Adjacency list

About: Adjacency list is a research topic. Over the lifetime, 4419 publications have been published within this topic receiving 78449 citations.

Sequence Scheduling With Genetic Algorithms

Abstract: The application of genetic algorithms to sequence scheduling problems grew out of attempts to use this method to solve Traveling Salesman Problems. A genetic recombination operator for the Traveling Salesman Problem which preserves adjacency (or edges between cities) was developed; this operator proved to be superior to previous genetic operators for this problem [15]. Recently, a new enhancement to the edge recombination operator has been developed which further improves performance when compared to the original operator. Using this operator in the context of the GENITOR algorithm we obtain best known solutions for 30 and 105 city problems with considerable consistency. Our first test of this approach to scheduling was optimization of a printed circuit production line at Hewlett Packard[16). Success with this problem led us to apply similar methods to production scheduling on a sequencing problem posed by the Coors Brewing Co. This work has resulted in new findings regarding sequencing operators and their emphasis on adjacency, order, and position.

Constructing Labeling Schemes through Universal Matrices

Abstract: Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs ℱ labels the vertices of all graphs in ℱ such that for every graph G∈ℱ and every two vertices u,v∈G, f(u,v) can be inferred by merely inspecting the labels of u and v. The size of a labeling scheme is the maximum number of bits used in a label of any vertex in any graph in ℱ. This paper illustrates that the notion of universal matrices can be used to efficiently construct f-labeling schemes. Let ℱ(n) be a family of connected graphs of size at most n and let $\mathcal{C}(\mathcal{F},n)$denote the collection of graphs of size at most n, such that each graph in $\mathcal{C}(\mathcal{F},n)$is composed of a disjoint union of some graphs in ℱ(n). We first investigate methods for translating f-labeling schemes for ℱ(n) to f-labeling schemes for $\mathcal{C}(\mathcal{F},n)$. In particular, we show that in many cases, given an f-labeling scheme of size g(n) for a graph family ℱ(n), one can construct an f-labeling scheme of size g(n)+log log n+O(1) for $\mathcal{C}(\mathcal{F},n)$. We also show that in several cases, the above mentioned extra additive term of log log n+O(1) is necessary. In addition, we show that the family of n-node graphs which are unions of disjoint circles enjoys an adjacency labeling scheme of size log n+O(1). This illustrates a non-trivial example showing that the above mentioned extra additive term is sometimes not necessary. We then turn to investigate distance labeling schemes on the class of circles of at most n vertices and show an upper bound of 1.5log n+O(1) and a lower bound of 4/3log n−O(1) for the size of any such labeling scheme.

Evaluation of an application of graph theory to the layout problem

Abstract: When designing a facility layout it is desirable to obtain an optimum design which satisfies certain necessary relationships among departments. Recent research has indicated that applying graph theory to the layout problem can result in the development of improved solutions, but little can be found to indicate how much better the resultant solutions are. The objectives of this paper are to develop a graph theory approach based on research experience and suggestions of Moore, Carrie and Seppanen, to develop a general computer program implementing the procedure and to compare its performance with that of CRAFT, CORELAP and ALDEP. The comparisons are based upon the adjacency relationships satisfied in the resultant layouts. The procedure finds the graph G equivalent to the problem being solved and then generates one of its maximal spanning trees, which after being transformed to its ‘string’ equivalent, is used to extract a maximal planar sub-graph of G. The dual of this sub-graph represents the desired solu...

Matrices and Graphs

Abstract: The present article is designed to be a contribution to the chapter 'Combinatorial Matrix Theory and Graphs' of the Handbook of Linear Algebra, to be published by CRC Press. The format of the handbook is to give just definitions, theorems, and examples; no proofs. In the five sections given below, we present the most important notions and facts about matrices related to (undirected) graphs. 1. Graphs. 2. The adjacency matrix and its eigenvalues. 3. Other matrix representations. 4. Graph parameters. 5. Association schemes.