Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Deitmar schemes, graphs and zeta functions   In [19] it was explained how one can naturally associate a Deitmar scheme
(which is a scheme defined over the field with one element, $\mathbb{F}_1$) to
a so-called "loose graph" (which is a generalization of a graph). Several
properties of the Deitmar scheme can be proven easily from the combinatorics of
the (loose) graph, and known realizations of objects over $\mathbb{F}_1$ such
as combinatorial $\mathbb{F}_1$-projective and $\mathbb{F}_1$-affine spaces
exactly depict the loose graph which corresponds to the associated Deitmar
scheme. In this paper, we first modify the construction of loc. cit., and show
that Deitmar schemes which are defined by finite trees (with possible end
points) are "defined over $\mathbb{F}_1$" in Kurokawa's sense; we then derive a
precise formula for the Kurokawa zeta function for such schemes (and so also
for the counting polynomial of all associated $\mathbb{F}_q$-schemes). As a
corollary, we find a zeta function for all such trees which contains
information such as the number of inner points and the spectrum of degrees, and
which is thus very different than Ihara's zeta function (which is trivial in
this case). Using a process called "surgery," we show that one can determine
the zeta function of a general loose graph and its associated { Deitmar,
Grothendieck }-schemes in a number of steps, eventually reducing the
calculation essentially to trees. We study a number of classes of examples of
loose graphs, and introduce the Grothendieck ring of $\mathbb{F}_1$-schemes
along the way in order to perform the calculations. Finally, we compare the new
zeta function to Ihara's zeta function for graphs in a number of examples, and
include a computer program for performing more tedious calculations.

19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th