Multi-edge trees as introduced in a recent paper of Dziemiańczuk are plane trees where multiple edges are allowed. We first show that d-ary multi-edge trees where the out-degrees are bounded by d are in bijection with classical d-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. Th...

A trivalent diagram is a connected, two-colored bipartite graph (parallel edges allowed but not loops) such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2, with a cyclic order imposed on every set of edges incident to to a same vertex. A rooted trivalent diagram is a trivalent diagram with a distinguished edge, its root. We shall describe and analyze an a...

In this paper, we begin by partitioning the edge (or arc) set of a circulant (di)graph according to which generator in the connection set leads to each edge. We then further refine the partition by subdividing any part that corresponds to an element of order less than n, according to which of the cycles generated by that element the edge is in. It is known that if the (di)graph is connected and...

Parsers that parametrize over wider scopes are generally more accurate than edge-factored models. For graph-based non-projective parsers, wider factorizations have so far implied large increases in the computational complexity of the parsing problem. This paper introduces a “crossing-sensitive” generalization of a third-order factorization that trades off complexity in the model structure (i.e....

Existing work on visualizing multivariate graphs is primarily concerned with representing the attributes of nodes. Even though edges are constitutive elements networks, there have been only few attempts to visualize edges. In this work, we focus critical importance edge for interpreting network visualizations and building trust in underlying data. We propose ‘unfolding edges’ as an interactive ...

the topological indices, defined as the sum of contributions of all pairs of vertices (among which are the wiener, harary, hyper–wiener indices, degree distance, and many others), are expressed in terms of contributions of edges and pairs of edges.

We shall consider graphs (hypergraphs) without loops and multiple edges . Let Ybe a family of so called prohibited graphs and ex (n, Y) denote the maximum number of edges (hyperedges) a graph (hypergraph) on n vertices can have without containing subgraphs from Y A graph (hypergraph) will be called supersaturated if it has more edges than ex (n, Y). If G has n vertices and ex (n, Y)=k edges (hy...

Let G and H be graphs. As usual, we say that H is a minor or subcontraction of G if V(G) contains disjoint subsets Wu , u # V(H), such that G[Wu] is connected for each u # V(H) and there is an edge in G between Wu and Wv whenever uv # E(H). (Here, G[Wu] stands for the subgraph of G induced by Wu ; our notation is standard and follows that of Bolloba s [2].) We write GoH if H is a minor of G, an...