Sign-Coherent Identities for Characteristic Polynomials of Matroids

1993 ◽  
Vol 2 (1) ◽  
pp. 33-51 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Explicit Formula ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Critical Problem ◽  
Whitney Numbers ◽  
Numerical Invariants ◽  
The Difference ◽  
Simple Matroid ◽  
Broken Circuit

We derive an explicit formula for the difference χ(G;λ) − χ(G|X;λ)χ(Tx(G); λ)/(λ − 1), where χ(G;λ) is the characteristic polynomial of a simple matroid G, G|X is the restriction of G to a flat X in G, and Tx(G) is the complete principal truncation of G at the flat X. Two counting proofs of this formula are given. The first uses the critical problem and the second uses the broken-circuit complex. We also derive several inequalities involving Whitney numbers of the first kind and other numerical invariants.

scholarly journals Characteristic polynomials of sample covariance matrices: The non-square case

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Holger Kösters
Keyword(s):  
Correlation Function ◽  
Characteristic Polynomial ◽  
Large Data ◽  
Covariance Matrices ◽  
Characteristic Polynomials ◽  
Complex Matrix ◽  
Sample Covariance Matrices ◽  
Sample Covariance ◽  
The Difference ◽  
Sine Kernel

AbstractWe consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given for real sample covariance matrices.

Laplace and Extended Adjacency Matrices for Isomorphism Detection of Kinematic Chains Using the Characteristic Polynomial Approach

2005 ◽  
Author(s):  
Rajesh Pavan Sunkari ◽  
Linda C. Schmidt
Keyword(s):  
Characteristic Polynomial ◽  
Spectral Properties ◽  
Kinematic Chain ◽  
Detection Algorithm ◽  
Isomorphism Problem ◽  
Characteristic Polynomials ◽  
Kinematic Chains ◽  
Laplace Matrix ◽  
Adjacency Matrices ◽  
Single Matrix

The kinematic chain isomorphism problem is one of the most challenging problems facing mechanism researchers. Methods using the spectral properties, characteristic polynomial and eigenvectors, of the graph related matrices were developed in literature for isomorphism detection. Detection of isomorphism using only the spectral properties corresponds to a polynomial time isomorphism detection algorithm. However, most of the methods used are either computationally inefficient or unreliable (i.e., failing to identify non-isomorphic chains). This work establishes the reliability of using the characteristic polynomial of the Laplace matrix for isomorphism detection of a kinematic chain. The Laplace matrix of a graph is used extensively in the field of algebraic graph theory for characterizing a graph using its spectral properties. The reliability in isomorphism detection of the characteristic polynomial of the Laplace matrix was comparable with that of the adjacency matrix. However, using the characteristic polynomials of both the matrices is superior to using either method alone. In search for a single matrix whose characteristic polynomial unfailingly detects isomorphism, novel matrices called the extended adjacency matrices are developed. The reliability of the characteristic polynomials of these matrices is established. One of the proposed extended adjacency matrices is shown to be the best graph matrix for isomorphism detection using the characteristic polynomial approach.

scholarly journals On the class of square Petrie matrices induced by cyclic permutations

2004 ◽  
Vol 2004 (31) ◽  
pp. 1617-1622
Author(s):  
Bau-Sen Du
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Dynamical Properties

Letn≥2be an integer and letP={1,2,…,n,n+1}. LetZpdenote the finite field{0,1,2,…,p−1}, wherep≥2is a prime. Then every mapσonPdetermines a realn×nPetrie matrixAσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσis acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to∑k=0nxk. As a consequence, we obtain that ifσis acyclicpermutation onP, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.

scholarly journals Asymptotics for the time of ruin in the war of attrition

2017 ◽  
Vol 49 (2) ◽  
pp. 388-410 ◽  
Author(s):  
Philip A. Ernst ◽  
Ilie Grigorescu
Keyword(s):  
Distribution Function ◽  
Explicit Formula ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Fluid Limit ◽  
War Of Attrition ◽  
Time Of Ruin ◽  
Standard Normal ◽  
The Difference ◽  
The Time Of Ruin

AbstractWe consider two players, starting withmandnunits, respectively. In each round, the winner is decided with probability proportional to each player's fortune, and the opponent loses one unit. We prove an explicit formula for the probabilityp(m,n) that the first player wins. Whenm~Nx0,n~Ny0, we prove the fluid limit asN→ ∞. Whenx0=y0,z→p(N,N+z√N) converges to the standard normal cumulative distribution function and the difference in fortunes scales diffusively. The exact limit of the time of ruin τNis established as (T- τN) ~N-βW1/β, β = ¼,T=x0+y0. Modulo a constant,W~ χ21(z02/T2).

scholarly journals Characteristic polynomials of some weighted graph bundles and its application to links

1994 ◽  
Vol 17 (3) ◽  
pp. 503-510 ◽  
Author(s):  
Moo Young Sohn ◽  
Jaeun Lee
Keyword(s):  
Characteristic Polynomial ◽  
Weighted Graph ◽  
Double Covering ◽  
Characteristic Polynomials ◽  
Weighted Graphs

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weightedK2(K¯2)-bundles over a weighted graphG?can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs areGAs an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

scholarly journals From Fibonacci Sequence to the Golden Ratio

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Alberto Fiorenza ◽  
Giovanni Vincenzi
Keyword(s):  
Characteristic Polynomial ◽  
Golden Ratio ◽  
Fibonacci Sequence ◽  
Maximum Modulus ◽  
Point Of View ◽  
General Point ◽  
Constant Coefficients ◽  
The Difference ◽  
New Perspective

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

On the solutions of a higher order difference equation

2020 ◽  
Vol 27 (2) ◽  
pp. 165-175 ◽  
Author(s):  
Raafat Abo-Zeid
Keyword(s):  
Difference Equation ◽  
Explicit Formula ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Forbidden Set ◽  
Order Difference Equation ◽  
The Difference

AbstractIn this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equationx_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots,where{a,b,c}are positive real numbers and the initial conditions{x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}}are real numbers. We show that when{a=b=c}, the behavior of the solutions depends on whetherkis even or odd.

scholarly journals Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1663
Author(s):  
Alexander Farrugia
Keyword(s):  
Characteristic Polynomial ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Reconstruction Problem ◽  
Vertex Set ◽  
Disconnected Graphs ◽  
Polynomial Reconstruction ◽  
Standing Problem

Let G be a simple graph and {1,2,…,n} be its vertex set. The polynomial reconstruction problem asks the question: given a deck P(G) containing the n characteristic polynomials of the vertex deleted subgraphs G−1, G−2, …, G−n of G, can ϕ(G,x), the characteristic polynomial of G, be reconstructed uniquely? To date, this long-standing problem has only been solved in the affirmative for some specific classes of graphs. We prove that if there exists a vertex v such that more than half of the eigenvalues of G are shared with those of G−v, then this fact is recognizable from P(G), which allows the reconstruction of ϕ(G,x). To accomplish this, we make use of determinants of certain walk matrices of G. Our main result is used, in particular, to prove that the reconstruction of the characteristic polynomial from P(G) is possible for a large subclass of disconnected graphs, strengthening a result by Sciriha and Formosa.

scholarly journals Sets with Few Differences in Abelian Groups

10.37236/5502 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Mitchell Lee
Keyword(s):  
Abelian Group ◽  
Explicit Formula ◽  
Abelian Groups ◽  
Difference Set ◽  
The Difference ◽  
Fixed Cardinality

Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is an elementary abelian $p$-group. This resolves a conjecture of Bajnok and Matzke on signed sumsets.

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