On groups all of whose undirected Cayley graphs of bounded valency are integral

A finite group $G$ is called Cayley integral if all undirected Cayley graphs over $G$ are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Kloster and Sander in the abelian case, and by Abdollahi and Jazaeri, and independently by Ahmady, Bell and Mohar in the non-abelian case. In this paper we generalize this class of groups by introducing the class $\mathcal{G}_k$ of finite groups $G$ for which all graphs $\mathrm{Cay}(G,S)$ are integral if $|S| \le k$. It will be proved that $\mathcal{G}_k$ consists of the Cayley integral groups if $k \ge 6;$ and the classes $\mathcal{G}_4$ and $\mathcal{G}_5$ are equal, and consist of:\ (1) the Cayley integral groups, (2) the generalized dicyclic groups $\mathrm{Dic}(E_{3^n} \times \mathbb{Z}_6),$ where $n \ge 1$

Indecomposable coverings with homothetic polygons

We prove that for any convex polygon $S$ with at least four sides, or a concave one with no parallel sides, and any $m>0$, there is an $m$-fold covering of the plane with homothetic copies of $S$ that cannot be decomposed into two coverings

Automorphism groups of rational circulant graphs through the use of Schur rings

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups

Dense point sets with many halving lines

A planar point set of $n$ points is called {\em $\gamma$-dense} if the ratio of the largest and smallest distances among the points is at most $\gamma\sqrt{n}$. We construct a dense set of $n$ points in the plane with $ne^{\Omega\left({\sqrt{\log n}}\right)}$ halving lines. This improves the bound $\Omega(n\log n)$ of Edelsbrunner, Valtr and Welzl from 1997. Our construction can be generalized to higher dimensions, for any $d$ we construct a dense point set of $n$ points in $\mathbb{R}^d$ with $n^{d-1}e^{\Omega\left({\sqrt{\log n}}\right)}$ halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.Comment: 18 page

Multiple coverings with closed polygons

A planar set $P$ is said to be cover-decomposable if there is a constant $k=k(P)$ such that every $k$-fold covering of the plane with translates of $P$ can be decomposed into two coverings. It is known that open convex polygons are cover-decomposable. Here we show that closed, centrally symmetric convex polygons are also cover-decomposable. We also show that an infinite-fold covering of the plane with translates of $P$ can be decomposed into two infinite-fold coverings. Both results hold for coverings of any subset of the plane.Comment: arXiv admin note: text overlap with arXiv:1009.4641 by other author

Star-galaxy separation strategies for WISE-2MASS all-sky infrared galaxy catalogs

We combine photometric information of the WISE and 2MASS all-sky infrared databases, and demonstrate how to produce clean and complete galaxy catalogs for future analyses. Adding 2MASS colors to WISE photometry improves star-galaxy separation efficiency substantially at the expense of loosing a small fraction of the galaxies. We find that 93% of the WISE objects within W1<15.2 mag have a 2MASS match, and that a class of supervised machine learning algorithms, Support Vector Machines (SVM), are efficient classifiers of objects in our multicolor data set. We constructed a training set from the SDSS PhotoObj table with known star-galaxy separation, and determined redshift distribution of our sample from the GAMA spectroscopic survey. Varying the combination of photometric parameters input into our algorithm we show that W1 - J is a simple and effective star-galaxy separator, capable of producing results comparable to the multi-dimensional SVM classification. We present a detailed description of our star-galaxy separation methods, and characterize the robustness of our tools in terms of contamination, completeness, and accuracy. We explore systematics of the full sky WISE-2MASS galaxy map, such as contamination from Moon glow. We show that the homogeneity of the full sky galaxy map is improved by an additional J<16.5 mag flux limit. The all-sky galaxy catalog we present in this paper covers 21,200 sq. degrees with dusty regions masked out, and has an estimated stellar contamination of 1.2% and completeness of 70.1% among 2.4 million galaxies with $z_{med}= 0.14$. WISE-2MASS galaxy maps with well controlled stellar contamination will be useful for spatial statistical analyses, including cross correlations with other cosmological random fields, such as the Cosmic Microwave Background. The same techniques also yield a statistically controlled sample of stars as well.Comment: 10 pages, 11 figures. Accepted for publication in MNRA

Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014), 679--693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called $0$-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism group. A $0$-type graph can be represented as the graph $\mathrm{BCay}(H,S),$ where $S$ is a subset of $H,$ the vertex set of which consists of two copies of $H,$ say $H_0$ and $H_1,$ and the edge set is $\{\{h_0,g_1\} : h,g \in H, g h^{-1} \in S\}$. A bi-Cayley graph $\mathrm{BCay}(H,S)$ is called a BCI-graph if for any bi-Cayley graph $\mathrm{BCay}(H,T),$ $\mathrm{BCay}(H,S) \cong \mathrm{BCay}(H,T)$ implies that $T = h S^\alpha$ for some $h \in H$ and $\alpha \in \mathrm{Aut}(H)$. It is also shown that every cubic connected arc-transitive $0$-type bi-Cayley graph over an abelian group is a BCI-graph

Cubic symmetric graphs having an abelian automorphism group with two orbits

Finite connected cubic symmetric graphs of girth 6 have been classified by K. Kutnar and D. Maru\v{s}i\v{c}, in particular, each of these graphs has an abelian automorphism group with two orbits on the vertex set. In this paper all cubic symmetric graphs with the latter property are determined. In particular, with the exception of the graphs K_4, K_{3,3}, Q_3, GP(5,2), GP(10,2), F40 and GP(24,5), all the obtained graphs are of girth 6.Comment: Keywords: cubic symmetric graph, Haar graph, voltage graph. This papaer has been withdrawn by the author because it is an outdated versio

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph $\mathrm{Cay}(\mathrm{Sym}_n,T_n),$ where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that Aut$(\mathrm{Cay}(\mathrm{Sym}_n,T_n))$ is the product of the left translation group by a dihedral group $\mathsf{D}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $\Gamma$ of $\mathrm{Cay}(\mathrm{Sym}_n,T_n)$ induced by the set $T_n$. In particular, $\Gamma$ is a $2(n-2)$-regular graph whose automorphism group is $\mathsf{D}_{n+1},$ $\Gamma$ has as many as $n+1$ maximal cliques of size $2,$ and its subgraph $\Gamma(V)$ whose vertices are those in these cliques is a $3$-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of $\mathsf{D}_{n+1}$ of order $n+1$ with regular Cayley maps on $\mathrm{Sym}_n$ is also discussed. It is shown that the product of the left translation group by the latter group can be obtained as the automorphism group of a non-$t$-balanced regular Cayley map on $\mathrm{Sym}_n$

A classification of nilpotent 3-BCI groups

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 $\{\{(x,0),(s x,1)\} : x \in G, s\in S \}$. A bi-Cayley graph \bcay(G,S) is called a BCI-graph if for any bi-Cayley graph \bcay(G,T), \bcay(G,S) \cong \bcay(G,T) implies that $T = g S^\alpha$ 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 paper we prove 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 the groups $\Z_{2^r},$ $\Z_2^r$ and \Q_8