We introduce and investigate different definitions of effective amenability, in terms of computability of Følner sets, Reiter functions, and Følner functions. As a consequence, we prove that recursively presented amenable groups have subrecursive Følner function, answering a question of Gromov; for the same class of groups we prove that solvability of the Equality Problem on a generic set (generic EP) is equivalent to solvability of the Word Problem on the whole group (WP), thus providing the first examples of finitely presented groups with unsolvable generic EP. In particular, we prove that for finitely presented groups, solvability of generic WP doesn't imply solvability of generic EP.
A well-known fact in Spectral Graph Theory is the existence of pairs of isospectral nonisomorphic graphs (known as PINGS). The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of isospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay-type routines developed for graphs, whose adjacency matrices are (0, 1)-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of −1's. We show that, with suitable adaption, such routines can be successfully ported to signed graphs, and we can built pairs of cospectral switching nonisomorphic signed graphs. (2010): 05C22, 05C50
Mathematics Subject Classification
ABSTRACT. We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich groups, finitely presented solvable groups with unsolvable Word Problem, have computable Følner sets. We also prove computability of Følner sets for extensions -with subrecursive distortion functions-of amenable groups with solvable Word Problem by finitely generated groups with computable Følner sets . Moreover we obtain some known and some new upper bounds for the Følner function for these particular extensions.
The wreath product of graphs is a graph composition inspired by the notion of wreath product of groups, with interesting connections with Geometric Group Theory and Probability. This paper is devoted to the description of some degree and distancebased invariants, of large interest in Chemical Graph Theory, for a wreath product of graphs. An explicit formula is obtained for the Zagreb indices, in terms of the Zagreb indices of the factor graphs. A detailed analysis of distances in a wreath product is performed, allowing to describe the antipodal graph and to provide a formula for the Wiener index. Finally, a formula for the Szeged index is obtained. Several explicit examples are given. (2010): 05C07, 05C12, 05C40, 05C76.
Mathematics Subject Classification
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