33 research outputs found
On strongly closed subgraphs of highly regular graphs
AbstractA geodetically closed induced subgraph Δ of a graph Γ is defined to be strongly closed if Γi(α) ∩ Γ1(β) stays in Δ for every i and α, β ϵ Δ with ∂(α, β) = i. We study the existence conditions of strongly closed subgraphs in highly regular graphs such as distance-regular graphs or distance-biregular graphs
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Distance-Biregular Graphs and Orthogonal Polynomials
This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems.
We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new
results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound
Diameter, Covering Index, Covering Radius and Eigenvalues
AbstractFan 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 Paige's 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 Tietäväinen
On Graph-Based Cryptography and Symbolic Computations
We have been investigating the cryptographical properties of
in nite families of simple graphs of large girth with the special colouring
of vertices during the last 10 years. Such families can be used for the
development of cryptographical algorithms (on symmetric or public key
modes) and turbocodes in error correction theory. Only few families of
simple graphs of large unbounded girth and arbitrarily large degree are
known.
The paper is devoted to the more general theory of directed graphs of
large girth and their cryptographical applications. It contains new explicit
algebraic constructions of in finite families of such graphs. We show that
they can be used for the implementation of secure and very fast symmetric
encryption algorithms. The symbolic computations technique allow us to
create a public key mode for the encryption scheme based on algebraic
graphs
Divergence in right-angled Coxeter groups
Let W be a 2-dimensional right-angled Coxeter group. We characterise such W
with linear and quadratic divergence, and construct right-angled Coxeter groups
with divergence polynomial of arbitrary degree. Our proofs use the structure of
walls in the Davis complex.Comment: This version incorporates the referee's comments. It contains the
complete appendix (which will be abbreviated in the journal version). To
appear in Transactions of the AM
Divergence in right-angled Coxeter groups
Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex