Notes on the connectivity of Cayley coset digraphs

Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.Comment: 15 page

Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version

Quantum Hall Ground States, Binary Invariants, and Regular Graphs

Extracting meaningful physical information out of a many-body wavefunction is often impractical. The polynomial nature of fractional quantum Hall (FQH) wavefunctions, however, provides a rare opportunity for a study by virtue of ground states alone. In this article, we investigate the general properties of FQH ground state polynomials. It turns out that the data carried by an FQH ground state can be essentially that of a (small) directed graph/matrix. We establish a correspondence between FQH ground states, binary invariants and regular graphs and briefly introduce all the necessary concepts. Utilizing methods from invariant theory and graph theory, we will then take a fresh look on physical properties of interest, e.g. squeezing properties, clustering properties, etc. Our methodology allows us to `unify' almost all of the previously constructed FQH ground states in the literature as special cases of a graph-based class of model FQH ground states, which we call \emph{accordion} model FQH states

Connectivité des graphes sommet-transitifs

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal

On the Metric Dimension for Snowflake Graph

The concept of metric dimension is derived from the resolving set of a graph, that is measure the diameter among vertices in a graph. For its usefulness in diverse fields, it is interesting to find the metric dimension of various classes of graphs. In this paper, we introduce two new graphs, namely snowflake graph and generalized snowflake graph. After we construct these graphs, aided with a lemma about the lower bound of the metric dimension on a graph that has leaves, and manually recognized the pattern, we found that dim(Snow) = 24 and dim(Snow(n,a,b,c)) = n(a+c+1)