Random Information Spread in Networks

Let G=(V,E) be an undirected loopless graph with possible parallel edges and s and t be two vertices of G. Assume that vertex s is labelled at the initial time step and that every labelled vertex copies its labelling to neighbouring vertices along edges with one labelled endpoint independently with probability p in one time step. In this paper, we establish the equivalence between the expected s-t first arrival time of the above spread process and the notion of the stochastic shortest s-t path. Moreover, we give a short discussion of analytical results on special graphs including the complete graph and s-t series-parallel graphs. Finally, we propose some lower bounds for the expected s-t first arrival time.Comment: 17 pages, 1 figur

Edge Contraction and Line Graphs

Given a family of graphs $\mathcal{H}$, a graph $G$ is $\mathcal{H}$-free if any subset of $V(G)$ does not induce a subgraph of $G$ that is isomorphic to any graph in $\mathcal{H}$. We present sufficient and necessary conditions for a graph $G$ such that $G/e$ is $\mathcal{H}$-free for any edge $e$ in $E(G)$. Thereafter, we use these conditions to characterize claw-free and line graphs.Comment: arXiv admin note: text overlap with arXiv:2203.0349

Surface topography of microtubule walls decorated with monomeric and dimeric kinesin constructs

The surface topography of opened-up microtubule walls (sheets) decorated with monomeric and dimeric kinesin motor domains was investigated by freeze-drying and unidirectional metal shadowing. Electron microscopy of surface-shadowed specimens produces images with a high signal/noise ratio, which enable a direct observation of surface features below 2 nm detail. Here we investigate the inner and outer surface of microtubules and tubulin sheets with and without decoration by kinesin motor domains. Tubulin sheets are flattened walls of microtubules, keeping lateral protofilament contacts intact. Surface shadowing reveals the following features: (i) when the microtubule outside is exposed the surface relief is dominated by the bound motor domains. Monomeric motor constructs generate a strong 8 nm periodicity, corresponding to the binding of one motor domain per beta -tubulin heterodimer. This surface periodicity largely disappears when dimeric kinesin motor domains are used for decoration, even though it is still visible in negatively stained or frozen hydrated specimens, This could be explained by disorder in the binding of the second (loosely tethered) kinesin head, and/or disorder in the coiled-coil tail. (ii) Both surfaces of undecorated sheets or microtubules, as well as the inner surface of decorated sheets, reveal a strong 4 nm repeat (due to the periodicity of tubulin monomers) and a weak 8 nm repeat (due to slight differences between alpha- and beta -tubulin). The differences between alpha- and beta -tubulin on the inner surface are stronger than expected from cryo-electron microscopy of unstained microtubules, indicating the existence of tubulin subdomain-specific surface properties that reflect the surface corrugation and hence metal deposition during evaporation. The 16 nm periodicity visible in some negatively stained specimens (caused by the pairing of cooperatively bound kinesin dimers) is not detected by surface shadowing