Laplace and Dirac operators on graphs

Dirac walks on regular trees

“…Then, in section 2.2.1, we will review necessary aspects of Fermions in R n to describe a Fermionic action on graphs and discuss the difficulties of defining a Dirac operator on general graphs. We will attempt to resolve this problem by defining a particular Dirac operator that has a combinatorial interpretation in terms of a particular kind of signed walk, which we will call the 'Dirac walk' in the section 2.2.2, as suggested by [28]. In the same section 2.2.2, we will illustrate the connection between the Dirac walk and the two-point function of a free Fermionic action.…”

“…Taking motivation from the Kähler-Dirac operator in the continuum, we define a graph Dirac operator [25,28] as a (|V| + |E|) squared matrix ( D :…”

“…A Dirac walk of one step ω τ =1 moves from a vertex to an incident edge or an edge to an incident vertex. We will define the sign of the walk ω τ =1 as introduced in [28]: if the edge involved in the walk is directed out of the vertex involved, then sgn(ω τ =1 ) = −1, and if the edge enters the vertex involved, then sgn(ω τ =1 ) = 1 (see the figure 1). The sign of a walk is defined as the product of all the signs of the individual steps taken.…”

“…Casiday et al [28] proposed an expansion of powers of a formal Dirac operator (i.e. square root of the graph Laplacian), whose quadratic form matches the Kähler-Dirac action on trees, in terms of superwalks and vertex-edge walks.…”

“…We plan our discussions as follows: We will start in section 2 by describing a framework to study quantum fields on general graphs. The propagation will be given by the square root of the Laplacian, in the formulation proposed by [28]. In the next section 3, specializing to the Bethe lattice, we will compute the generating function of weighted walks from vertices or edges.…”