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- R L Graham, L Lovasz
- 1978

Let G be a finite connected graph. If x and y are vertices of G, one may define a distance function d, on G by letting d&x, y) be the minimal length of any path between x and y in G (with d&, x) = 0). Thus, for example, d&x, y) = 1 if and only if {x, y} is an edge of G. Furthermore, we define the distance matrix D(G) for G to be the square matrix with rows… (More)

- M R Garey, R L Graham, D S Johnsonf, D E Knutht
- 1978

We present a linear-time algorithm for sparse symmetric matrices which converts a matrix into pentadiagonal form ("bandwidth 2"), whenever it is possible to do so using simultaneous row and column permutations. On the other hand when an arbitrary integer k and graph G are given, we show that it is NP-complete to determine whether or not there exists an… (More)

- Persi Diaconis, R L Graham, William M Kantor
- 1983

There are two ways to perfectly shuffle a deck of 2n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle 0 leaves the original top card on top. The in shuffle I leaves the original top card second from the top. Applications to the design of computer networks and card tricks are reviewed. The main result is the determination of… (More)

- F R K Chung, R L Graham
- 1990

- Persi Diaconis, R L Graham
- 1985

Suppose G is a finite group and / is a function mapping G into the set of real numbers R. For a subset S c G, define the Radon transform F s of/mapping G into R by: Fs(χ)-Σ f(y) y^S + x where S + x denotes the setf-s + xiseS). Thus, the Radon transform can be thought of as a way of replacing/by a "smeared out" version of /. This form of the transform… (More)

- N Alon, F R K Chung, R L Graham
- 1997

We consider a class of routing problems on connected graphs G. Initially, each vertex v of G is occupied by a " pebble " which has a unique destination T(V) in G, so that m is a permutation of the vertices of G. It is required to route all the pebbles to their respective destinations by performing a sequence of moves of the following type: A dis-joint set… (More)

- F R K Chung, R L Graham, J Shearer
- 1981

For a class V of graphs, denote by a(@?) the least value of m so that for some graph II on m vertices, every GE Q occurs as a subgraph of U. In this note we obtain rather sharp bounds on u(q) when Q is the class of caterpillars on n vertices, i.e., tree with property that the vertices of degree exceeding one induce a path. Recently several of the authors… (More)

- A R Calderbank, P Frankl, R L Graham, W.-C W Li, L A Shepp
- 1992

Shannon introduced the concept of zero-error capacity of a discrete memoryless channel. The channel determines an undirected graph on the symbol alphabet, where adjacency means that symbols cannot be confused at the receiver. The zero-error or Shannon capacity is an invariant of this graph. Gargano, Korner, and Vaccaro have recently extended the concept of… (More)

- F R K Chung, P Diaconis, R L Graham, C L Mallows
- 1981

- B P Bernays, Bg, V Blomer, A Granville, Bf, J Bourgain +8 others
- 2014

A proof of the positive density conjecture for integer Apollonian circle packings,