Random spread on the family of small-world networks

We present the analytical and numerical results of a random walk on the family of small-world graphs. The average access time shows a crossover from the regular to random behavior with increasing distance from the starting point of the random walk. We introduce an {\em independent step approximation}, which enables us to obtain analytic results for the average access time. We observe a scaling relation for the average access time in the degree of the nodes. The behavior of average access time as a function of $p$, shows striking similarity with that of the {\em characteristic length} of the graph. This observation may have important applications in routing and switching in networks with large number of nodes.Comment: RevTeX4 file with 6 figure

Six signed Petersen graphs, and their automorphisms

Up to switching isomorphism there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process we develop new properties of signed graphs, especially of their switching automorphism groups.Comment: 39 pp., 7 fi

Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original

Ultra-sharp soliton switching in a directional coupler

By a numerical investigation it is shown that a directional coupler, described by two linearly coupled non-linear Schro¨dinger equations, can be used to construct a soliton switch with an extremely narrow transition region