We define the class of conjunctive queries in relational data bases, and the generalized join operator on relations.The generalized join plays an important part in answering conjunctive queries, and it can be implemented using matrix multiplication.It is shown that while answering conjunctive queries is NP complete (general queries are PSPACE complete), one can find an implementation that is within a constant of optimal.The main lemma used to show this is that each conjunctive query has a unique minimal equivalent query (much like minimal finite automata).

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Abstract. View an n-vertex, m-edge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with the lengths of random walks on the graph. For example, the commute time between two vertices s and t (the expected length of a random walk from s to t and back) is precisely characterized by the eective resistance R st between s and t: commute time = 2mR st . As a corollary, the cover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistance R in the graph to within a factor of log n: mR cover time O(mR log n). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, we improve known bounds on cover times for high-degree graphs and expanders, and give new proofs of known results for multidimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.

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