Showing papers on "Spectral graph theory published in 1997"
TL;DR: In this paper, it was shown that for every nonnegative integer k there is a unique connected graph T(k) that has Cheeger constant k, but removing any edge from it reduces the cheeger constant.
Abstract: It is shown that every (infinite) graph with a positive Cheeger constant contains a tree with a positive Cheeger constant. Moreover, for every nonnegative integer k there is a unique connected graph T(k) that has Cheeger constant k, but removing any edge from it reduces the Cheeger constant. This minimal graph, T(k), is a tree, and every graph G with Cheeger constant \( h(G) \geq k \) has a spanning forest in which each component is isomorphic to T(k).
84 citations
TL;DR: In this article, an upper bound for the eigenvalues of the Laplacian matrix of a graph was established for the line graph of a given graph and the largest eigenvalue of the graph reached the upper bound.
65 citations
18 Dec 1997
TL;DR: In this article, the analysis of Hill's operator D 2 + q(x) for qeven and periodic is extended from the real line to homogeneous trees T. The spectrum is exactly described when the degree of the tree is greater than two, in which case there are both spectral bands and eigenvalues.
Abstract: The analysis of Hill’s operator D 2 + q(x)for qeven and periodic is extended from the real line to homogeneous trees T. Generalizing the classical problem, a detailed analysis of Hill’s equation and its related operatortheoryon L 2 (T)isprovided. Themultipliersforthisnewversion of Hill’s equation are identied and analyzed. An explicit description of theresolventis given. The spectrumis exactly describedwhen thedegree of the tree is greater than two, in which case there are both spectral bands and eigenvalues. Spectral projections are computed by means of an eigenfunction expansion. Long time asymptotic expansions for the associated semigroup kernel are also described. A summation formula expresses the resolvent for a regular graph as a function of the resolvent of its covering homogeneous tree and the covering map. In the case of a nite regular graph, a trace formula relates the spectrum of the Hill’s operator to the lengths of closed paths in the graph.
59 citations
TL;DR: A large family of non-isomorphic Laplacian isospectral graphs are known, see as mentioned in this paper for a recent survey of nonisomorphic graph isomorphism.
Abstract: Two graphs are isomorphic only if they are Laplacian isospectral, that is, their Laplacian matrices share the same multiset of eigenvalues. Large families of nonisomorphic Laplacian isospectral gra...
26 citations
21 Apr 1997
TL;DR: This paper provides links between the young field of attractor coding and the well-established fields of systems theory and graph theory and investigates the patterns of interdependency between signal elements (or image pixels) using concepts from graph and matrix theory.
Abstract: This paper provides links between the young field of attractor coding and the well-established fields of systems theory and graph theory. Attractor decoders are modeled as linear systems whose stability is both necessary and sufficient for convergence of the decoder. This stability is dictated by the location of the eigenvalues of the sparse state transition matrix of the system. The relationship between these eigenvalues, spatial causality of the system, and the patterns of interdependency between signal elements (or image pixels) is investigated for several cases using concepts from graph and matrix theory.
5 citations
19 Dec 1997
TL;DR: In this paper, a newa priori estimate for very weak p{harmonic mappings when pis close to two is given, which sheds some light on a conjecture posed by Iwaniec and Sbordone.
Abstract: We prove a newa priori estimate for very weak p{harmonic mappings when pis close to two. This sheds some light on a conjecture posed by Iwaniec and Sbordone.
2 citations