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The cutting number of a point of a connected graph is a measure of the extent to which the removal of that point cuts the graph. The cutting center of the graph is the set of points of maximal cutting number. In an earlier paper [1] all possible structures for the cutting center of a tree were determined and examples constructed which realize them. In this paper we extend those results to all graphs.

This study explores the structure of graphs which together with their complements are edge symmetric.

In the usual development of dimension theory in metric spaces, the equivalence of covering and large inductive dimension plays a central role. In this paper we develope the basic theory of dimension directly from the notion of covering dimension. Several of the basic theorems are extended to non-metrizable spaces. For those results which do require the full strength of metrizability, the proofs are new and rather different from the usual ones. With two minor exceptions, the paper is self-contained.

We present an elementary proof of a transfinite symmetric form of Philip Hall's theorem on systems of distinct representatives and derive some wellknown mapping theorems from it to illustrate the ease with which this form of the theorem may be used.

The radius and diameter of a graph are known to satisfy the relation rad G ≤ diam G ≤ 2 rad G. We show that this is the only restriction on these parameters and construct all non-isomorphic graphs of minimal order having a specified radius and diameter.

is any connected graph of order 6 in List C.Of the 171 graphs which appear in these lists, 116 have x = x = 3. Inaddition to these the complements of the 51 graphs in List E areconnected graphs of order 7 through 9 with x = x = 3. And Corollary 2dimplies that there are many other graphs of order 7 through 9 with

Every discrete group G generates a C*-algebra 9Q1(G) of operators on the Hilbert space L2(G) of square summable complex valued functions on G. 9Q1(G), in which elements are written as formal (generally infinite) complex linear combinations ExE -GaXx, is the closure in operator norm of the group algebra L(G) of finite linear combinations, which acts on L2(G) by left multiplication (the left regular representation of G). Our main result is a formula for the explicit calculation of the norm of certain operators in L(G). We show that, if xl, . . ., x, satisfy a certain "freedom condition" (in particular if they freely generate a free subgroup of G), then for arbitrary complex scalars al,,... ,an5 a~ ~ ~ ~ ~~~Z Z21 i1 _Z JZ>A7