The results presented in this paper are parts of my doctoral thesis [ 193; the algorithm and some of the characterizations in Section 7 have been introduced in [ 181. Motivation for isometric embeddings into Hamming graphs has come from communication theory (Graham and Pollak [ 121) an-d linguistics (Firsov 1181). Isometric subgraphs of Hamming graphs also appear in biology as “quasi-species” (...

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...

Finite, undirected graphs without self-loops are considered in the paper. The goal is to nd a protocol for local computation which, starting with a uniform labelling of graph nodes, eventually terminates with a labelling being an enumeration of nodes. It is known from 4] that for some (\ambiguous") graphs of the considered family such a protocol does not exist; in the present paper a protocol i...

Let $G$ be a graph with eigenvalues $lambda_1(G)geqcdotsgeqlambda_n(G)$. In this paper we find all simple graphs $G$ such that $G$ has at most twelve vertices and $G$ has exactly two non-negative eigenvalues. In other words we find all graphs $G$ on $n$ vertices such that $nleq12$ and $lambda_1(G)geq0$, $lambda_2(G)geq0$ and $lambda_3(G)0$, $lambda_2(G)>0$ and $lambda_3(G)

Function-Described Graphs (FDGs) have been introduced as a representation of an ensemble of Attributed Graphs (AGs) for structural pattern recognition and a distance measure using restrictions between AGs and FDGs has been reported. Nevertheless, in real applications, AGs can be distorted by some external noise, and therefore some constraints have to be relaxed. To gain more flexibility and rob...

A total labelling of a graph with v vertices and e edges is a one-to-one map taking the vertices and edges onto the set {1, 2, 3, . . . , v + e}. A labelling can be used to define a weight for each vertex and edge. For a vertex the weight is the sum of the label of the vertex and the labels of the incident edges. For an edge {x, y} the weight is the sum of the label of the edge and the labels o...

Graphs and graph transformations are natural means to describe systems of processes. Graphs represent structure of the system, and graph rewriting rules model dynamic behaviour. In this chapter, we illustrate the technique by describing Petri nets, statecharts, parallel logic programming, and systems of processes. Whereas description of Petri nets is based on usual graphs, statecharts lead us t...

The Kneser graph Kn:k for positive integers n ≥ k has as its vertex set the k-element subsets of some n-set, with disjoint sets being adjacent. Every finite simple graph can be found as an induced subgraph of some Kneser graph; this can be viewed as a way of representing graphs by labelling their vertices with sets. We explore questions of finding the smallest representation (both in terms of n...

Function-Described Graphs (FDGs) have been introduced by the authors as a representation of an ensemble of Attributed Graphs (AGs) for structural pattern recognition alternative to first-order random graphs. Both optimal and approximate algorithms for error-tolerant graph matching, which use a distance measure between AGs and FDGs, have been reported elsewhere. Also, the supervised synthesis of...

A graph G is a k-dot product graph if there exists a vector labelling u : V (G)→ Rk such that u(i)Tu(j) ≥ 1 if and only if ij ∈ E(G). Fiduccia, Scheinerman, Trenk and Zito [4] asked whether every planar graph is a 3-dot product graph. We show that the answer is “no”. On the other hand, every planar graph is a 4-dot product graph. We also answer the corresponding questions for planar graphs of p...