KPPY 61

A graph X is {\it symmetric} (or {\it arc-transitive}) if, given any two pairs of adjacent vertices $(u,v)$ and $(u',v')$, there is an automorphism $f$ of $X$, such that $f(u)=u'$ and $f(v)=v'$. A {\it bicirculant} is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. The most famous example of a symmetric biciruculant is the Petersen graph, which is cubic. All cubic and tetravalent symmetric bicirculants are known. In this talk I will present a classification of pentavalent symmetric bicirculants. This is a joint work with Iva Antončič and Klavdija Kutnar.

Given a finite group $G$ and a subset $S \subseteq G$, the bi-Cayley graph $BCay(G, S)$ is the graph whose vertex set is $G \times \{0, 1\}$ and edge set is $\{\{(x, 0), (sx, 1)\} : x \in G, s \in S\}$. A bi-Cayley graph $BCay(G, S)$ is called a BCI-graph if whenever a bi-Cayley graph $BCay(G, T)$ is isomorphic to $BCay(G, S)$, we have that $T = gS^{\alpha}$ for some $g \in G$ and $ \alpha \in Aut(G)$. A group $G$ is called an m-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs. In this talk, we present some general result about BCI-groups and we show that a finite nilpotent group is a 3-BCI-group if and only if it is in the form $U \times V$, where $U$ is a homocyclic group of odd order, and $V$ is trivial or one of following groups: $Z_{2^r} , Z^r_2$ and the quaternion group $Q_8$.

Let $F$ be a field of positive characteristic $p$ and $V$ a finite dimensional vector space over $F$. The Jordan-Chevalley decomposition of a linear operator on $V$ is a unique sum of the commuting semisimple and nilpotent parts. It is well-known that, if $F$ is perfect, any linear operator on $V$ has such a decomposition. But if $F$ is non-perfect there is a counter example, i.e., a linear operator which does not have the Jordan-Chevalley decomposition. Over a non-perfect field (or a general field) we can extend the notion of the Jordan-Chevalley decomposition to that of the Jordan-Chevalley-Seligman decomposition. In this talk I give a criterion for the existence of the Jordan-Chevalley-Seligman decomposition of a given linear operator on $V$ over a non-perfect field.

For a finite graph, we can consider modular representation and product representation. In this talk, we give a lower bound of a product dimension of graphs, and apply that bound to some graphs.

A fundamental result in the evolutionary-game paradigm of cyclic competition in spatially extended ecological systems, as represented by the classic Reichenbach-Mobilia-Frey (RMF) model, is that high mobility tends to hamper or even exclude species coexistence. This result was obtained under the hypothesis that individuals move randomly without taking into account the suitability of their local environment. We incorporate local habitat suitability into the RMF model and investigate its effect on coexistence. In particular, we hypothesize the use of basic instinct of an individual to determine its movement at any time step. That is, an individual is more likely to move when the local habitat becomes hostile and is no longer favorable for survival and growth. We show that, when such local habitat suitability is taken into account, robust coexistence can emerge even in the high-mobility regime where extinction is certain in the RMF model. A surprising finding is that coexistence is accompanied by the occurrence of substantial empty space in the system. Reexamination of the RMF model confirms the necessity and the important role of empty space in coexistence. Our study implies that adaptation/movements according to local habitat suitability are a fundamental factor to promote species coexistence and, consequently, biodiversity.