Census of quadrangle groups inclusions

In a classical result of 1972 Singerman classifies the inclusions between triangle groups. We extend the classification to a broader family of triangle and quadrangle groups forming a particular subfamily of Fuchsian groups. With two exceptions, each inclusion determines a finite bipartite map (hypermap) on a 2-dimensional spherical orbifold that encodes the complete information and gives a graphical visualisation of the inclusion. A complete description of all the inclusions is contained in the attached tables.

Impact of node mobility in user-centric routing

Recent paradigms in wireless communication architectures describe environments where nodes present a highly dynamic behavior (e.g., User Centric Networks). In such environments, routing is still performed based on the regular packet-switched behavior of store-and-forward. Albeit sufficient to compute at least an adequate path between a source and a destination, such routing behavior cannot adequately sustain the highly nomadic lifestyle that Internet users are today experiencing. This thesis aims to analyse the impact of the nodes’ mobility on routing scenarios. It also aims at the development of forwarding concepts that help in message forwarding across graphs where nodes exhibit human mobility patterns, as is the case of most of the user-centric wireless networks today. The first part of the work involved the analysis of the mobility impact on routing, and we found that node mobility significance can affect routing performance, and it depends on the link length, distance, and mobility patterns of nodes. The study of current mobility parameters showed that they capture mobility partially. The routing protocol robustness to node mobility depends on the routing metric sensitivity to node mobility. As such, mobility-aware routing metrics were devised to increase routing robustness to node mobility. Two categories of routing metrics proposed are the time-based and spatial correlation-based. For the validation of the metrics, several mobility models were used, which include the ones that mimic human mobility patterns. The metrics were implemented using the Network Simulator tool using two widely used multi-hop routing protocols of Optimized Link State Routing (OLSR) and Ad hoc On Demand Distance Vector (AODV). Using the proposed metrics, we reduced the path re-computation frequency compared to the benchmark metric. This means that more stable nodes were used to route data. The time-based routing metrics generally performed well across the different node mobility scenarios used. We also noted a variation on the performance of the metrics, including the benchmark metric, under different mobility models, due to the differences in the node mobility governing rules of the models.

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs G and H. Let H^k[G] be the lexicographic product of H^k and G, where H^k is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H^k[G]H and H^k when G and H are regular and the Laplacian spectrum of H^k[G] and H^k for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10^100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.

Bethe graphs attached to the vertices of a connected graph: a spectral approach

A weighted Bethe graph $B$ is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family $F$ of graphs. The operation of identifying the root vertex of each of $r$ weighted Bethe graphs to the vertices of a connected graph $\mathcal{R}$ of order $r$ is introduced as the $\mathcal{R}$-concatenation of a family of $r$ weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when $F$ has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in $F$ are all regular) of the $\mathcal{R}$-concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.

A lower bound for the energy of symmetric matrices and graphs

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric partitioned matrix into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized composition of a family of arbitrary graphs are obtained. A lower bound for the energy of a graph with a bridge is given. Some computational experiments are presented in order to show that, in some cases, the obtained lower bound is incomparable with the well known lower bound $2\sqrt{m}$, where $m$ is the number of edges of the graph.