Properties of Catlin's reduced graphs and supereulerian graphs

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained from $G$ by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs $G$ of order $n$ with $d(u)+d(v)\ge 2(n/p-1)$ for any $uv\in E(G)$ where $p>0$ are given, we show how such graphs change if they have no spanning Eulerian subgraphs when $p$ is increased from $p=1$ to 10 then to $15$

Cluster synchronization in networks of coupled non-identical dynamical systems

In this paper, we study cluster synchronization in networks of coupled non-identical dynamical systems. The vertices in the same cluster have the same dynamics of uncoupled node system but the uncoupled node systems in different clusters are different. We present conditions guaranteeing cluster synchronization and investigate the relation between cluster synchronization and the unweighted graph topology. We indicate that two condition play key roles for cluster synchronization: the common inter-cluster coupling condition and the intra-cluster communication. From the latter one, we interpret the two well-known cluster synchronization schemes: self-organization and driving, by whether the edges of communication paths lie at inter or intra-cluster. By this way, we classify clusters according to whether the set of edges inter- or intra-cluster edges are removable if wanting to keep the communication between pairs of vertices in the same cluster. Also, we propose adaptive feedback algorithms on the weights of the underlying graph, which can synchronize any bi-directed networks satisfying the two conditions above. We also give several numerical examples to illustrate the theoretical results

Singularity categories of skewed-gentle algebras

Let $K$ be an algebraically closed field. Let $(Q,Sp,I)$ be a skewed-gentle triple, $(Q^{sg},I^{sg})$ and $(Q^g,I^{g})$ be its corresponding skewed-gentle pair and associated gentle pair respectively. It proves that the skewed-gentle algebra $KQ^{sg}/$ is singularity equivalent to $KQ/$. Moreover, we use $(Q,Sp,I)$ to describe the singularity category of $KQ^g/$. As a corollary, we get that $\mathrm{gldim} KQ^{sg}/<\infty$ if and only if $\mathrm{gldim} KQ/<\infty$ if and only if $\mathrm{gldim} KQ^{g}/< I^{g}><\infty$.Comment: 13 pages, 1 figur