14976506
Average independence polynomials
The independence polynomial of a graph G is the function i(G, x) = Σ k ≥ 0 ik xk, where ik is the number of independent sets of vertices in G of cardinality k. We investigate here the average independence polynomial, where the average is taken over all independence polynomials of (labeled) graphs of order n. We prove that while almost every independence polynomial has a nonreal root, the average independence polynomials always have all real, simple roots.
5845717
Graph coloring satisfying restraints
Abstract For an integer k ⩾2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k -colors. A graph G is amenably k-colorable if, for each nonconstant proper k -restraint r on G , there is a k -coloring c of G with c ( v )≠ r ( v ) for each vertex v of G . A graph G is amenable if it is amenably k -colorable and k is the chromatic number of G . For any k ≠3, there are infinitely many amenable k -critical graphs. For k ⩾ 3, we use a construction of B. Toft and amenable graphs to associate a k -colorable graph to any k -colorable finite hypergraph. Some constructions for amenable graphs are given. We also consider a related property—being strongly critical —that is satisfied by many critical graphs, including complete graphs. A strongly critical graph is critical and amenable, but the converse is not always true. The Dirac join operation preserves both amenability and the strongly critical property. In addition, the Hajos construction applied to a single edge in each of two strongly k -critical graphs yields an amenable graph. However, for any k ⩾5, there are amenable k -critical graphs for which the Hajos construction on two copies is not amenable.
45045031
Well-Covered Vector Spaces of Graphs
For any field ${\bf F}$, the set of all functions $f : V(G) \rightarrow {\bf F}$ whose sum on each maximal independent set is constant forms a vector space over ${\bf F}$. In this paper, we show that the dimension can vary depending on the characteristic of the field. We also investigate the dimensions of these vector spaces and show that while some families, such as chordal graphs, have unbounded dimension, other families, such as nonempty circulant graphs of prime order, have bounded dimension.
16943733
Chromatic polynomials and order ideals of monomials
Abstract One expansion of the chromatic polynomial π ( G , x ) of a graph G relies on spanning trees of a graph. In fact, for a connected graph G of order n , one can express π(G,x)=(−1) n−1 x ∑ i=1 n−1 t i (1−x) i ), where t i is the number of spanning trees with external activity 0 and internal activity i . Moreover, it is known (via commutative ring theory) that t i arises as the number of monomials of degree n − i − 1 in a set of monomials closed under division. We describe here how to explicitly carry out this construction algebraically. We also apply this viewpoint to prove a new bound for the roots of chromatic polynomials.
35054660
The Ultimate Categorical Independence Ratio of a Graph
Let $\beta(G)$ denote the independence number of a graph $G$. We introduce $A(G) = \lim_{k \rightarrow \infty}\beta(G^{k})/|V(G)|^{k}$, where the categorical graph product is used. This limit, surprisingly, lies in the range $(0,1/2]\cup\{1\}$. We can show that this limit can take any such rational number, but is there any $G$ for which $A(G)$ is irrational? A useful technique for bounding $A(G)$ is to consider special spanning subgraphs. These bounds allow us to efficiently compute $A(G)$ for many $G$. We give a condition which if true for $G$ shows that $A(G)>\beta(G)/|V(G)|$. This brings up the question: for which $G$ does $A(G)=\beta(G)/|V(G)|$? This happens if $G$ is a Cayley graph of an Abelian group or if $G$ is a connected graph that has an automorphism which has a single orbit.
28649866
A ramsey type problem concerning vertex colourings
Abstract Let H → k v G denote the fact that for every function π: V ( H ) → {1, …, k } there is an induced subgraph G ′ of H with G ′ ≅ G and V ( G ′) ⊆ π −1 ( i ) for some i . Folkman has shown that for all graphs G and for all positive integers k such a graph H exists. We examine here f ( G , k ), the minimum order of a graph H for which H → k v G . We show that for any fixed integer k ≥ 2 there are positive constants C 1 and C 2 such that C 1 n 2 ≤ max{ f ( G , k ): | V ( G )| = n } ≤ C 2 n 2 log 2 n .
1800673
Why AM and Eurisko Appear to Work
Seven years ago, the AM program was constructed as an experiment in learning by discovery. Its source of power was a large body of heuristics, rules which guided it toward fruitful topics of investigation, toward profitable experiments to perform, toward plausible hypotheses and definitions. "Other heuristics evaluated those discoveries for utility and "interestingness", and they were added to AM's vocabulary of concepts. AM's ultimate limitation apparently was due to its inability to discover new, powerful, domain-specific heuristics for the various new fields it uncovered. At that time, it seemed straight-forward to simply add Heuretics (the study of heuristics) as one more field in which to let AM explore, observe, define, and develop. That task - learning new heuristics by discovery - turned out to be much more difficult than was realized initially, and we have just now achieved some successes at it. Along the way, it became clearer why AM had succeeded in the first place, and why it was so difficult to use the same paradigm to discover new heuristics. This paper discusses those recent insights. They spawn questions about "where the meaning really resides" in the concepts discovered by AM. This leads to an appreciation of the crucial and unique role of representation in theory formation, a role involving the relationship between Form and Content.
10597458
Computational and neural mechanisms of task switching
Switching between tasks that overlap in perceptual and response characteristics is assumed to rely upon the maintenance of task representations in prefrontal cortex (PFC). However, task-switching studies demonstrate ''switch costs,'' even when there is sufficient time to prepare for a new task. These costs suggest that task-switching performance reflects a complex interplay between priming and the updating and maintenance of task representations. We describe a computational model in which this interaction is made explicit and linked to the dynamics of PFC. Simulation results account for a variety of empirical phenomena and predict a double dissociation in lateral PFC that was subsequently identified.
45989851
The Social Life of Documents
Documents are evolving as means for structuring and navigating information space, and resources for constructing and negotiating social space
5243593
Steps Toward Automatic Theory Formation
This paper describes a theory formation system which can discover a partial axiomization of a data base represented as extensionally defined binary relations.- The system first discovers all possible intensional definitions of each binary relation in terms of the others. It then determines a minimal set of these relations from which the others can be defined. It then attempts to discover all the ways the relations of this minimal set can interact with each other, thus generating a set of inference rules. Although the system was originally designed to explore automatic techniques for theory construction for question-answering systems, it is currently being expanded to function as a symbiotic system to help social scientists explore certain kinds of data bases.
7675324
Applications of Artificial Intelligence
One of the aims of artificial intelligence has been to make machines behave intelligently as humans do. Thus the study of artificial intelligence includes the study of how humans acquire and apply knowledge, reason under uncertainty and in complex environments, and how they do planning and solve problems. Knowledge-based automation deals with the application of artificial intelligence to a production environment in order to reduce the involvement of human beings to a minimum.