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The problem of finding a maximal dense subgraph of a power-law random graph is considered for every value of density and for every . It is shown that in case a maximal -dense subgraph has size , in case it is limited whp, and in case it is whp less than .
For nonnegative integers , we prove a combinatorial identity for the -binomial coefficient based on abelian -groups. A purely combinatorial proof of this identity is not known. While proving this identity, for and a prime, we present a purely combinatorial formula for the number of subgroups of of finite index with quotient isomorphic to the finite abelian -group of type , which is a partition of into at most parts. This purely combinatorial formula is similar to that for the enumeration of subgroups of a certain type in a finite abelian -group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise to many enumeration formulae that involve polynomials in with nonnegative integer coefficients.
Motivated by long-standing conjectures on the discretization of classical inequalities in the geometry of numbers, we investigate a new set of parameters, which we call packing minima, associated to a convex body and a lattice . These numbers interpolate between the successive minima of and the inverse of the successive minima of the polar body of and can be understood as packing counterparts to the covering minima of Kannan & Lovász (1988).
As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.
Let be an -element set, where is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family of -element subsets of , one can partition into disjoint pairs in such a way that no matter how we pick one element from each of the first pairs, the set formed by them can always be completed to a member of by adding an element of the last pair.
The above problem is related to classical questions in extremal set theory. For any , we call a family of sets -separable if there is a -element subset such that for every ordered pair of elements of , there exists such that . For a fixed , , and , we establish asymptotically tight estimates for the smallest integer such that every family with is -separable.
Modularity is designed to measure the strength of a division of a network into clusters. For , consider a set of elements. Let be the set of all subsets of of size . Consider the graph with vertices and edges that connect two vertices if and only if their intersection has size . In this article, we find the exact modularity of for .
The modularity of a graph is a value that shows how well the graph can be split into clusters. It is a key part in many clustering algorithms. In this paper, we improve the lower bound on the modularity of Johnson graphs significantly.
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