A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a nite number of isolated vertices to it will always yield a sum graph, and the sum number (G) of G is the smallest number of isolated vertices that...

A unique-path labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u, v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on n vertices that admits a unique-path labelling has at most n log2(n)/2 edges, and that this bound is tight for infinitely many values of n. Thus we signi...

omputer 012.10.00 Abstract Many schema labelling approaches have been designed to facilitate querying of XML documents. The proposed algorithms are based on the fact that ancestor–descendant relationships among nodes can be quickly determined. Schema labelling is a family of technologies widely used in indexing tree, graph, or structured XML graph, in which a unique identifier is assigned to ea...

Let SG be the group and a collection of subgroups Then union graph is denoted by U(SG) whose vertex set itself two vertices Li Lj;i? j in are adjacent if only either ? Lj or for i ? j. A G said to on ? some SG. This paper describes simple labelling applied job assignment problem. The task problem related one fundamental combinatorial problems branch optimization. Key words-union graph, groups, ...

A good edge-labelling of a graph G is a labelling of its edges such that for any two distinct vertices u, v, there is at most one (u, v)-path with non-decreasing labels. This notion was introduced in [3] to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at characterizing the class of graphs that admit a good edge-labelling. First, we exhibit infini...

An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. Thomassen has conjectured that a countable graph is bounded if and only if its edges can be oriented, possibly both ways, so that every vertex has finite out-degree and every ray has a forwar...

For a connected graph G of order |V (G)| ≥ 3 and a k-labelling c : E(G) → {1, 2, . . . , k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (l1, l2, . . . , lk), where li is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detect...

We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph C∗-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an AF algebra are discussed, and one sufficient condition is proved that if the row-finite directed graph is constructed by possibly attaching some 1-loops to a row-finite directed graph w...