342 research outputs found
Nilpotent adjacency matrices, random graphs, and quantum random variables
International audienceFor fixed , the space of finite graphs on vertices is canonically associated with an abelian, nilpotent-generated subalgebra of the -particle fermion algebra. using the generators of the subalgebra, an algebraic probability space of "nilpotent adjacency matrices" associated with finite graphs is defined. Each nilpotent adjacency matrix is a quantum random variable whose moment corresponds to the number of -cycles in the graph . Each matrix admits a canonical "quantum decomposition" into a sum of three algebraic random variables: , where is classical while a^\Lambdan^4$ multiplications within the algebra
Integral Cayley graphs and groups
We solve two open problems regarding the classification of certain classes of
Cayley graphs with integer eigenvalues. We first classify all finite groups
that have a "non-trivial" Cayley graph with integer eigenvalues, thus solving a
problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups
was introduced by Klotz and Sander. These are groups for which every Cayley
graph has only integer eigenvalues. In the second part of the paper, all Cayley
integral groups are determined.Comment: Submitted June 18 to SIAM J. Discrete Mat
Operator Calculus Algorithms for Multi-Constrained Paths
Classical approaches to multi-constrained routing problems generally require construction of trees and the use of heuristics to prevent combinatorial explosion. Introduced here is the notion of constrained path algebras and their application to multi-constrained path problems. The inherent combinatorial properties of these algebras make them useful for routing problems by implicitly pruning the underlying tree structures. Operator calculus (OC) methods are generalized to multiple non-additive constraints in order to develop algorithms for the multi constrained path problem and multi constrained optimization problem. Theoretical underpinnings are developed first, then algorithms are presented. These algorithms demonstrate the tremendous simplicity, flexibility and speed of the OC approach. Algorithms are implemented in Mathematica and Java and applied to a problem first proposed by Ben Slimane et al. as an example
Forest matrices around the Laplacian matrix
We study the matrices Q_k of in-forests of a weighted digraph G and their
connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the
total weight of spanning converging forests (in-forests) with k arcs such that
i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated
recursively and expressed by polynomials in the Laplacian matrix; they provide
representations for the generalized inverses, the powers, and some eigenvectors
of L. The normalized in-forest matrices are row stochastic; the normalized
matrix of maximum in-forests is the eigenprojection of the Laplacian matrix,
which provides an immediate proof of the Markov chain tree theorem. A source of
these results is the fact that matrices Q_k are the matrix coefficients in the
polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's
matrices for -L.
Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest
theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection;
Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic
Graph Theor
A graph signal processing solution for defective directed graphs
The main purpose of this thesis is to nd a method that allows to systematically adapt GSP
techniques so they can be used on most non-diagonalizable graph operators.
In Chapter 1 we begin by presenting the framework in which GSP is developed, giving
some basic de nitions in the eld of graph theory and in relation with graph signals. We also
present the concept of a Graph Fourier Tranform (GFT), which will be of great importance
in the proposed solution.
Chapter 2 presents the actual motivation of the research: Why the computation of the
GFT is problematic for some directed graphs, and the speci c cases in which this happen. We
will see that the issue can not be assigned to a very speci c graph topography, and therefore
it is important to develop solutions that can be applied to any directed graph.
In Chapter 3 we introduce our proposed new method, which can be used to form, based on
the spectral decomposition of a matrix obtained through its Schur decomposition, a complete
basis of vectors that can be used as a replacement of the previously mentioned Graph Fourier
Transform. The proposed method, the Graph Schur Transform (GST), aims to o er a valid
operator to perform a spectral decomposition of a graph that can be used even in the case of
defective matrices.
Finally, in Chapter 4 we study the main properties of the proposed method and compare
them with the corresponding properties o ered by the Di usion Wavelets design. In the last
section we prove, for a large set of directed graphs, that the GST provides a valid solution for
the proble
Groups all of whose undirected Cayley graphs are integral
Let be a finite group, be a set such that if
, then , where denotes the identity element of .
The undirected Cayley graph of over the set is the graph
whose vertex set is and two vertices and are adjacent whenever
. The adjacency spectrum of a graph is the multiset of all
eigenvalues of the adjacency matrix of the graph. A graph is called integral
whenever all adjacency spectrum elements are integers. Following Klotz and
Sander, we call a group Cayley integral whenever all undirected Cayley
graphs over are integral. Finite abelian Cayley integral groups are
classified by Klotz and Sander as finite abelian groups of exponent dividing
or . Klotz and Sander have proposed the determination of all non-abelian
Cayley integral groups. In this paper we complete the classification of finite
Cayley integral groups by proving that finite non-abelian Cayley integral
groups are the symmetric group of degree , and
for some integer , where is the
quaternion group of order .Comment: Title is change
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