Norms and Generating Functions in Clifford Algebras View Full Text


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Article Info

DATE

2008-02

AUTHORS

G. Stacey Staples

ABSTRACT

Generating functions are commonly used in combinatorics to recover sequences from power series expansions. Convergence of formal power series in Clifford algebras of arbitrary signature is discussed. Given , powers of u are recovered by expanding (1 − tu)−1 as a polynomial in t with Clifford-algebraic coefficients. It is clear that (1 − tu)(1 + tu + t2u2 + ...) = 1, provided the sum (1 + tu + t2u2 + ...) exists, in which case um is the Cliffordalgebraic coefficient of tm in the series expansion of (1 − tu)−1. In this paper, conditions on for the existence of (1 − tu)−1 are given, and an explicit formulation of the generating function is obtained. Allowing A to be an m × m matrix with entries in , a “Clifford-Frobenius” norm of A is defined. Norm inequalities are then considered, and conditions for the existence of (I − tA)−1 are determined. As an application, adjacency matrices for graphs are defined with vectors of as entries. For positive odd integer k > 3, k-cycles based at a fixed vertex of a graph are enumerated by considering the appropriate entry of Ak. Moreover, k-cycles in finite graphs are enumerated and expected numbers of k-cycles in random graphs are obtained from the norm of the degree-2k part of tr(1 − tu)−1. Unlike earlier work using commutative subalgebras of , this approach represents a “true” application of Clifford algebras to graph theory. More... »

PAGES

75-92

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00006-007-0063-6

DOI

http://dx.doi.org/10.1007/s00006-007-0063-6

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1006910548


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