Recently, Smilansky expressed the determinant of the bond scattering matrix of a graph by means of the determinant of its Laplacian. We present another proof for this Smilansky’s formula by using some weighted zeta function of a graph. Furthermore, we reprove a weighted version of Smilansky’s formula by Bass’ method used in the determinant expression for the Ihara zeta function of a graph.

This paper describes a novel procedure based on Core Path Graphs to generate continuously differentiable sub-optimal paths for wheeled mobile robots in the presence of obstacles. The operational scenario is first discretized with a finite dimensional grid of positions-directions pairs. A weighted and oriented graph is then defined whose nodes are the above mentioned grid points, and whose arcs ...

The main contributions of this paper are three-fold. First, we use a dynamic approach based on Reiter’s pioneering work on Karp-Miller computation graphs [19] to give a new and short proof of Mohar’s Minty-type Theorem [15]. Second, we bridge circular colorings and discrete event dynamic systems to show that the Barbosa and Gafni’s results on circular chromatic number [5, 21] can be generalized...

Given a vertex-weighted oriented graph, we can associate to it set of monomials. We consider the toric ideal whose defining map is given by these find generating for certain classes graphs which depends on combinatorial structure and weights graph. provide result analogous unweighted, unoriented graph case, show that when associated simple has only trivial even closed walks, zero ideal. Moreove...