2,574 research outputs found
The spectra of generalized Paley graphs of powers and applications
We consider a special class of generalized Paley graphs over finite fields,
namely the Cayley graphs with vertex set and connection set
the nonzero -th powers in , as well as their
complements. We explicitly compute the spectrum of these graphs. As a
consequence, the graphs turn out to be (with trivial exceptions) simple,
connected, non-bipartite, integral and strongly regular (of Latin square type
in half of the cases). By using the spectral information we compute several
invariants of these graphs. We exhibit infinite families of pairs of
equienergetic non-isospectral graphs. As applications, on the one hand we solve
Waring's problem over for the exponents , for each
and for infinite values of and . We obtain that the Waring's
number or , depending on and , thus tackling
some open cases. On the other hand, we construct infinite towers of Ramanujan
graphs in all characteristics.Comment: 27 pages, 3 tables. A little modification of the title. Corollary 4.8
removed. Added Section 6 on "Energy". Minor typos corrected. Ihara zeta
functions at the end correcte
New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix
The purpose of this article is to improve existing lower bounds on the
chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency
matrix sorted in non-increasing order.
First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / -
sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower
bound which only involves the maximum and minimum eigenvalues, i.e., the case
. We provide several examples for which the new bound exceeds the {\sc
Hoffman} lower bound.
Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^-
are the sums of the squares of positive and negative eigenvalues, respectively.
To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We
show that the conjectured lower bound is tight for several families of graphs.
We also performed various searches for a counter-example, but none was found.
Our proofs rely on a new technique of converting the adjacency matrix into
the zero matrix by conjugating with unitary matrices and use majorization of
spectra of self-adjoint matrices.
We also show that the above bounds are actually lower bounds on the
normalized orthogonal rank of a graph, which is always less than or equal to
the chromatic number. The normalized orthogonal rank is the minimum dimension
making it possible to assign vectors with entries of modulus one to the
vertices such that two such vectors are orthogonal if the corresponding
vertices are connected.
All these bounds are also valid when we replace the adjacency matrix A by W *
A where W is an arbitrary self-adjoint matrix and * denotes the Schur product,
that is, entrywise product of W and A
Partitions of graphs into small and large sets
Let be a graph on vertices. We call a subset of the vertex set
\emph{-small} if, for every vertex , . A subset is called \emph{-large} if, for every vertex
, . Moreover, we denote by the
minimum integer such that there is a partition of into -small
sets, and by the minimum integer such that there is a
partition of into -large sets. In this paper, we will show tight
connections between -small sets, respectively -large sets, and the
-independence number, the clique number and the chromatic number of a graph.
We shall develop greedy algorithms to compute in linear time both
and and prove various sharp inequalities
concerning these parameters, which we will use to obtain refinements of the
Caro-Wei Theorem, the Tur\'an Theorem and the Hansen-Zheng Theorem among other
things.Comment: 21 page
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