The Metric Dimension of Graph with Pendant Edges

For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1), d(v,w_2),..., d(v,w_k)) where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every two vertices of G have distinct representations. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by dim(G), is the number of vertices in a basis of G. In this paper, we determine the dimensions of some corona graphs G⊙K_1, and G⊙K_m for any graph G and m ≥ 2, and a graph with pendant edges more general than corona graphs G⊙K_m

Eulerian digraphs and toric Calabi-Yau varieties

We investigate the structure of a simple class of affine toric Calabi-Yau varieties that are defined from quiver representations based on finite eulerian directed graphs (digraphs). The vanishing first Chern class of these varieties just follows from the characterisation of eulerian digraphs as being connected with all vertices balanced. Some structure theory is used to show how any eulerian digraph can be generated by iterating combinations of just a few canonical graph-theoretic moves. We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples. We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.Comment: 27 pages, 8 figure

The multidimensional Manhattan networks

The $n$-dimensional Manhattan network $M_n$---a special case of $n$-regular digraph---is formally defined and some of its structural properties are studied. In particular, it is shown that $M_n$ is a Cayley digraph, which can be seen as a subgroup of the $n$-dim version of the wallpaper group $pgg$. These results induce a useful new presentation of $M_n$, which can be applied to design a (shortest-path) local routing algorithm and to study some other metric properties. Also it is shown that the $n$-dim Manhattan networks are Hamiltonian and, in the standard case (that is, dimension two), they can be decomposed in two arc-disjoint Hamiltonian cycles. Finally, some results on the connectivity and distance-related parameters of $M_n$, such as the distribution of the node  distances and the diameter are presented