In this note, the octonion multiplication table is recovered from a regular tesselation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map.

In the present paper, we classify groups with the same order and degree pattern as an almost simple group related to the projective special linear simple group $L_2\left(49\right)$. As a consequence of this result we can give a positive answer to a conjecture of W. J. Shi and J. X. Bi, for all almost simple groups related to $L_2\left(49\right)$ except $L_2\left(49\right)·2^2$. Also, we prove that if $M$ is an almost simple group related to $L_2\left(49\right)$ except $L_2\left(49\right)·2^2$ and $G$ is a finite group such that $\left|G\right|=\left|M\right|$ and $\Gamma \left(G\right)=\Gamma \left(M\right)$, then $G\cong M$.

We deal with the graph operator $\overline{Pow₂}$ defined to be the complement of the square of a graph: $\overline{Pow₂}\left(G\right)=\overline{Pow₂\left(G\right)}$. Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m,n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective geometries...

We revisit the problem of deciding whether a finitely generated subgroup $H$ is a free factor of a given free group $F$. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of $H$ and exponential in the rank of $F$. We show that the latter dependency can be made exponential in the rank difference rank$\left(F\right)$ - rank$\left(H\right)$, which often makes a significant change.

We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in the rank of F. We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change.

We give necessary and sufficient conditions for various vertex-transitivity of Cayley graphs of the class of completely 0-simple semigroups and its several subclasses. Moreover, the question when the Cayley graphs of completely 0-simple semigroups are undirected is considered.

We prove that if the Walsh bipartite map $ℳ=𝒲\left(\mathscr{H}\right)$ of a regular oriented hypermap $\mathscr{H}$ is also orientably regular then both $ℳ$ and $\mathscr{H}$ have the same chirality group, the covering core of $ℳ$ (the smallest regular map covering $ℳ$) is the Walsh bipartite map of the covering core of $\mathscr{H}$ and the closure cover of $ℳ$ (the greatest regular map covered by $ℳ$) is the Walsh bipartite map of the closure cover of $\mathscr{H}$. We apply these results to the family of toroidal chiral hypermaps ${(3,3,3)}_{b,c}={𝒲}^{-1}{\{6,3\}}_{b,c}$ induced by the family of toroidal bipartite maps...

In this paper, we have studied the connectedness of the graphs on the direct product of the Weyl groups. We have shown that the number of the connected components of the graph on the direct product of the Weyl groups is equal to the product of the numbers of the connected components of the graphs on the factors of the direct product. In particular, we show that the graph on the direct product of the Weyl groups is connected iff the graph on each factor of the direct product is connected.

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number ${\chi }_D\left(G\right)$ of G. Extending these concepts to infinite graphs we prove that $D\left(Q_{\aleph }₀\right)=2$ and ${\chi }_D\left(Q_{\aleph }₀\right)=3$, where $Q_{\aleph }₀$ denotes the hypercube of countable dimension. We also show that ${\chi }_D\left(Q₄\right)=4$, thereby completing the investigation of finite hypercubes with respect to ${\chi }_D$. Our results...