To a simple graph $G=(V,E)$, we correspond a simple graph $G_{triangle,square}$ whose vertex set is ${{x,y}: x,yin V}$ and two vertices ${x,y},{z,w}in G_{triangle,square}$ are adjacent if and only if ${x,z},{x,w},{y,z},{y,w}in Vcup E$. The graph $G_{triangle,square}$ is called the $(triangle,square)$-edge graph of the graph $G$. In this paper, our ultimate goal is to provide a link between the ...

During the last decade, a lot of progress has been made in the enumerative branch of topological graph theory. Enumeration formulas were developed for a large class of graph covering projections. The purpose of this paper is to count graph covering projections of graphs such that the corresponding covering space is a connected graph. The main tool of the enumeration is Pólya’s theorem.

The main thrust of my research is the use of techniques from information theory, probability and combinatorics to study structural, enumerative and algorithmic aspects of graph homomorphisms and related models, both in particular instances (independent sets and colorings, for example) and in general. Graph homomorphisms are important objects in graph theory, where they generalize a number of ce...

A solid object in 3-dimensional space may be described by a collection of all its topologically distinct 2-dimensional appearances, its aspect graph. In this paper, we study the complexity of aspect graphs of piecewise-smooth algebraic objects using the modern tools of algebraic geometry, i.e. intersection theory, multiple-point theory and enumerative geometry. We give a bound on the number of ...

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniq...

We present exact results for the delta peak at zero energy in the spectral density of the random graph incidence matrix model as a function of the average connectivity. We give an analytic expression for the height of this peak, and a detailed description of the localized eigenvectors. Their total contribution to the peak is given. This allows to study analytically a delocalization and a reloca...