### $(2,n)$ – звёздно транзитивные графы

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We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

The order of every finite group $G$ can be expressed as a product of coprime positive integers ${m}_{1},\cdots ,{m}_{t}$ such that $\pi \left({m}_{i}\right)$ is a connected component of the prime graph of $G$. The integers ${m}_{1},\cdots ,{m}_{t}$ are called the order components of $G$. Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups ${C}_{2}\left(q\right)$ where $q>5$ are also uniquely determined by their order components. As corollaries of this result, the validities of a...

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, tetravalent one-regular graphs of order 3p², where p is a prime, are classified.

Research on combinatorial properties of sequences in groups and semigroups originates from Bernhard Neumann's theorem answering a question of Paul Erd"{o}s. For results on related combinatorial properties of sequences in semigroups we refer to the book [3]. In 2000 the authors introduced a new combinatorial property and described all groups satisfying it. The present paper extends this result to all semigroups.

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G\left(R\right)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...

The problem of finding minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic groups. This problem was considered earlier by other authors. We give a construction of an undirected graph having $4n+6$ vertices and automorphism group cyclic of order $4n$, $n\ge 1$. As a special case we get graphs with ${2}^{k}+6$ vertices and cyclic automorphism groups of order ${2}^{k}$. It can revive interest in related problems.

A graph $X$, with a group $G$ of automorphisms of $X$, is said to be $(G,s)$-transitive, for some $s\ge 1$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs. Let $X$ be a connected $(G,s)$-transitive graph of prime valency $p\ge 5$, and ${G}_{v}$ the vertex stabilizer of a vertex $v\in V\left(X\right)$. Suppose that ${G}_{v}$ is solvable. Weiss (1974) proved that $|{G}_{v}{|\mid p(p-1)}^{2}$. In this paper, we prove that ${G}_{v}\cong ({\mathbb{Z}}_{p}\u22ca{\mathbb{Z}}_{m})\times {\mathbb{Z}}_{n}$ for some positive integers $m$ and $n$ such that $ndivm$ and $m\mid p-1$.