Related papers: Graphs, spectral triples and Dirac zeta functions

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and…

In this paper we study spectral zeta functions associated to finite and infinite graphs. First we establish a meromorphic continuation of these functions under some general conditions. Then we study special values in the case of standard…

Conjecturally, almost all graphs are determined by their spectra. This problem has also been studied for variants such as the spectra of the Laplacian and signless Laplacian. Here we consider the problem of determining graphs with Ihara and…

We construct spectral zeta functions for the Dirac operator on metric graphs. We start with the case of a rose graph, a graph with a single vertex where every edge is a loop. The technique is then developed to cover any finite graph with…

A Dirac operator is presented that will yield a 1+ summable regular even spectral triple for all noncommutative compact surfaces defined as subalgebras of the Toeplitz algebra. Connes' conditions for noncommutative spin geometries are…

In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalize the Perron-Frobenius-Ruelle theorem and obtain a polynomial decay of the operator,…

A fundamental tool in noncommutative geometry is Connes' character formula. This formula is used in an essential way in the applications of noncommutative geometry to index theory and to the spectral characterisation of manifolds. A…

As an outgrowth of our investigation of non-regular spaces within the context of quantum gravity and non-commutative geometry, we develop a graph Hilbert space framework on arbitrary (infinite) graphs and use it to study spectral properties…

We compute the quadratic embedding constant for complete bipartite graphs with disjoint edges removed. Moreover, we study the quadratic embedding property for theta graphs, i.e., graphs consisting of three paths with common initial points…

We introduce a numerical invariant $\zeta(\Sigma)$ measuring the end-complexity of $\Sigma$ and use it to organize coarse-geometric features of Map($\Sigma$). Our main tool is the \emph{non-peripheral curve graph} $C_{\rm np}(\Sigma)$,…

We define the zeta function of a noncommutative K3 surface over a finite field, an invariant under Fourier-Mukai equivalence that can be used to define point counts in this noncommutative setting. These point counts can be negative, and can…

We initiate the study of spectral zeta functions $\zeta_{X}$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions.…

Spectral triples and quantum statistical mechanical systems are two important constructions in noncommutative geometry. In particular, both lead to interesting reconstruction theorems for a broad range of geometric objects, including number…

In this paper, we explore the properties of zeta functions associated with infinite graphs of groups that arise as quotients of cuspidal tree-lattices, including all non-uniform arithmetic quotients of the tree of rank one Lie groups over…

Let $k\ge 3$ be an integer, $q$ be a prime power, and $\mathbb{F}_q$ denote the field of $q$ elements. Let $f_i, g_i\in\mathbb{F}_q[X]$, $3\le i\le k$, such that $g_i(-X) = -\, g_i(X)$. We define a graph $S(k,q) =…

Stark and Terras introduced the edge zeta function of a finite graph in 1996. The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time. We look at graph…

In previous papers, we constructed smooth (1,\infty)-summable semfinite spectral triples for graph algebras with a faithful trace, and (k,\infty)-summable semifinite spectral triples for k-graph algebras. In this paper we identify classes…

Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities:…

Continuing previous work we develop a certain piece of functional analysis on general graphs and use it to create what Connes calls a 'spectral triple', i.e. a Hilbert space structure, a representation of a certain (function) algebra and a…

We introduce a ``non-orientable'' variation of Serre's definition of a graph, which we call an abstract isogeny graph. These objects capture the combinatorics of the graphs $G(p,\ell,H)$, the $\ell$-isogeny graphs of supersingular elliptic…