Cornelis Hoede

Clique polynomials and independent set polynomials of graphs

This paper introduces two kinds of graph polynomials, clique polynomial andindependent set polynomial. The paper focuses on expansions of these polynomials. Some open problems are mentioned.

A characterization of consistent marked graphs

A marked graph is obtained from a graph by giving each point either a positive or a negative sign. Beineke and Harary raised the problem of characterzing consistent marked graphs in which the product of the signs of the points is positive for every cycle. In this paper a characterization is given in terms of fundamental cycles of a cycle basis.

Integral complete r-partite graphs

A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we give a useful sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinite many new classes of such integral graphs. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving some Diophantine equations. The discovery of these integral complete r-partite graphs is a new contribution to the search of such integral graphs. Finally, we propose several basic open problems for further study.

Word Graphs: The Second Set

In continuation of the paper of Hoede and Li on word graphs for a set of prepositions, word graphs are given for adjectives, adverbs and Chinese classifier words. It is argued that these three classes of words belong to a general class of words that may be called adwords. These words express the fact that certain graphs may be brought into connection with graphs that describe the important classes of nouns and verbs. Some subclasses of adwords are discussed as well as some subclasses of Chinese classifier words.

On the Ontology of Knowledge Graphs

Knowledge graphs are a special kind of conceptual graphs. They were developed independently from 1982 on and are special in that a very restricted set of types of relationships is used. In this paper the ontology of knowledge graphs and the philosophy behind its choice are discussed.

Degree sums and subpancyclicity in line graphs

A graph is called subpancyclic if it contains a cycle of length k for each k between 3 and the circumference of the graph. In this paper, we show that if the degree sum of the vertices along each 2-path of a graph G exceeds (n+6)/2, or if the degree sum of the vertices along each 3-path of G exceeds (2n+16)/3, then its line graph L(G) is subpancyclic. Simple examples show that these bounds are best possible. Our results shed some light on the content of a famous Metaconjecture of Bondy.

Hard graphs for the maximum clique problem

The maximum clique problem is one of the NP-complete problems. There are graphs for which a reduction technique exists that transforms the problem for these graphs into one for graphs with specific properties in polynomial time. The resulting graphs do not grow exponentially in order and number. Graphs that allow such a reduction technique are called soft. Hard graphs are those graphs for which none of the reduction techniques can be applied. A list of properties of hard graphs is determined.

Word Graphs: The Third Set

This is the third paper in a series of natural language processing in term of knowledge graphs. A word is a basic unit in natural language processing. This is why we study word graphs. Word graphs were already built for prepositions and adwords (including adjectives, adverbs and Chinese quantity words) in two other papers [2], [3]. In this paper, we propose the concept of the logic word and classify logic words into groups in terms of semantics and the way they are used in describing reasoning processes.