Let G $G$ be a finite group and recall that the Frattini subgroup Frat ( ) ${\rm Frat}(G)$ is intersection of all maximal subgroups . In this paper, we investigate number , denoted α $\alpha (G)$ which minimal whose coincides with earlier work, studied in special case where simple here extend analysis to almost groups. particular, prove ⩽ 4 (G) \leqslant 4$ for every best possible. We also esta...

The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...

The celebrated Erdös, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on v points has chromatic index at most v. We will introduce the linear intersection number of a graph, and use this number to give an alternative formulation of the Erdös, Faber, Lovász conjecture. Finally, first results about the linear intersection number will be proved. For example, the definit...

The geometric intersection number of a curve on a surface is the minimal number of selfintersections of any homotopic curve, i.e., of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most ` on a combinatorial surface of complexity n we describe simple algorithms to (1) compute the geometric intersection number of c in O(n+ `2) time, (2) con...