Constructing segments of quadratic length in Spec(T_n) through segments of linear length

2024/03/30 by Artem Kravchuk, Kravchuk, Artem · 1 citation
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2404.00410

Abstract

A Transposition graph T_n is defined as a Cayley graph over the symmetric group Sym_n generated by all transpositions. It is known that the spectrum of T_n consists of integers, but it is not known exactly how these numbers are distributed. In this paper we prove that integers from the segment [-n, n] lie in the spectrum of T_n for any n\geqslant 31. Using this fact we also prove the main result of this paper that a segment of quadratic length with respect to n lies in the spectrum of T_n.

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• Proof of a conjecture on eigenvalues of transposition graph

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• Spectrum of Cayley graphs on the symmetric group generated by transpositions
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