Counting pure \(k\)-cycles in sequences of Cayley graphs
type: Item
instance of: scholarly article
title: Counting pure \(k\)-cycles in sequences of Cayley graphs
zbMATH Open document ID: 0844.05050
DOI: 10.1016/0012-365X(94)00295-T
author: Pedro Berrizbeitia
author: Reinaldo E. Giudici
published in: Discrete Mathematics
publication date: 1996-03-25T00:00:00Z
review text: For each positive integer \(n\), let \(G_n\) be a finite abelian group and \(S_n\) a symmetric subset of \(G_n\), i.e. \(S_n= - S_n\). \(\text{Cay}(G_n, S_n)\) is the Cayley graph of \(G_n\) with respect to \(S_n\). In this paper, sequences \(X_n\) of Cayley graphs \(\text{Cay}(G_n, S_n)\) are considered. The sequence \(X_n\) has the multiplicative arithmetic property if for each pair \((m, n)\) of positive integers with \(\text{gcd}(m, n)= 1\) there is a group isomorphism \(\varphi_{m, n}\) from \(G_{nm}\) to \(G_n\times G_m\) such that \(\varphi_{m, n}\) maps \(S_{nm}\) onto \(S_n\times S_m\). For a sequence \(X_n\) of Cayley graphs, \(p_k(n)\) denotes the number of induced \(k\)-cycles of \(X_n\). The main result is the following theorem: If a sequence \(X_n\) has the multiplicative arithmetic property, then the function \(2kp_k\) is a linear combination (with integer coefficients) of multiplicative arithmetic functions.
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 858097
zbMATH Keywords: Cayley graph
zbMATH Keywords: sequences
zbMATH Keywords: multiplicative arithmetic property
zbMATH Keywords: group isomorphism
zbMATH Keywords: \(k\)-cycles
zbMATH Keywords: arithmetic functions
reviewed by: Ulrike Baumann
MaRDI profile type: Publication
cites work: Chromatic Polynomials
cites work: A bibliography on chromatic polynomials
cites work: On chromatic coefficients
cites work: Q5283993
cites work: On the chromatic uniqueness of bipartite graphs
cites work: Coefficients of chromatic polynomials
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description: scientific article; zbMATH DE number 858097