The equidistant dimension of graphs

A subset S of vertices of a connected graph G is a distance-equalizer set if for every two distinct vertices x,y¿V(G)\S there is a vertex w¿S such that the distances from x and y to w are the same. The equidistant dimension of G is the minimum cardinality of a distance-equalizer set of G. This paper is devoted to introduce this parameter and explore its properties and applications to other mathematical problems, not necessarily in the context of graph theory. Concretely, we first establish some bounds concerning the order, the maximum degree, the clique number, and the independence number, and characterize all graphs attaining some extremal values. We then study the equidistant dimension of several families of graphs (complete and complete multipartite graphs, bistars, paths, cycles, and Johnson graphs), proving that, in the case of paths and cycles, this parameter is related to 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.Peer ReviewedPostprint (published version

Graphical Designs and Gale Duality

A graphical design is a subset of graph vertices such that the weighted averages of certain graph eigenvectors over the design agree with their global averages. We use Gale duality to show that positively weighted graphical designs in regular graphs are in bijection with the faces of a generalized eigenpolytope of the graph. This connection can be used to organize, compute and optimize designs. We illustrate the power of this tool on three families of Cayley graphs -- cocktail party graphs, cycles, and graphs of hypercubes -- by computing or bounding the smallest designs that average all but the last eigenspace in frequency order. We also prove that unless NP = coNP, there cannot be an efficient description of all minimal designs that average a fixed number of eigenspaces in a graph.Comment: 30 pages, 14 figures, 1 tabl

An introduction of the theory of nonlinear error-correcting codes

Nonlinear error-correcting codes are the topic of this thesis. As a class of codes, it has been investigated far less than the class of linear error-correcting codes. While the latter have many practical advantages, it the former that contain the optimal error-correcting codes. In this project the theory (with illustrative examples) of currently known nonlinear codes is presented. Many definitions and theorems (often with their proofs) are presented thus providing the reader with the opportunity to experience the necessary level of mathematical rigor for good understanding of the subject. Also, the examples will give the reader the additional benefit of seeing how the theory can be put to use. An introduction to a technique for finding new codes via computer search is presented

On Linear Codes over F2 x F2

A code of length n and size M consist of a set of M vectors of n components. The components being taken from some alphabet set S. So a code C is a set of n-tuples subset of Sn. If S has a ring structure then C is called a linear code over S if it is an S-module. To every linear code C there corresponds its dual C⊥, if C C⊥, then C is called self-orthogonal. If C = C⊥ then C is called self-dual. In this thesis we will study linear and self-dual codes over the rings of four alphabets and in more details over the ring F2 x F2, this ring is isomorphic to the ring F2 + vF2 where v2 = v and F2 = {0; 1}. We would also study linear and self-dual codes for other rings in the form Fp + vFp for different primes p. Also we will construct simplex code over the ring F2 + vF2≃ F2 x F2

Efficient Channel Modeling Methods for Mobile Communication Systems

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