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We associate a graph $\mathcal{C}_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | < x,y> \text{is cyclic for all} y\in G\}$ is called the cyclicizer of $G$, and join two vertices if they do not generate a cyclic subgroup. For a simple graph $\Gamma$, $w(\Gamma)$ denotes the clique number of $\Gamma$, which is the maximum size (if it exists) of a complete subgraph of $\Gamma$. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group $G$ is solvable whenever $w(\mathcal{C}_G)<31$ and the equality for a non-solvable group $G$ holds if and only if $G/Cyc(G)\cong A_5$ or $S_5$., Comment: to appear in Journal of Algebra and its Applications

We associate a graph $\Gamma_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | \left \text{is cyclic for all} y\in G\}$, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of $\Gamma_G$ is finite if and only if $\Gamma_G$ has no infinite clique. We prove that if $G$ is a finite nilpotent group and $H$ is a group with $\Gamma_G\cong\Gamma_H$ and $|Cyc(G)|=|Cyc(H)|=1$, then $H$ is a finite nilpotent group. We give some examples of groups $G$ whose non-cyclic graphs are ``unique'', i.e., if $\Gamma_G\cong \Gamma_H$ for some group $H$, then $G\cong H$. In view of these examples, we conjecture that every finite non-abelian simple group has a unique non-cyclic graph. Also we give some examples of finite non-cyclic groups $G$ with the property that if $\Gamma_G \cong \Gamma_H$ for some group $H$, then $|G|=|H|$. These suggest the question whether the latter property holds for all finite non-cyclic groups.

In this paper we prove that a set of points $B$ of PG(n,2) is a minimal blocking set if and only if $=PG(d,2)$ with $d$ odd and $B$ is a set of $d+2$ points of $PG(d,2)$ no $d+1$ of them in the same hyperplane. As a corollary to the latter result we show that if $G$ is a finite 2-group and $n$ is a positive integer, then $G$ admits a $\mathfrak{C}_{n+1}$-cover if and only if $n$ is even and $G\cong (C_2)^{n}$, where by a $\mathfrak{C}_m$-cover for a group $H$ we mean a set $\mathcal{C}$ of size $m$ of maximal subgroups of $H$ whose set-theoretic union is the whole $H$ and no proper subset of $\mathcal{C}$ has the latter property and the intersection of the maximal subgroups is core-free. Also for all $n<10$ we find all pairs $(m,p)$ ($m>0$ an integer and $p$ a prime number) for which there is a blocking set $B$ of size $n$ in $PG(m,p)$ such that $=PG(m,p)$.