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1993 Discrete Applied Mathematics  
The enumeration of N, and N + from the characteristic polynomial. A graph-theoretical classification of conjugated hydrocarbons based on their spectral characteristics).  ...  Methods based on graphic polynomials (The characteristic polynomial. The Acyclic polynomial. The permanental polynomial). The method based on the coefficients of nonbonding molecular orbitals.  ... 
doi:10.1016/0166-218x(93)90134-a fatcat:guesbjpz7fcb7nex3cwh2gfg4e

Generalized characteristic polynomials of graph bundles

Dongseok Kim, Hye Kyung Kim, Jaeun Lee
2008 Linear Algebra and its Applications  
We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph.  ...  In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles.  ...  Its adjacency matrix, known as a "kite", is one of important examples in matrix analysis. The outline of this paper is as follows.  ... 
doi:10.1016/j.laa.2008.03.023 fatcat:lhs7hodvizf5looulutb2gv6ya

Shift-Enabled Graphs: Graphs Where Shift-Invariant Filters are Representable as Polynomials of Shift Operations

Liyan Chen, Samuel Cheng, Vladimir Stankovic, Lina Stankovic
2018 IEEE Signal Processing Letters  
However, the characteristic and minimum polynomials of the adjacency matrix must be identical for the property to hold.  ...  When extended to graph signal processing (GSP), this would mean that a shift-invariant graph filter can be represented as a polynomial of the adjacency (shift) matrix of the graph.  ...  A graph G is shift-enabled † if the characteristic and minimal polynomials of its corresponding shift matrix S are equivalent, i.e., p S (λ) = m S (λ).  ... 
doi:10.1109/lsp.2018.2849685 fatcat:6nikkuarhvb6leb4b2pez55j2i

Thermodynamic characterization of networks using graph polynomials

Cheng Ye, César H. Comin, Thomas K. DM. Peron, Filipi N. Silva, Francisco A. Rodrigues, Luciano da F. Costa, Andrea Torsello, Edwin R. Hancock
2015 Physical Review E  
Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function  ...  ) characterization of graph structure.  ...  A more convenient route is to adopt an alternative graph representation based on a characteristic polynomial.  ... 
doi:10.1103/physreve.92.032810 pmid:26465531 fatcat:5srvx2ay3bh6lmkswh4kob33se

Efficient computation of Ihara coefficients using the Bell polynomial recursion

Samuel Rota Bulò, Edwin R. Hancock, Furqan Aziz, Marcello Pelillo
2012 Linear Algebra and its Applications  
It is determined by the prime cycles of a finite graph G = (V, E) and can be characterized in terms of a quasi characteristic polynomial of the adjacency matrix T of the oriented line graph associated  ...  The Ihara zeta function has proved to be a powerful tool in the analysis of graph structures.  ...  It can be expressed in terms of a quasi characteristic polynomial of the adjacency matrix T of the oriented line graph L(G) associated to G.  ... 
doi:10.1016/j.laa.2011.08.017 fatcat:znqlw4glgzgi7l7a5b77l4muxu

The Companion Model – a Canonical Model in Graph Signal Processing [article]

John Shi, Jose M. F. Moura
2022 arXiv   pre-print
We further show that, surprisingly, in the canonical model, convolution of graph signals is fast convolution by the DSP FFT.  ...  This paper introduces a canonical graph signal model defined by a canonical graph and a canonical shift, the companion graph and the companion shift.  ...  Since 𝐶 comp is determined by the characteristic polynomial Δ 𝐴 (𝑥), it only depends on the graph frequencies or eigenvalues of 𝐴, not on the spectral modes or eigenvectors of 𝐴.  ... 
arXiv:2203.13791v1 fatcat:mci63yfa2vbelb6i6cxtygjlcu

Visual Exploration of Polynomial Combination Based on Variant Measurement Model

Feng Deng, Jeffrey Zheng
2019 Journal of Applied Mathematics and Physics  
It can be seen from the variant maps of the polynomial that the graphical results under multiple conditions exhibit spatial symmetry, as well as special changes and fixed characteristics of the points.  ...  The visualization of the final polynomial combination formula is presented in a two-dimensional color map mode.  ...  Acknowledgements I would like to thank my instructor, Professor Jeffrey Zheng, for his careful guidance on this paper and his care and concern.  ... 
doi:10.4236/jamp.2019.711203 fatcat:dv4eygk5lnfmdiziogkdgw33ly

Symbolic OBDD-Based Reachability Analysis Needs Exponential Space [chapter]

Beate Bollig
2010 Lecture Notes in Computer Science  
Sketch of proof: a pathological graph instance G n : symbolic OBDD-representation is polynomial with respect to the number of Boolean variables the characteristic function of R, the set that consists of  ...  Sketch of proof: a pathological graph instance G n : symbolic OBDD-representation is polynomial with respect to the number of Boolean variables the characteristic function of R, the set that consists of  ...  Polynomial upper bound on the OBDD size of χ E (v i 1 ,...,in , v ℓ 1 ,...  ... 
doi:10.1007/978-3-642-11266-9_19 fatcat:dibjikvj6rdhdmmwlysur2ibpu

Multi-graph learning of spectral graph dictionaries

Dorina Thanou, Pascal Frossard
2015 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)  
In particular, we concentrate on graph atoms that are constructed on polynomials of the graph Laplacian.  ...  We propose to learn graph atoms and build graph dictionaries that provide sparse representations for classes of signals, which share common spectral characteristics but reside on the vertices of different  ...  These characteristics are reflected in a common dictionary structure that is constructed to be a polynomial function of the graph Laplacian matrix, and the coefficients of the polynomials are jointly learned  ... 
doi:10.1109/icassp.2015.7178601 dblp:conf/icassp/ThanouF15 fatcat:ydiwq65bybderlfbada2tnqfga

Weighted graph characteristics from oriented line graph polynomials

Peng Ren, Richard C Wilson, Edwin R Hancock
2009 2009 IEEE 12th International Conference on Computer Vision  
We develop a novel method for extracting graph characteristics from edge-weighted graphs, based on an extension of the Ihara zeta function from unweighted to edgeweighted graphs.  ...  We also present a spectral analysis of the edge-weighted graph Ihara coefficients and indicate their advantages over graph spectral methods.  ...  Although the the zeta function draws on the characteristic polynomial of a graph and is hence akin to methods from algebraic graph theory, it first relies on a graph transformation.  ... 
doi:10.1109/iccv.2009.5459449 dblp:conf/iccv/RenWH09 fatcat:rfq77ktitjdkjmnzmfr4evbrg4

Spectral Analysis of Complex Laplacian Matrices [chapter]

Richard C. Wilson, Edwin R. Hancock
2004 Lecture Notes in Computer Science  
Our spectral representation is based on the eigendecomposition of the resulting Hermitian property matrix. The eigenvalues of the matrix are real while the eigenvectors are complex.  ...  This paper explores how to extend the spectral analysis of graphs to the case where the nodes and edges are attributed. To do this we introduce a complex Hermitian variant of the Laplacian matrix.  ...  characteristics to perform pattern analysis on sets of graphs and trees.  ... 
doi:10.1007/978-3-540-27868-9_5 fatcat:xvowttehzjcohef7675pask6mi

Discrete Signal Processing on Graphs

Aliaksei Sandryhaila, José M. F. Moura
2013 IEEE Transactions on Signal Processing  
This paper extends to signals on graphs DSP and its basic tenets, including filters, convolution, z-transform, impulse response, spectral representation, Fourier transform, frequency response, and illustrates  ...  DSP on graphs by classifying blogs, linear predicting and compressing data from irregularly located weather stations, or predicting behavior of customers of a mobile service provider.  ...  The degree of the minimal polynomial satisfies deg m A (x) = N A = M−1 m=0 R m ≤ N. (42) The characteristic polynomial of the matrix A is defined as Fig. 1 . 1 Graph representations for different datasets  ... 
doi:10.1109/tsp.2013.2238935 fatcat:er2wivkywnapnfhfepx4lbxwmq

Model Development for Analysis of Steam Power Plant Reliability [chapter]

Bernard Tonderayi Mangara
2017 Recent Improvements of Power Plants Management and Technology  
The graph theoretical analysis and a graph's characteristic polynomial are deployed as the basis for a system approach employed to develop a model for estimating the reliability index and evaluating the  ...  Computing a graph's characteristic polynomial using three different methods, namely, the linearly independent cycles, the figure equation and the adjacent matrix, the approximate reliability of the system  ...  The characteristic polynomial of the graph is the characteristic polynomial of its adjacency matrix.  ... 
doi:10.5772/intechopen.68285 fatcat:ak25o6oiyrh2fbxvvta5zdcxve

Symmetric Determinantal Representations in characteristic 2

Bruno Grenet, Thierry Monteil, Stéphan Thomassé
2013 Linear Algebra and its Applications  
This paper studies Symmetric Determinantal Representations (SDR) in characteristic 2, that is the representation of a multivariate polynomial P by a symmetric matrix M such that P=det(M), and where each  ...  We conclude by describing the case of Alternating Determinantal Representations in any characteristic.  ...  We once again work with the graph representation instead of the matrix representation.  ... 
doi:10.1016/j.laa.2013.04.022 fatcat:3n2hn7bi6zeblijittbvgklv4y

Polynomial Matrix Inequalities within Tame Geometry [article]

Christos Aravanis and Johannes Aspman and Georgios Korpas and Jakub Marecek
2022 arXiv   pre-print
Polynomial matrix inequalities can be solved using hierarchies of convex relaxations, pioneered by Henrion and Lassere.  ...  In some cases, this might not be practical, and one may need to resort to methods with local convergence guarantees, whose development has been rather ad hoc, so far.  ...  J.A. is supported by the Government of Ireland Postgraduate Scholarship Programme GOIPG/2020/910 of the Irish Research Council.  ... 
arXiv:2206.03941v1 fatcat:qycpk6bxnbd4vaytz2646c7nbu
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