Charles Delorme

Diameter, Covering Index, Covering Radius and Eigenvalues

Fan Chung has recently derived an upper bound on the diameter of a regular graph as a function of the second largest eigenvalue in absolute value. We generalize this bound to the case of bipartite biregular graphs, and regular directed graphs.We also observe the connection with the primitivity exponent of the adjacency matrix. This applies directly to the covering number of Finite Non Abelian Simple Groups (FINASIG). We generalize this latter problem to primitive association schemes, such as the conjugacy scheme of Paiges simple loop.By noticing that the covering radius of a linear code is the diameter of a Cayley graph on the cosets, we derive an upper bound on the covering radius of a code as a function of the scattering of the weights of the dual code. When the code has even weights, we obtain a bound on the covering radius as a function of the dual distance dl which is tighter, for d? large enough, than the recent bounds of Tietavainen.

Optimal schedules for d-D grid graphs with communication delays

We consider a task graph model taking into account the communication among tasks of a parallel system. First, we assume that the available number of processors is adequate for dealing with the whole width of the task graph (i.e. the number of processors is unbounded), and we propose a schedule, called Line-Schedule, which executes the tasks of a d-dimensional grid graph (d-D grid in short) in the optimal time. We continue by proving that Line-Schedule is the only strategy able to execute a d-D grid in the optimal time. Furthermore, we compute the minimum number of processors required to execute a d-D grid optimally.

Table of large (D, D )-graphs

Bermond, J.-C., C. Delcwme and J.J. Quisquater, Table of large (4 D )-graphs, Discrete Applied Mathematics 37/38 (1992) 575-577. Given is a table of large (d, D)-graphs (Table 1).

Cycles with a chord are graceful

In this paper, we prove that every cycle plus a chord is graceful, thus answering a conjecture of R. Bodendiek, H. Schumacher, and H. Wegner.

Two sets of graceful graphs

We give graceful numberings to the following graphs: (a) the union of n K4 having one edge in common, in other words the join of K2 and the union of n disjoint K2 and (b) the union of n C4 having one edge in common, in other words the product of K2 and K1,n, with n + 1 not a multiple of 4.

Infinite generalized friendship graphs

We give necessary and sufficient conditions for the existence of infinite generalized friendship graphs and show that there are 2^o non-isomorphic ones of each admissible order c and chromatic number. Further we prove that such graphs and their complements are almost always regular of degree equal to the order and that various generalizations of the Friendship Theorem do not hold for infinite generalized friendship graphs.

A note on partial Cayley graphs

Abstract In this paper we propose new results about constructions of bipartite graphs of given degree and diameter. These constructions yield graphs with high symmetry.

Rubik Routing Permutations on Graphs

We consider the permutation routing problem on undirected graphs when the routings correspond to products of elements of some fixed set. Previous work has concerned only two-way links. We introduce here the Rubik routing model in order to study the one-way case. We give a complete characterization of Rubik rearrangeable graphs and we study how the 2D grid and the hypercube behave with respect to Rubik routing of permutations.