593 research outputs found
Spectrally degenerate graphs: Hereditary case
It is well known that the spectral radius of a tree whose maximum degree is D
cannot exceed 2sqrt{D-1}. Similar upper bound holds for arbitrary planar
graphs, whose spectral radius cannot exceed sqrt{8D}+10, and more generally,
for all d-degenerate graphs, where the corresponding upper bound is sqrt{4dD}.
Following this, we say that a graph G is spectrally d-degenerate if every
subgraph H of G has spectral radius at most sqrt{d.Delta(H)}. In this paper we
derive a rough converse of the above-mentioned results by proving that each
spectrally d-degenerate graph G contains a vertex whose degree is at most
4dlog_2(D/d) (if D>=2d). It is shown that the dependence on D in this upper
bound cannot be eliminated, as long as the dependence on d is subexponential.
It is also proved that the problem of deciding if a graph is spectrally
d-degenerate is co-NP-complete.Comment: Updated after reviewer comments. 14 pages, no figure
Growth Series and Random Walks on Some Hyperbolic Graphs
Consider the tesselation of the hyperbolic plane by m-gons, l per vertex. In
its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of
geodesics with common extremities. We also introduce and enumerate "holly
trees", a family of reduced loops in these graphs. We then apply Grigorchuk's
result relating cogrowth and random walks to obtain lower estimates on the
spectral radius of the Markov operator associated with a symmetric random walk
on these graphs.Comment: 21 pages. to appear in monash. mat
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
The -invariant massive Laplacian on isoradial graphs
We introduce a one-parameter family of massive Laplacian operators
defined on isoradial graphs, involving elliptic
functions. We prove an explicit formula for the inverse of , the
massive Green function, which has the remarkable property of only depending on
the local geometry of the graph, and compute its asymptotics. We study the
corresponding statistical mechanics model of random rooted spanning forests. We
prove an explicit local formula for an infinite volume Boltzmann measure, and
for the free energy of the model. We show that the model undergoes a second
order phase transition at , thus proving that spanning trees corresponding
to the Laplacian introduced by Kenyon are critical. We prove that the massive
Laplacian operators provide a one-parameter
family of -invariant rooted spanning forest models. When the isoradial graph
is moreover -periodic, we consider the spectral curve of the
characteristic polynomial of the massive Laplacian. We provide an explicit
parametrization of the curve and prove that it is Harnack and has genus . We
further show that every Harnack curve of genus with
symmetry arises from such a massive
Laplacian.Comment: 71 pages, 13 figures, to appear in Inventiones mathematica
Phase Transitions on Nonamenable Graphs
We survey known results about phase transitions in various models of
statistical physics when the underlying space is a nonamenable graph. Most
attention is devoted to transitive graphs and trees
Martin boundary of killed random walks on isoradial graphs
Avec un appendice d'Alin BostanWe consider killed planar random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and are spatially non-homogeneous. Despite these crucial differences, we compute the asymptotics of the Martin kernel, deduce the Martin boundary and show that it is minimal. Similar results on the grid are derived in a celebrated work of Ney and Spitzer
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