377 research outputs found

    Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

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    A maximal minor MM of the Laplacian of an nn-vertex Eulerian digraph Γ\Gamma gives rise to a finite group Zn−1/Zn−1M\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M known as the sandpile (or critical) group S(Γ)S(\Gamma) of Γ\Gamma. We determine S(Γ)S(\Gamma) of the generalized de Bruijn graphs Γ=DB(n,d)\Gamma=\mathrm{DB}(n,d) with vertices 0,…,n−10,\dots,n-1 and arcs (i,di+k)(i,di+k) for 0≤i≤n−10\leq i\leq n-1 and 0≤k≤d−10\leq k\leq d-1, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime pp and an nn-cycle permutation matrix X∈GLn(p)X\in\mathrm{GL}_n(p) we show that S(DB(n,p))S(\mathrm{DB}(n,p)) is isomorphic to the quotient by ⟨X⟩\langle X\rangle of the centralizer of XX in PGLn(p)\mathrm{PGL}_n(p). This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field Fpn\mathbb{F}_{p^n} from spanning trees in DB(n,p)\mathrm{DB}(n,p).Comment: I+24 page

    Learning Local Receptive Fields and their Weight Sharing Scheme on Graphs

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    We propose a simple and generic layer formulation that extends the properties of convolutional layers to any domain that can be described by a graph. Namely, we use the support of its adjacency matrix to design learnable weight sharing filters able to exploit the underlying structure of signals in the same fashion as for images. The proposed formulation makes it possible to learn the weights of the filter as well as a scheme that controls how they are shared across the graph. We perform validation experiments with image datasets and show that these filters offer performances comparable with convolutional ones.Comment: To appear in 2017, 5th IEEE Global Conference on Signal and Information Processing, 5 pages, 3 figures, 3 table

    Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments

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    A digraph such that every proper induced subdigraph has a kernel is said to be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI for short) resp.) if the digraph has a kernel (does not have a kernel resp.). The unique CKI-tournament is C→3\overrightarrow{C}_3 and the unique KP-tournaments are the transitive tournaments, however bipartite tournaments are KP. In this paper we characterize the CKI- and KP-digraphs for the following families of digraphs: locally in-/out-semicomplete, asymmetric arc-locally in-/out-semicomplete, asymmetric 33-quasi-transitive and asymmetric 33-anti-quasi-transitive TT3TT_3-free and we state that the problem of determining whether a digraph of one of these families is CKI is polynomial, giving a solution to a problem closely related to the following conjecture posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for locally in-semicomplete digraphs.Comment: 13 pages and 5 figure

    An algebraic analysis of the graph modularity

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    One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix MM, defined in terms of the adjacency matrix and a rank one null model matrix. That matrix could be posed inside the set of relevant matrices involved in graph theory, alongside adjacency, incidence and Laplacian matrices. This is the reason we propose a graph analysis based on the algebraic and spectral properties of such matrix. In particular, we propose a nodal domain theorem for the eigenvectors of MM; we point out several relations occurring between graph's communities and nonnegative eigenvalues of MM; and we derive a Cheeger-type inequality for the graph optimal modularity
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