Characteristic polynomials of large graphs. On alternate form of characteristic polynomial [1]

1984 ◽  
Vol 65 (3) ◽  
pp. 191-197 ◽  
Author(s):  
Sherif El-Basil
Keyword(s):  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Large Graphs ◽  
Alternate Form

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 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 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 On coefficients of circuit polynomials and characteristic polynomials

1985 ◽  
Vol 8 (4) ◽  
pp. 697-705
Author(s):  
E. J. Farrell
Keyword(s):  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Simple Circuit ◽  
Explicit Formulae

Results are given from which expressions for the coefficients of the simple circuit polynomial of a graph can be obtained in terms of subgraphs of the graph. From these are deduced parallel results for the coefficients of the characteristic polynomial of a graph. Some specific results are presented on the parities of the coefficients of characteristic polynomials. A characterization is then determined for graphs in which the number of sets of independent edges is always even. This leads to an interesting link between matching polynomials and characteristic polynomials. Finally explicit formulae are derived for the number of ways of covering two well known families of graphs with node disjoint circuits, and for the first few coefficients of their characteristic polynomials.

Linearized systems and a generalized Cayley–Hamilton theorem

2017 ◽  
Vol 16 (06) ◽  
pp. 1750120
Author(s):  
Jeffrey Lang ◽  
Daniel Newland
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Minimum Distance ◽  
Solution Space ◽  
Arbitrary Field ◽  
Characteristic Polynomials ◽  
Systems Of Equations ◽  
Square Matrix

We study linearized systems of equations in characteristic [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a square matrix and [Formula: see text]. We present algorithms for calculating their solutions and for determining the minimum distance of their solution spaces. In the case when [Formula: see text] has entries in [Formula: see text], the finite field of [Formula: see text] elements, we explore the relationships between the minimal and characteristic polynomials of [Formula: see text] and the above mentioned features of the solution space. In order to extend and generalize these findings to the case when [Formula: see text] has entries in an arbitrary field of characteristic [Formula: see text], we obtain generalizations of the characteristic polynomial of a matrix and the Cayley–Hamilton theorem to square linearized systems.

scholarly journals Computing the Characteristic Polynomials of a Class of Hyperelliptic Curves for Cryptographic Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Lin You ◽  
Guangguo Han ◽  
Jiwen Zeng ◽  
Yongxuan Sang
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Hyperelliptic Curve ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Common Method ◽  
Characteristic Polynomials ◽  
Computational Data ◽  
Prime Factors

Hyperelliptic curves have been widely studied for cryptographic applications, and some special hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems, and the most common method used for the computation of Jacobian group orders is by computing the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the hyperelliptic curveCq:v2=up+au+bover the fieldFqwithqbeing a power of an odd primep, Duursma and Sakurai obtained its characteristic polynomial forq=p,a=−1,andb∈Fp. In this paper, we determine the characteristic polynomials ofCqover the finite fieldFpnforn=1, 2 anda,b∈Fpn. We also give some computational data which show that many of those curves have large prime factors in their Jacobian group orders, which are both practical and vital for the constructions of efficient and secure hyperelliptic curve cryptosystems.

A deletion-contraction algorithm for the characteristic polynomial of a multigraph

1987 ◽  
Vol 105 (1) ◽  
pp. 153-160 ◽  
Author(s):  
Peter Rowlinson
Keyword(s):  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Largest Eigenvalues

SynopsisThe characteristic polynomial of a finite multigraph G is expressed in terms of characteristic polynomials oflocal modifications of G. The resulting formula is used to investigate the largest eigenvalues of certain theta graphs.

Characteristic Polynomial in Assessment of Carbon-Nano Structures

2017 ◽  
pp. 122-147
Author(s):  
Sorana D. Bolboacă ◽  
Lorentz Jäntschi
Keyword(s):  
Characteristic Polynomial ◽  
Chemical Structure ◽  
Chair Conformation ◽  
Point Of View ◽  
Characteristic Polynomials ◽  
Graph Invariant ◽  
Real Roots ◽  
Nano Structures ◽  
Central Form ◽  
Number Of Real Roots

Six dodecahedrane assemblies as multiple of five and respectively six structures were constructed and investigated from the topological point of view. The investigation was conducted using characteristic polynomials, graph invariant encoding important properties of the graph of the chemical structure. The assemblies of 5, 6, 15 and 25 dodecahedranes proved to have the center in the same plane while the assemblies of 12 and 24 dodecahedranes degenerated from the planar central form to a chair conformation. Generally, the number of real roots of characteristic polynomials is equal to the number of atoms in the assembly. The obtained roots of the characteristic polynomial were split into intervals and the frequency apparition spectra were simulated. The obtained spectra were used to investigate the behavior of investigated assembly.

On the Laplacian characteristic polynomials of mixed graphs

2015 ◽  
Vol 30 ◽  
Author(s):  
Dariush Kiani ◽  
Maryam Mirzakhah
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Characteristic Polynomials ◽  
Laplacian Spectral Radius ◽  
Mixed Graph ◽  
First Derivative ◽  
Signless Laplacian ◽  
Mixed Graphs

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial of a mixed graph is reconstructible from the collection of the Laplacian characteristic polynomials of its edge deleted subgraphs. Then, it is investigated how graph modifications affect the mixed Laplacian characteristic polynomial. Also, a connection between the Laplacian characteristic polynomial of a non-singular connected mixed graph and the signless Laplacian characteristic polynomial is provided, and it is used to establish a lower bound for the spectral radius of L(G). Finally, using Coates digraphs, the perturbation of the mixed Laplacian spectral radius under some graph transformations is discussed.

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