Arabian Journal of Mathematics Pub Date : 2019-11-19 , DOI: 10.1007/s40065-019-00271-z Ekaterina Khomyakova , Elena V. Konstantinova
The Star graph \(S_n\), \(n\geqslant 2\), is the Cayley graph over the symmetric group \(\mathrm {Sym}_n\) generated by transpositions \((1~i),\,2\leqslant i \leqslant n\). This set of transpositions plays an important role in the representation theory of the symmetric group. The spectrum of \(S_n\) contains all integers from \(-(n-1)\) to \(n-1\), and also zero for \(n\geqslant 4\). In this paper we observe methods for getting explicit formulas of eigenvalue multiplicities in the Star graphs \(S_n\), present such formulas for the eigenvalues \(\pm (n-k)\), where \(2\leqslant k \leqslant 12\), and finally collect computational results of all eigenvalue multiplicities for \(n\leqslant 50\) in the catalogue.
中文翻译:
星图特征值多重性的目录
星形图\(S_n \),\(n \ geqslant 2 \)是由换位\((1〜i ),\,2生成的对称组\(\ mathrm {Sym} _n \)上的Cayley图\ leqslant我\ leqslant n \)。这组换位在对称组的表示理论中起着重要作用。\(S_n \)的频谱包含从\(-(n-1)\)到\(n-1 \)的所有整数,并且对于\(n \ geqslant 4 \)也为零。在本文中,我们观察了在星图\(S_n \)中获得特征值多重性的明确公式的方法,并给出了特征值\(\ pm(nk)\)的此类公式,其中\(2 \ leqslant k \ leqslant 12 \),最后在目录中收集\(n \ leqslant 50 \)的所有特征值多重性的计算结果。