spectral Radius of Graphs
This paper is a survey on the upper and lower bounds for the largest eigenvalue of the Laplacian matrix, known as the Laplacian spectral radius, of a graph. The bounds are given as functions of graph parameters like the number of vertices, the number of edges, degree sequence, average 2-degrees, diameter, covering number, domination number, independence number and other parameters
New bounds for the signless Laplacian spread
Let be an undirected simple graph. The signless Laplacian spread of is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian spread. Several of these bounds depend on invariant parameters of the graph. We also use a minmax principle to find several lower bounds for this spectral invariant.publishe
Majorantes para a ordem de subgrafos induzidos k-regulares
Doutoramento em MatemáticaMuitos dos problemas de otimização em grafos reduzem-se à determinação
de um subconjunto de vértices de cardinalidade máxima que induza
um subgrafo k-regular. Uma vez que a determinação da ordem
de um subgrafo induzido k-regular de maior ordem é, em geral, um
problema NP-difícil, são deduzidos novos majorantes, a determinar em
tempo polinomial, que em muitos casos constituam boas aproximações
das respetivas soluções ótimas. Introduzem-se majorantes espetrais
usando uma abordagem baseada em técnicas de programação convexa
e estabelecem-se condições necessárias e suficientes para que sejam
atingidos. Adicionalmente, introduzem-se majorantes baseados no espetro
das matrizes de adjacência, laplaciana e laplaciana sem sinal. É
ainda apresentado um algoritmo não polinomial para a determinação de
umsubconjunto de vértices de umgrafo que induz umsubgrafo k-regular
de ordem máxima para uma classe particular de grafos. Finalmente,
faz-se um estudo computacional comparativo com vários majorantes e
apresentam-se algumas conclusões.Many optimization problems on graphs are reduced to the determination
of a subset of vertices of maximum cardinality inducing a k-regular
subgraph. Since the determination of the order of a k-regular induced
subgraph of highest order is in general a NP-hard problem, new upper
bounds, determined in polynomial time which in many cases are good
approximations of the respective optimal solutions are deduced. Using
convex programming techniques, spectral upper boundswere introduced
jointly with necessary and sufficient conditions for those upper bounds
be achieved. Additionally, upper bounds based on adjacency, Laplacian
and signless Laplacian spectrum are introduced. Furthermore, a nonpolynomial
time algorithm for determining a subset of vertices of a graph
which induces a maximum k-regular induced subgraph for a particular
class is presented. Finally, a comparative computational study is provided
jointly with a few conclusions