A ROUTING AND WAVELENGTH ASSIGNMENT STRATEGY FOR SUCCESSFUL TRANSMISSION IN OPTICAL NETWORKS

Optical Burst and Packet Switching (OBS/OPS) are techniques designed to serve higher-layer packet-based communication protocols by allowing statistical multiplexing. Since OBS and OPS networks provide connectionless transport, they both suffer from contention, which occurs when multiple communications want to use simultaneously the same wavelength in a link. This paper proposes a Routing and Wavelength Assignment (RWA) strategy based on the concept of (rooted) collision-free digraph, which represents all paths assigned by the routing to those communications sharing a wavelength. Using the proposed RWA strategy, the contention problem can be successfully solved by using simple mechanisms based on adding a suitable additional delay to burst/packet transmissions. Here we define and characterize the routing-antipodal networks, in which we can define [n/2] pairs of arc-disjoint collision-free digraphs (with n being the number of nodes) that altogether include all arcs of the network. This implies that, using [n/2] wavelengths, we can achieve connectivity between any pair of nodes under the wavelength-continuity constraint. Solutions with fewer wavelengths are also feasible. In particular, if the routing-antipodal network has a trail that passes through all vertices at least once, one wavelength is enough to ensure connectivity between each pair of nodes. We also show that the line digraph technique provides us with a simple tool for obtaining proper collision-free digraphs. The proposed method works in either a synchronous or an asynchronous transmission environment. Also, the arriving and length burst/packet distributions can be of any type, provided that the maximum theoretical offered load is not exceeded.

On Oriented Arc-Coloring of Subcubic Graphs

A homomorphism from an oriented graph $G$ to an oriented graph $H$ is a mapping $\varphi$ from the set of vertices of $G$ to the set of vertices of $H$ such that $\overrightarrow{\varphi(u)\varphi(v)}$ is an arc in $H$ whenever $\overrightarrow{uv}$ is an arc in $G$. The oriented chromatic index of an oriented graph $G$ is the minimum number of vertices in an oriented graph $H$ such that there exists a homomorphism from the line digraph $LD(G)$ of $G$ to $H$ (Recall that $LD(G)$ is given by $V(LD(G))=A(G)$ and $ \overrightarrow{ab}\in A(LD(G))$ whenever $a=\overrightarrow{uv}$ and $b=\overrightarrow{vw}$). We prove that every oriented subcubic graph has oriented chromatic index at most $7$ and construct a subcubic graph with oriented chromatic index $6$.

A Matrix Representation of Graphs and its Spectrum as a Graph Invariant

We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined by the spectrum of the adjacency matrix of the original graph. We then show by computation that the isomorphism classes of many known families of strongly regular graphs (up to 64 vertices) are characterized by the spectrum of this matrix. We conjecture that this is always the case for strongly regular graphs and we show that the conjecture is not valid for general graphs. We verify that the smallest regular graphs which are not distinguished with our method are on 14 vertices.

TRANSMISSION FAULT-TOLERANCE OF ITERATED LINE DIGRAPHS

The main result of this paper states that, if every cyclic modification of a d-regular digraph has super line-connectivity d, then the line digraph also has super line-connectivity d. Since many well-known interconnection network topologies, such as the Kautz digraphs, the de Bruijn digraphs, etc., can be constructed by iterating the line digraph construction, our result leads to several known and new connectivity results for these topologies, as shown later in the paper.

Algebraic properties of a digraph and its line digraph

Let G be a digraph, LG its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows us to relate properties of G and LG, specifically the number of cycles of a given length.