Raziskovalni matematični seminar - Arhiv

Let G be a graph with n vertices, S=\mathbb{K}[x_1,\dots,x_n] be the polynomial ring in n variables over a field \mathbb{K} and I(G) denote the edge ideal of G.

We survey a number of recent studies of the Castelnuovo-Mumford regularity of the edge ideal of G. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated graphs.

In addition, for every collection \mathcal{H} of connected graphs with K_2\in \mathcal{H}, we introduce the notions of ind-match_{\mathcal{H}}(G) and min-match_{\mathcal{H}}(G). We will improve the inequalities for regularity of S/I(G).

Let G be a group and S\subseteq G. A Haar graph of G with connection set S has vertex set \Z_2\times G and edge set \{(0,g)(1,gs):g\in G{\rm\ and\ }s\in S\}. Haar graphs are then natural bipartite analogues of Cayley digraphs. We first examine the relationship between the automorphism group of a Cayley digraph of G with connection set S and a Haar graph of G with connection set S. We establish that the automorphism group of a Haar graph contains a natural subgroup isomorphic to the automorphism group of the corresponding Cayley digraph. In the case where G is abelian, we then give four situations in which the automorphism group of the Haar graph can be larger than the natural subgroup corresponding to the automorphism group of the Cayley digraph together with a specific involution, and analyze the full automorphism group in each of these cases. As an application, we show that all s-transitive Cayley graphs of generalized dihedral groups have a quasiprimitive automorphism group, can be ``reduced" to s-arc-transitive graphs of smaller order, or are Haar graphs of abelian groups whose automorphism groups have a particular permutation group theoretic property.

Roughly speaking, the celebrated central limit theorem says that a sum of many small independent random variables with sufficiently nice distributions approximately follows the normal (Gaussian) distribution. An important issue is the estimation of the error in the normal approximation. Numerous approaches have been suggested. In 1962, Slepian suggested elegant gradual replacement of the original sum with a Gaussian random variable. This approach is essentially equivalent to Stein's method introduced in 1970, which arises from a different idea.

Though relatively old, Stein's method is still a highly active field of research. This is highly due to its flexibility. In particular, Stein's method does not only work for sums of independent random variables, but also for families with various kind of dependence structure. We shall focus on the so called local dependence.

We discuss a new family of cubic graphs, which we call $SGP$-graphs, that bears a close resemblance to the family of generalized Petersen graphs; both in definition and properties. The focus of our paper is on determining the algebraic properties of graphs from our new family. We look for highly symmetric graphs, e.g., graphs with large automorphism groups, vertex- or arc-transitive graphs. In particular, we present arithmetic conditions for the defining parameters that guarantee that graphs with these parameters are vertex-transitive or Cayley, and we find one arc-transitive $SGP$-graph which is neither a $CQ$ graph of Feng and Wang, nor a generalized Petersen graph. 

Joint work with Katarina Jasencakova and Robert Jajcay.