Given a vertex-labeled tree on n vertices we show how to obtain a straight-line, crossings-free drawing of it on a set of n labeled concentric tracks, such that the vertex labels match the track labels. The tracks can be defined by conic sections (such as circles, ellipses, circular arcs) or other smooth convex curves. We show that this type of embedding can be used to simultaneously embed tree...

We define a latent structure random graph as dot product (RDPG) in which the position distribution incorporates both probabilistic and geometric constraints, delineated by family of underlying distributions on some fixed Euclidean space, structural support submanifold from are drawn positions for graph. For one-dimensional model with known support, we extend existing results consistency spectra...

A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termed sign-balanced if it equivalent to an unsigned under operation of sign switching; otherwise called sign-unbalanced. nut a one dimensional kernel with corresponding eigenvector that full. In this paper we generalise notion graphs graphs. Orders for which regular exist were determined recently degrees up $11$. By extendi...

A plane graph is a planar graph with a fixed embedding. In a no-bend orthogonal drawing of a plane graph, each vertex is drawn as a point and each edge is drawn as a single horizontal or vertical line segment. A planar graph is said to have a no-bend orthogonal drawing if at least one of its plane embeddings has a no-bend orthogonal drawing. In this paper we consider a class of planar graphs, c...

The partial representation extension problem is a recently introduced generalization of the recognition problem. A circle graph is an intersection graph of chords of a circle. We study the partial representation extension problem for circle graphs, where the input consists of a graph G and a partial representation R′ giving some pre-drawn chords that represent an induced subgraph of G. The ques...

Let Col n denote the set of all `-colored graphs on n vertices and let G be a graph drawn uniformly at random from this set. Provided that `→∞, we prove that all color classes of G deviate roughly at most √ 2 log ` from ⌊ n ` ⌋ , and there is at least one color class with such deviation. Furthermore, we prove similar results for the class Col n,M , which contains all `-colored graphs with preci...

We study drawings of graphs on the torus with crossings allowed. A question posed in [4], specialized to the case of the torus, asks, whether for every disconnected graph there is a drawing in the torus with the minimal number of crossings, such that one of the graphs is drawn in a planar disc. We reduce the problem to an interesting question from the geometry of numbers and solve a special case.

Message passing algorithms are popular in many combinatorial optimization problems. Forexample, experimental results show that survey propagation (a certain message passing algorithm)is effective in finding proper k-colorings of random graphs in the near-threshold regime. Gallagerin 1962 introduced the concept of Low Density Parity Check (LDPC) codes, and suggested asimple d...