7,955 research outputs found
Invertibility of graph translation and support of Laplacian Fiedler vectors
The graph Laplacian operator is widely studied in spectral graph theory
largely due to its importance in modern data analysis. Recently, the Fourier
transform and other time-frequency operators have been defined on graphs using
Laplacian eigenvalues and eigenvectors. We extend these results and prove that
the translation operator to the 'th node is invertible if and only if all
eigenvectors are nonzero on the 'th node. Because of this dependency on the
support of eigenvectors we study the characteristic set of Laplacian
eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish
on large neighborhoods and then explicitly construct a family of non-planar
graphs that do exhibit this property.Comment: 21 pages, 7 figure
On the Invertibility of Storage Systems
The invertibility of single-input single output storage systems (network of reservoirs) is considered in this paper. The analysis shows that cascade and feedback connections of invertible subsystems give rise to invertible systems, and that parallel connections are invertible provided that the network is not too diversified topologically and that the reservoirs have comparable dynamics. These results often allow one to ascertain the invertibility of a complex storage system by direct inspection of a graph
Invertible families of sets of bounded degree
Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then
hypergraph H is invertible iff there exists a permutation pi of V such that for
all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility
critical if H is not invertible but every hypergraph obtained by removing an
edge from H is invertible. The degree of H is d if |{E belongs to H(edges)|x
belongs to E}| =< d for each x belongs to V Let i(d) be the maximum number of
edges of an invertibility critical hypergraph of degree d. Theorem: i(d) =<
(d-1) {2d-1 choose d} + 1. The proof of this result leads to the following
covering problem on graphs: Let G be a graph. A family H is subset of (2^{V(G)}
is an edge cover of G iff for every edge e of G, there is an E belongs to
H(edge set) which includes e. H(edge set) is a minimal edge cover of G iff for
H' subset of H, H' is not an edge cover of G. Let b(d) (c(d)) be the maximum
cardinality of a minimal edge cover H(edge set) of a complete bipartite graph
(complete graph) where H(edge set) has degree d. Theorem: c(d)=< i(d)=<b(d)=<
c(d+1) and 3. 2^{d-1} - 2 =< b(d)=< (d-1) {2d-1choose d} +1. The proof of this
result uses Sperner theory. The bounds b(d) also arise as bounds on the maximum
number of elements in the union of minimal covers of families of sets.Comment: 9 page
- …