342 research outputs found

    Nilpotent adjacency matrices, random graphs, and quantum random variables

    Get PDF
    International audienceFor fixed n>0n>0, the space of finite graphs on nn vertices is canonically associated with an abelian, nilpotent-generated subalgebra of the 2n2n-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 mthm^th moment corresponds to the number of mm-cycles in the graph GG. Each matrix admits a canonical "quantum decomposition" into a sum of three algebraic random variables: a=aΔ+aΥ+aLambdaa = a^\Delta+ a^\Upsilon+a^Lambda, where aΔa^\Delta is classical while aΥanda^\Upsilon and a^\Lambdaarequantum.Moreover,withinthealgebraiccontext,theNPproblemofcycleenumerationisreducedtomatrixmultiplication,requiringnomorethan are quantum. Moreover, within the algebraic context, the NP problem of cycle enumeration is reduced to matrix multiplication, requiring no more than n^4$ multiplications within the algebra

    Integral Cayley graphs and groups

    Full text link
    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

    Get PDF
    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

    Get PDF
    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

    Get PDF
    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

    Full text link
    Let GG be a finite group, S⊆G∖{1}S\subseteq G\setminus\{1\} be a set such that if a∈Sa\in S, then a−1∈Sa^{-1}\in S, where 11 denotes the identity element of GG. The undirected Cayley graph Cay(G,S)Cay(G,S) of GG over the set SS is the graph whose vertex set is GG and two vertices aa and bb are adjacent whenever ab−1∈Sab^{-1}\in S. 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 GG Cayley integral whenever all undirected Cayley graphs over GG are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing 44 or 66. 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 S3S_{3} of degree 33, C3⋊C4C_{3} \rtimes C_{4} and Q8×C2nQ_{8}\times C_{2}^{n} for some integer n≥0n\geq 0, where Q8Q_8 is the quaternion group of order 88.Comment: Title is change
    • …
    corecore