TY: UNPB
T1 - Spectral properties of the n-Queens' graphs
A1 - Cardoso, Domingos M.
A1 - Costa, Inês Serôdio
A1 - Duarte, Rui
N2 - The n-Queens? graph, Q(n), is the graph associated to the n×n chessboard (a generalization of the classical 8×8 chessboard), with n
2
vertices,
each one corresponding to a square of the chessboard. Two vertices of
Q(n) are adjacent if and only if they are in the same row, in the same
column or in the same diagonal of the chessboard. After a short overview
on the main combinatorial properties of Q(n), its spectral properties are
investigated. First, a lower bound on the least eigenvalue of an arbitrary
graph is obtained using clique edge partitions and a sufficient condition for
this lower bound be attained is deduced. For the particular case of Q(n),
we prove that for every n, its least eigenvalue is not less than ?4 and it
is equal to ?4 with multiplicity (n ? 3)2
, for every n ? 4. Furthermore,
n ? 4 is also an eigenvalue of Q(n), with multiplicity at least n?2
2 when n
is even and at least n+1
2 when n is odd. A conjecture about the integer
eigenvalues of Q(n) is presented. We finish this article with an algorithm
to determine an equitable partition of the n-Queens? graph, Q(n), for
n ? 3, concluding that such equitable partition has (?n/2?+1)?n/2?
2
cells.
UR - https://ria.ua.pt/handle/10773/30086
Y1 - 2020
PB - arXiv