384 research outputs found
Spectral problem on graphs and L-functions
The scattering process on multiloop infinite p+1-valent graphs (generalized
trees) is studied. These graphs are discrete spaces being quotients of the
uniform tree over free acting discrete subgroups of the projective group
. As the homogeneous spaces, they are, in fact, identical to
p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the
finite subgraph-the reduced graph containing all loops of the generalized tree.
We study the spectral problem on these graphs, for which we introduce the
notion of spherical functions-eigenfunctions of a discrete Laplace operator
acting on the graph. We define the S-matrix and prove its unitarity. We present
a proof of the Hashimoto-Bass theorem expressing L-function of any finite
(reduced) graph via determinant of a local operator acting on this
graph and relate the S-matrix determinant to this L-function thus obtaining the
analogue of the Selberg trace formula. The discrete spectrum points are also
determined and classified by the L-function. Numerous examples of L-function
calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur
The enumeration of planar graphs via Wick's theorem
A seminal technique of theoretical physics called Wick's theorem interprets
the Gaussian matrix integral of the products of the trace of powers of
Hermitian matrices as the number of labelled maps with a given degree sequence,
sorted by their Euler characteristics. This leads to the map enumeration
results analogous to those obtained by combinatorial methods. In this paper we
show that the enumeration of the graphs embeddable on a given 2-dimensional
surface (a main research topic of contemporary enumerative combinatorics) can
also be formulated as the Gaussian matrix integral of an ice-type partition
function. Some of the most puzzling conjectures of discrete mathematics are
related to the notion of the cycle double cover. We express the number of the
graphs with a fixed directed cycle double cover as the Gaussian matrix integral
of an Ihara-Selberg-type function.Comment: 23 pages, 2 figure
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