Let G = (V, E) be a graph of order 2n. If A ⊆ V and hAi ∼= hV \Ai, then A is said to be isospectral. If for every n-element subset A of V we have hAi ∼= hV \Ai, then we say that G is ...spectral-equipartite. In 1, Igor Shparlinski communicated with Bibak et al., proposing a full characterization of spectral-equipartite graphs. In this paper, we gave a characterization of disconnected spectral-equipartite graphs. Moreover, we introduced the concept eccentricity-equipartite graphs.
Signal processing on graphs using adjacency matrix (as opposed to more traditional graph Laplacian) results in an algebraic framework for graph signals and shift invariant filters. This can be seen ...as an example of the algebraic signal processing theory. In this study, the authors examine the concepts of homomorphism and isomorphism between two graphs from a signal processing point of view and refer to them as GSP isomorphism and GSP homomorphism, respectively. Collectively, they refer to these concepts as structure preserving maps (SPMs). The fact that linear combination of signals and linear transforms on signals are meaningful operations has implications on the GSP isomorphism and GSP homomorphism, which diverges from the topological interpretations of the same concepts (i.e. graph isomorphism and graph homomorphism). When SPMs exist between two graphs, signals and filters can be mapped between them while preserving spectral properties. They examine conditions on adjacency matrices for such maps to exist. They also show that isospectral graphs form a special case of GSP isomorphism and that GSP isomorphism and GSP homomorphism is intrinsic to resampling and downsampling process.
Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. An infinite series of trace ...formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrodinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.
Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ type. Under one additional assumption, the inverse ...topology problem is treated. Using the apparatus of boundary triples, we generalize and extend existing results on necessary conditions of isospectrality of two Laplacians defined on different graphs. A result is also given covering the case of Schrodinger operators.
Laplacian graph eigenvectors Merris, Russell
Linear algebra and its applications,
1998, Letnik:
278, Številka:
1
Journal Article
Recenzirano
Odprti dostop
If
G is a graph, its Laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix. The main thrust of the present article is to prove several Laplacian ...eigenvector “principles” which in certain cases can be used to deduce the effect on the spectrum of contracting, adding or deleting edges and/or of coalescing vertices. One application is the construction of two isospectral graphs on 11 vertices having different degree sequences, only one of which is bipartite, and only one of which is decomposable.
In this article, we develop a perturbative technique to construct families of non-isomorphic discrete graphs which are isospectral for the standard (also called normalised) Laplacian and its signless ...version. We use vertex contractions as a graph perturbation and spectral bracketing with auxiliary graphs which have certain eigenvalues with high multiplicity. There is no need to know explicitly the eigenfunctions of the corresponding graphs. In principle, one only needs to know the multiplicity of the eigenvalues of the auxiliary graphs, and that these eigenvalues are all different. We illustrate the method by presenting several families of examples of isospectral graphs including fuzzy balls, complete bipartite graphs and subdivision graphs obtained from the previous examples. All the examples constructed turn out to be also isospectral for the standard (Kirchhoff) Laplacian on the associated equilateral metric graph.
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 ...graphs are exhibited for which the common multiset of eigenvalues consists entirely of integers.
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