Isoparametric Problems on Graphs and Posets Isoparametric Problems on Graphs and Posets

We consider extremal graph-theoretic problems on regular structures. The problems ask to maximize a certain parameter of the structures in question under the condition that the values of some other parameter(s) are constant. This is where the term isoparametric is taken from. The graph problems we consider are edge-isoperimetric problems, and the poset problems are dealing with the notion of Macaulay posets. The research is split into two parts theoretical and practical. The practical part of research is developing computer programs that are used to find examples of structures satisfying certain properties. The theoretical part involves proving that the found examples build infinite series of new graph/poset classes. Part I: Introduction and Summary of Results 1 Edge-isoperimetric Problems in Graphs Let G = (VG, EG) be a graph and A ⊆ VG. Denote θG(A) = {(u, v) ∈ EG | u ∈ A, v 6∈ A} EG(A) = {(u, v) ∈ EG | u, v ∈ A} θG(m) = { min A⊆EG |A|=m θG(A)} EG(m) = {max A⊆EG |A|=m |E(A)|}. The Edge-Isoperimetric Problem (EIP) consists of finding for a given m, 1 ≤ m ≤ |VG|, a set A ⊆ EG such that |EG(A)| = EG(m). Such sets are called isoperimetric. There are two versions of EIP: maximizing |EG(A)| and minimizing |θG(A)|. In this paper we are most concerned with maximizing |IG(A)|. For k-regular graphs both version of EIP are equivalent due to the following identity: 2 · |EG(A)|+ |θG(A)| = k · |A|. (1) We say that a graph G with |VG| = p admits an isoperimetric order if there exists a numbering of VG by numbers 0, 1, . . . , p such that for every m, 1 ≤ m ≤ p the initial segment Im of length m of this numbering constitutes an isoperimetric set. In other words, |EG(Im)| = EG(m). If an isoperimetric order exists for a graph G then G is called isoperimetric.