# Spanning tree generating functions and Mahler measures

@article{Guttmann2012SpanningTG, title={Spanning tree generating functions and Mahler measures}, author={Anthony J. Guttmann and Mathew Rogers}, journal={arXiv: Mathematical Physics}, year={2012} }

We define the notion of a spanning tree generating function (STGF) $\sum a_n z^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form…

## 17 Citations

Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2021

A spanning tree generating function T(z) for infinite periodic vertex-transitive (vt) lattices L vt has been proposed by Guttmann and Rogers (2012 J. Phys. A: Math. Theor. 45 494001). Their spanning…

On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

- Mathematics
- 2018

Let $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ be the generating function for the number $\tau(n)$ of spanning trees in the circulant graphs $C_{n}(s_1,s_2,\ldots,s_k).$ We show that $F(x)$ is a…

The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic

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The number of rooted forests in circulant graphs

- MathematicsArs Mathematica Contemporanea
- 2022

In this paper, we develop a new method to produce explicit formulas for the number $f_{G}(n)$ of rooted spanning forests in the circulant graphs $ G=C_{n}(s_1,s_2,\ldots,s_k)$ and $…

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

- Mathematics
- 2012

We show that the n-fold integrals χ(n) of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the ‘Ising class’, or n-fold integrals from enumerative…

Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure

- Mathematics
- 2014

The space C of conservative vertex colorings (over a field F) of a countable, locally finite graph G is introduced. The subspace of based colorings is shown to be isomorphic to the bicycle space of…

Mahler/Zeta Correspondence

- Mathematics
- 2022

The Mahler measure was introduced by Mahler in the study of number theory. It is known that the Mahler measure appears in different areas of mathematics and physics. On the other hand, we have been…

Complexity of discrete Seifert foliations over a graph

- MathematicsДоклады Академии наук
- 2019

In the present paper, we study the complexity of an infinite family of graphs Hn = Hn(G1, G2, ..., Gm) that are discrete Seifert foliations over a graph H on m vertices with fibers G1, G2, ..., Gm.…

The hypergeometric series for the partition function of the 2D Ising model

- Mathematics
- 2015

In 1944 Onsager published the formula for the partition function of the Ising model for the infinite square lattice. He was able to express the internal energy in terms of a special function, but he…

Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs

- Mathematics
- 2018

Abstract—We study analytical and arithmetical properties of the complexity function for infinite families of circulant Cn(s1, s2,…, sk) C2n(s1, s2,…, sk, n). Exact analytical formulas for the…

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