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We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…

We prove an upper bound for the number of cyclic transitive subgroups in a finite permutation group and clarify the structure of the groups for which this bound becomes sharp. We also give an application in the theory of number fields.

We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the…

We study a two-species bidirectional exclusion process, and a single species variant, which is motivated by the motion of organelles and vesicles along microtubules. Specifically, we are interested in the clustering of the particles and…

We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn…

Let $F(N,m)$ denote a random forest on a set of $N$ vertices, chosen uniformly from all forests with $m$ edges. Let $F(N,p)$ denote the forest obtained by conditioning the Erdos-Renyi graph $G(N,p)$ to be acyclic. We describe scaling limits…

We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the…

Denote by $G$ a finite group and let $\psi(G)$ denote the sum of element orders in $G$. In 2009, H.Amiri, S.M.Jafarian Amiri and I.M.Isaacs proved that if $|G|=n$ and $G$ is non-cyclic, then $\psi(G)<\psi(C_n)$, where $C_n$ denotes the…

We consider the slow movement of randomly biased random walk $(X_n)$ on a supercritical Galton--Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of…

It is shown that a refined version of a q-analogue of the Eulerian numbers together with the action, by conjugation, of the subgroup of the symmetric group $S_n$ generated by the $n$-cycle $(1,2,...,n)$ on the set of permutations of fixed…

It is shown, from $\sigma$-centered Martin's Axiom, that there exists a proper dense subgroup of the symmetric group on a countably infinite set whose natural action on sufficiently flexible relational structures is transitive. This allows…

Behavior of hysteretic trajectories for cyclical input is investigated as a function of the internal structure of a system modeled by the classical random network of binary spins. Different regimes of hysteretic behavior are discovered for…

Recently Csikv\'ari \cite{csik} proved a conjecture of Nikiforov concerning the number of closed walks on trees. Our aim is to extend his theorem to all walks. In addition, we give a simpler proof of Csikv\'ari's result and answer one of…

We study positive $m$-divisible non-crossing partitions and their positive Kreweras maps. In classical types, we describe their combinatorial realisations as certain non-crossing set partitions. We also realise these positive Kreweras maps…

We consider a recent model of random walk that recursively grows the network on which it evolves, namely the Tree Builder Random Walk (TBRW). We introduce a bias $\rho \in (0,\infty)$ towards the root, and exhibit a phase transition for…

Cyclic coverings produce many examples of topologically contractible smooth affine complex varieties. In this paper, we study the motivic cohomology groups of cyclic coverings over algebraically closed fields of characteristic $0$. In…

In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the…

In this article, we prove that if all non-trivial cyclic subgroups of a group $G$ are self normalizing and $G$ satisfies the implication $$ \ o(x)\neq o(y)\Rightarrow o(xy)\neq o(x), o(y), $$ for all non-trivial elements $x$ and $y$, then…

We provide a bijection between the set of factorizations, that is, ordered (n-1)-tuples of transpositions in ${\mathcal S}_{n}$ whose product is (12...n), and labelled trees on $n$ vertices. We prove a refinement of a theorem of D\'{e}nes…

In this article we study the K- and L-theory of groups acting on trees. We consider the problem in the context of the fibered isomorphism conjecture of Farrell and Jones. We show that in the class of residually finite groups it is enough to…