33,358 research outputs found
The Relativized Second Eigenvalue Conjecture of Alon
We prove a relativization of the Alon Second Eigenvalue Conjecture for all
-regular base graphs, , with : for any , we show that
a random covering map of degree to has a new eigenvalue greater than
in absolute value with probability .
Furthermore, if is a Ramanujan graph, we show that this probability is
proportional to , where
is an integer depending on , which can be computed by a finite algorithm for
any fixed . For any -regular graph, , is
greater than .
Our proof introduces a number of ideas that simplify and strengthen the
methods of Friedman's proof of the original conjecture of Alon. The most
significant new idea is that of a ``certified trace,'' which is not only
greatly simplifies our trace methods, but is the reason we can obtain the
estimate above. This estimate represents an
improvement over Friedman's results of the original Alon conjecture for random
-regular graphs, for certain values of
List homomorphism problems for signed graphs
We consider homomorphisms of signed graphs from a computational perspective.
In particular, we study the list homomorphism problem seeking a homomorphism of
an input signed graph , equipped with lists , of allowed images, to a fixed target signed graph . The
complexity of the similar homomorphism problem without lists (corresponding to
all lists being ) has been previously classified by Brewster and
Siggers, but the list version remains open and appears difficult. We illustrate
this difficulty by classifying the complexity of the problem when is a tree
(with possible loops). The tools we develop will be useful for classifications
of other classes of signed graphs, and we illustrate this by classifying the
complexity of irreflexive signed graphs in which the unicoloured edges form
some simple structures, namely paths or cycles. The structure of the signed
graphs in the polynomial cases is interesting, suggesting they may constitute a
nice class of signed graphs analogous to the so-called bi-arc graphs (which
characterize the polynomial cases of list homomorphisms to unsigned graphs).Comment: various changes + rewritten section on path- and cycle-separable
graphs based on a new conference submission (split possible in future
Totally frustrated states in the chromatic theory of gain graphs
We generalize proper coloring of gain graphs to totally frustrated states,
where each vertex takes a value in a set of `qualities' or `spins' that is
permuted by the gain group. (An example is the Potts model.) The number of
totally frustrated states satisfies the usual deletion-contraction law but is
matroidal only for standard coloring, where the group action is trivial or
nearly regular. One can generalize chromatic polynomials by constructing spin
sets with repeated transitive components.Comment: 14 pages, 2 figure
An elementary chromatic reduction for gain graphs and special hyperplane arrangements
A gain graph is a graph whose edges are labelled invertibly by "gains" from a
group. "Switching" is a transformation of gain graphs that generalizes
conjugation in a group. A "weak chromatic function" of gain graphs with gains
in a fixed group satisfies three laws: deletion-contraction for links with
neutral gain, invariance under switching, and nullity on graphs with a neutral
loop. The laws lead to the "weak chromatic group" of gain graphs, which is the
universal domain for weak chromatic functions. We find expressions, valid in
that group, for a gain graph in terms of minors without neutral-gain edges, or
with added complete neutral-gain subgraphs, that generalize the expression of
an ordinary chromatic polynomial in terms of monomials or falling factorials.
These expressions imply relations for chromatic functions of gain graphs.
We apply our relations to some special integral gain graphs including those
that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining
new evaluations of and new ways to calculate the zero-free chromatic polynomial
and the integral and modular chromatic functions of these gain graphs, hence
the characteristic polynomials and hypercubical lattice-point counting
functions of the arrangements. We also calculate the total chromatic polynomial
of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page
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