Page 295 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 295
PHS AND GROUPS, GEOMETRIES AND GAP - G2G2 (MS-7)
multiplicity function of Sn for eigenvalues ±(n − k) where 1 k n+1 . The main result is
2
given by the following theorem.
Theorem 1. Let n, k ∈ Z, n 2 and 1 k n+1 , then the multiplicity mul(n − k) of
2
eigenvalue (n − k) of the Star graph Sn is calculated by the formula:
mul(n − k) = n2(k−1) + P (n),
(k − 1)!
where P (n) is a polynomial of degree 2k − 3.
In 2019, E. N. Khomyakova and E. V. Konstantinova [5] presented explicit formulas for cal-
culating multiplicities of eigenvalues ±(n − k) where 2 k 12 and firstly collected
computational results of all eigenvalue multiplicities for n 50 in a catalogue (https:
//link.springer.com/article/10.1007/s40065-019-00271-z). This exact
values show that Theorem 1 holds for any n 2 and 1 k n. Authors used computa-
tional results to get diagrams with plotting them on a logarithmic scale with base 2 such that
the abscissa corresponds to the eigenvalues of the Star graphs Sn for a fixed n and the ordinate
corresponds to the multiplicities [4]. In case k is fixed, diagram looks like a polynomial func-
tion by Theorem 1. In case n is fixed, diagram in normal scale contains exponential rises and
falls appear. Thus the function may be a straight exponent for sufficiently large n, but it is just
conjecture.
Acknowledgments. The work has been supported by RFBR Grant 18-501-51021.
References
[1] S. B. Akers, B. Krishnamurthy, A group–theoretic model for symmetric interconnection
networks. IEEE Trans. Comput. 38 (4) (1989) 555–566.
[2] S. V. Avgustinovich, E. N. Khomyakova, E. V. Konstantinova, Multiplicities of eigenval-
ues of the Star graph. Siberian Electronic Mathematical Report 13 (2016) 1258–1270.
[3] G. Chapuy, V. Feray, A note on a Cayley graph of Symn. arXiv:1202.4976v2 (2012) 1–3.
[4] E. N. Khomyakova, E. V. Konstantinova, Note on exact values of multiplicities of eigen-
values of the Star graph. Siberian Electronic Mathematical Report 12 (2015), 92–100.
[5] E. N. Khomyakova, E. V. Konstantinova, Catalogue of the Star graph eigenvalue multi-
plicities. Arab. J. Math. (2019) DOI 10.1007/s40065-019-00271-z.
[6] E. N. Khomyakova, On the eigenvalues multiplicity function of the Star graph. Siberian
Electronic Mathematical Report 15 (2018) 1416–1425.
[7] R. Krakovski, B. Mohar, Spectrum of Cayley graphs on the symmetric group generated
by transposition. Linear Algebra and its Applications 437 (2012) 1033–1039.
293
multiplicity function of Sn for eigenvalues ±(n − k) where 1 k n+1 . The main result is
2
given by the following theorem.
Theorem 1. Let n, k ∈ Z, n 2 and 1 k n+1 , then the multiplicity mul(n − k) of
2
eigenvalue (n − k) of the Star graph Sn is calculated by the formula:
mul(n − k) = n2(k−1) + P (n),
(k − 1)!
where P (n) is a polynomial of degree 2k − 3.
In 2019, E. N. Khomyakova and E. V. Konstantinova [5] presented explicit formulas for cal-
culating multiplicities of eigenvalues ±(n − k) where 2 k 12 and firstly collected
computational results of all eigenvalue multiplicities for n 50 in a catalogue (https:
//link.springer.com/article/10.1007/s40065-019-00271-z). This exact
values show that Theorem 1 holds for any n 2 and 1 k n. Authors used computa-
tional results to get diagrams with plotting them on a logarithmic scale with base 2 such that
the abscissa corresponds to the eigenvalues of the Star graphs Sn for a fixed n and the ordinate
corresponds to the multiplicities [4]. In case k is fixed, diagram looks like a polynomial func-
tion by Theorem 1. In case n is fixed, diagram in normal scale contains exponential rises and
falls appear. Thus the function may be a straight exponent for sufficiently large n, but it is just
conjecture.
Acknowledgments. The work has been supported by RFBR Grant 18-501-51021.
References
[1] S. B. Akers, B. Krishnamurthy, A group–theoretic model for symmetric interconnection
networks. IEEE Trans. Comput. 38 (4) (1989) 555–566.
[2] S. V. Avgustinovich, E. N. Khomyakova, E. V. Konstantinova, Multiplicities of eigenval-
ues of the Star graph. Siberian Electronic Mathematical Report 13 (2016) 1258–1270.
[3] G. Chapuy, V. Feray, A note on a Cayley graph of Symn. arXiv:1202.4976v2 (2012) 1–3.
[4] E. N. Khomyakova, E. V. Konstantinova, Note on exact values of multiplicities of eigen-
values of the Star graph. Siberian Electronic Mathematical Report 12 (2015), 92–100.
[5] E. N. Khomyakova, E. V. Konstantinova, Catalogue of the Star graph eigenvalue multi-
plicities. Arab. J. Math. (2019) DOI 10.1007/s40065-019-00271-z.
[6] E. N. Khomyakova, On the eigenvalues multiplicity function of the Star graph. Siberian
Electronic Mathematical Report 15 (2018) 1416–1425.
[7] R. Krakovski, B. Mohar, Spectrum of Cayley graphs on the symmetric group generated
by transposition. Linear Algebra and its Applications 437 (2012) 1033–1039.
293
