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45
The laplacian spectrum of graphs”.
 In Graph Theory, Combinatorics,
, 1991
"... Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, ..."
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Cited by 228 (2 self)
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Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added.
INVERSE SPECTRAL PROBLEM FOR NORMAL MATRICES AND THE GAUSSLUCAS THEOREM
"... Abstract. We establish an analog of the CauchyPoincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss–Lucas theorem and prove the old conjecture of de B ..."
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Cited by 28 (0 self)
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Abstract. We establish an analog of the CauchyPoincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss–Lucas theorem and prove the old conjecture of de BruijnSpringer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle’s theorem, conjectured by Schoenberg. 1.
Algebraic Graph Theory Without Orientation
"... Let G be an undirected graph with vertices {v 1 , v 2 , . . . , v # } and edges {e 1 , e 2 , . . . , e # }. Let M be the # × # matrix whose ijth entry is 1 if e j is a link incident with v i , 2 if e j is a loop at v i , and 0 otherwise. The matrix obtained by orienting the edges of a loo ..."
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Cited by 17 (1 self)
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Let G be an undirected graph with vertices {v 1 , v 2 , . . . , v # } and edges {e 1 , e 2 , . . . , e # }. Let M be the # × # matrix whose ijth entry is 1 if e j is a link incident with v i , 2 if e j is a loop at v i , and 0 otherwise. The matrix obtained by orienting the edges of a loopless graph G (i.e., changing one of the 1s to a 1 in each column of M) has been studied extensively in the literature. The purpose of this paper is to explore the substructures of G and the vector spaces associated with the matrix M without imposing such an orientation. We describe explicitly bases for the kernel and range of the linear transformation from R # to R # defined by M. Our main results are determinantal formulas, using the unoriented Laplacian matrix MM t , to count certain spanning substructures of G. These formulas may be viewed as generalizations of the Matrix Tree Theorem. The point of view adopted in this paper also gives rise to a matroid structure on the edges o...
Information Hiding in Product Development: The Design Churn Effect, Forthcoming Research
 in Engineering Design, Volume 14. The Liar’s Club: Concealing Rework in Concurrent Development 219 + [25.9.2003–8:45am] [211–220] [Page No. 219] REVISE PROOFS I:/Sage/Cer/Cer113/CER38028.3d (CER) Paper: CER38028 Keyword
, 2003
"... Execution of a complex product development project is facilitated through its decomposition into an interrelated set of localized development tasks. When a local task is completed, its output is integrated through an iterative cycle of systemwide integration activities. Integration is often accompa ..."
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Cited by 15 (5 self)
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Execution of a complex product development project is facilitated through its decomposition into an interrelated set of localized development tasks. When a local task is completed, its output is integrated through an iterative cycle of systemwide integration activities. Integration is often accompanied by inadvertent information hiding due to the asynchronous information exchanges. We show that information hiding leads to persistent recurrence of problems (termed as the design churn effect) such that progress oscillates between being on schedule and falling behind. The oscillatory nature of the PD process confounds progress measurement and makes it difficult to judge whether the project is on schedule or slipping. We develop a dynamic model of work transformation to derive conditions under which churn is observed as an unintended consequence of information hiding due to local and system task decomposition. We illustrate these conditions with a case example from an automotive development project and discuss strategies to mitigate design churn.
On the discounted penalty function in a Markovdependent risk model
"... We present a unified approach to the analysis of several popular models in collective risk theory. Based on the analysis of the discounted penalty function in a semiMarkovian risk model by means of LaplaceStieltjes transforms, we rederive and extend some recent results in the field. In particular, ..."
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Cited by 13 (4 self)
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We present a unified approach to the analysis of several popular models in collective risk theory. Based on the analysis of the discounted penalty function in a semiMarkovian risk model by means of LaplaceStieltjes transforms, we rederive and extend some recent results in the field. In particular, the classical compound Poisson model, Sparre Andersen models with phasetype interclaim times and models with causal dependence of a certain Markovian type between claim sizes and interclaim times are contained as special cases.
Möbius Transformations of Matrix Polynomials
, 2014
"... We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information co ..."
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Cited by 11 (1 self)
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We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated.
Smith Forms of Palindromic Matrix Polynomials
, 2011
"... Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry, a structure we call palindromic. Several properties of scalar palindromic polynomials are derived, and together with properties of compound matrices, used to establish the Smith form of regular an ..."
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Cited by 9 (3 self)
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Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry, a structure we call palindromic. Several properties of scalar palindromic polynomials are derived, and together with properties of compound matrices, used to establish the Smith form of regular and singular Tpalindromic matrix polynomials over arbitrary fields. The invariant polynomials are shown to inherit palindromicity, and their structure is described in detail. Jordan structures of palindromic matrix polynomials are characterized, and necessary conditions for the existence of structured linearizations established. In the odd degree case, a constructive procedure for building palindromic linearizations shows that the necessary conditions are sufficient as well. The Smith form for ∗palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of characteristic two are presented.
FLOWS ON GRAPHS APPLIED TO DIAGONAL SIMILARITY AND DIAGONAL EQUIVALENCE FOR MATRICES
, 1978
"... Three equivalence relations are considered on the set of n x n matrices with elements in F o ' an · abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for ma ..."
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Cited by 9 (8 self)
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Three equivalence relations are considered on the set of n x n matrices with elements in F o ' an · abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for matrices with elements in a field, but only multiplication is involved. Thus our formulation in terms of an abelian group with 0 is natural. Moreover, if F is chosen to be an additive group, diagonal similarity is characterized in terms of flows on the pattern graph of the matrices and diagonal equivalence in terms of flows on the bipartite graph of the matrices. For restricted diagonal equivalence a pseudodiagonal of the graph must also be considered. When no pseudodiagonal is present, the divisibility properties of the grO\iP F playa role. We show that the three relations are characterized by cyclic, polygonal, and pseudodiagonal products for multiplicative F. Thus, our method of reducing propositions concering the three equivalence relations to propositions concerning flows on graphs, provides a unified approach to problems previously considered independently, and yields some new or improved results. Our consideration of cycles rather than circuits eliminates certain restrictions (e.g., the complete reducibility of the matrices) which have previously been imposed. Our results extend theorems in Engel and Schneider [5], where however the group F is permitted to be noncommutative.
SkewSymmetric Matrix Polynomials and their Smith Forms
, 2012
"... We characterize the Smith form of skewsymmetric matrix polynomials over an arbitrary field F, showing that all elementary divisors occur with even multiplicity. Restricting the class of equivalence transformations to unimodular congruences, a Smithlike skewsymmetric canonical form for skewsymm ..."
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Cited by 8 (2 self)
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We characterize the Smith form of skewsymmetric matrix polynomials over an arbitrary field F, showing that all elementary divisors occur with even multiplicity. Restricting the class of equivalence transformations to unimodular congruences, a Smithlike skewsymmetric canonical form for skewsymmetric matrix polynomials is also obtained. These results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skewsymmetric matrices, as well as the existence of structured linearizations for skewsymmetric matrix polynomials. By contrast with other classes of structured matrix polynomials (e.g., alternating or palindromic polynomials), every regular skewsymmetric matrix polynomial is shown to have a structured strong linearization. While there are singular skewsymmetric polynomials of even degree for which a structured linearization is impossible, for each odd degree we develop a skewsymmetric companion form that uniformly provides a structured linearization for every regular and singular skewsymmetric polynomial of that degree. Finally, the results are applied to the construction of minimal symmetric factorizations of skewsymmetric rational matrices.
Continuity for MultiType Branching Processes With Varying Environments: Example
, 1997
"... Gamma1 ; 2 n\Gamma1 p 3) + Vn ] and En+1 = En [ [(2 n ; 0) +En ] [ [(2 n\Gamma1 ; 2 n\Gamma1 p 3) +En ] taking the sums elementwise over the given sets. Let V = V1 [ [\GammaV 1 ] and E = E1 [ [\GammaE 1 ] and write G 0 for the graph (V; E) and Gn for 2 \Gamman G 0 . The Sierpinski gas ..."
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Cited by 8 (2 self)
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Gamma1 ; 2 n\Gamma1 p 3) + Vn ] and En+1 = En [ [(2 n ; 0) +En ] [ [(2 n\Gamma1 ; 2 n\Gamma1 p 3) +En ] taking the sums elementwise over the given sets. Let V = V1 [ [\GammaV 1 ] and E = E1 [ [\GammaE 1 ] and write G 0 for the graph (V; E) and Gn for 2 \Gamman G 0 . The Sierpinski gasket G is the closure of the set [ 1 n=0 2 \Gamman V