Results 1  10
of
30
Distributed coverage verification in sensor networks without location information
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2008
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Simplicial matrixtree theorems
, 2008
"... We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrarydimensional simplicial complexes ∆, extending an idea due to G. Kalai. We prove a simplicial version of the MatrixTree Theorem that counts simplicial spanning trees, weighted by the squa ..."
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Cited by 25 (3 self)
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We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrarydimensional simplicial complexes ∆, extending an idea due to G. Kalai. We prove a simplicial version of the MatrixTree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their topdimensional integral homology groups, in terms of the Laplacian matrix of ∆. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of ∆ and replacing the entries of the Laplacian with Laurent monomials. When ∆ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.
Control Using Higher Order Laplacians in Network Topologies
 Proc. of 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto
, 2006
"... This paper establishes the proper notation and precise interpretation for Laplacian flows on simplicial complexes. In particular, we have shown how to interpret these flows as timevarying discrete differential forms that converge to harmonic forms. The stability properties of the corresponding dyna ..."
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Cited by 24 (3 self)
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This paper establishes the proper notation and precise interpretation for Laplacian flows on simplicial complexes. In particular, we have shown how to interpret these flows as timevarying discrete differential forms that converge to harmonic forms. The stability properties of the corresponding dynamical system are shown to be related to the topological structure of the underlying simplicial complex. Finally, we discuss the relevance of these results in the context of networked control and sensing. I.
Shifted set families, degree sequences, and plethysm
, 2008
"... We study, in three parts, degree sequences of kfamilies (or kuniform hypergraphs) and shifted kfamilies. • The first part collects for the first time in one place, various implications such as Threshold ⇒ Uniquely Realizable ⇒ DegreeMaximal ⇒ Shifted which are equivalent concepts for 2families ..."
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Cited by 12 (4 self)
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We study, in three parts, degree sequences of kfamilies (or kuniform hypergraphs) and shifted kfamilies. • The first part collects for the first time in one place, various implications such as Threshold ⇒ Uniquely Realizable ⇒ DegreeMaximal ⇒ Shifted which are equivalent concepts for 2families ( = simple graphs), but strict implications for kfamilies with k ≥ 3. The implication that uniquely realizable implies degreemaximal seems to be new. • The second part recalls Merris and Roby’s reformulation of the characterization due to Ruch and Gutman for graphical degree sequences and shifted 2families. It then introduces two generalizations which are characterizations of shifted kfamilies. • The third part recalls the connection between degree sequences of kfamilies of size m and the plethysm of elementary symmetric functions em[ek]. It then
A Common Recursion For Laplacians of Matroids and Shifted Simplicial Complexes
 DOCUMENTA MATH.
, 2005
"... A recursion due to Kook expresses the Laplacian eigenvalues of a matroid M in terms of the eigenvalues of its deletion M − e and contraction M/e by a fixed element e, and an error term. We show that this error term is given simply by the Laplacian eigenvalues of the pair (M −e,M/e). We further show ..."
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Cited by 8 (2 self)
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A recursion due to Kook expresses the Laplacian eigenvalues of a matroid M in terms of the eigenvalues of its deletion M − e and contraction M/e by a fixed element e, and an error term. We show that this error term is given simply by the Laplacian eigenvalues of the pair (M −e,M/e). We further show that by suitably generalizing deletion and contraction to arbitrary simplicial complexes, the Laplacian eigenvalues of shifted simplicial complexes satisfy this exact same recursion. We show that the class of simplicial complexes satisfying this recursion is closed under a wide variety of natural operations, and that several specializations of this recursion reduce to basic recursions for natural invariants. We also find a simple formula for the Laplacian eigenvalues of an arbitrary pair of shifted complexes in terms of a kind of generalized degree sequence.
CELLULAR SPANNING TREES AND LAPLACIANS OF CUBICAL COMPLEXES
, 2009
"... We prove a MatrixTree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigen ..."
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Cited by 6 (3 self)
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We prove a MatrixTree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifted cubical complexes, in particular, these eigenvalues are also integers. Finally, we recover Adin’s enumeration of spanning trees of a complete colorful simplicial complex from the cellular MatrixTree Theorem together with a result of Kook, Reiner and Stanton.
A majorization bound for the eigenvalues of some graph Laplacians
 SIAM J. DISCRETE MATH
, 2004
"... It is conjectured that the Laplacian spectrum of a graph is majorized by its conjugate degree sequence. In this paper, we prove that this majorization holds for a class of graphs including trees. We also show that a generalization of this conjecture to graphs with Dirichlet boundary conditions is e ..."
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Cited by 5 (0 self)
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It is conjectured that the Laplacian spectrum of a graph is majorized by its conjugate degree sequence. In this paper, we prove that this majorization holds for a class of graphs including trees. We also show that a generalization of this conjecture to graphs with Dirichlet boundary conditions is equivalent to the original conjecture.
Laplacian spectrum of weakly quasithreshold graphs
 GRAPHS COMBIN
, 2008
"... In this paper we study the class of weakly quasithreshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a n ..."
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Cited by 5 (0 self)
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In this paper we study the class of weakly quasithreshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old vertex. This class contains the class of quasithreshold graphs. We show that weakly quasithreshold graphs are precisely the comparability graphs of a forest consisting of rooted trees with each vertex of a tree being replaced by an independent set. We also supply a quadratic time algorithm in the the size of the vertex set for recognizing such a graph. We completely determine the Laplacian spectrum of weakly quasithreshold graphs. It turns out that weakly quasithreshold graphs are Laplacian integral. As a corollary we obtain a closed formula for the number of spanning trees in such graphs. A conjecture of Grone and Merris asserts that the spectrum of the Laplacian of any graph is majorized by the conjugate of the degree sequence of the graph. We show that the conjecture holds for cographs.