The regular continuous image of a minimal regular space is not necessarily minimal regular

On (minimal) regular graphs of girth 6

Minimal Union-Free Decompositions of Regular Languages

A regular language is called union-free if it can be represented by a regular expression that does not contain the union operation. Every regular language can be decomposed into a union of a finite number of union-free languages (the so-called union-free decomposition). We call the number of components in the minimal union-free decomposition of a regular language the union width of the regular ...

Minimal Sense of Direction in Regular Networks

A network is said to have Sense of Direction when the port labeling satisses a particular set of global consistency constraints. In this paper we study the link between the topology of a system and the number of labels that are necessary to have a Sense of Direction in that system. We consider systems whose topology is a regular graph and we study the relationship between structural properties ...

Tree-minimal Graphs Are Almost Regular

For all fixed trees T and any graph G we derive a counting formula for the number N̂T (G) of homomorphisms from T to G in terms of the degree sequence of G. As a consequence we obtain that any n-vertex graph G with edge density p = p(n) n−1/(t−2), which contains at most (1 + o(1))pt−1nt copies of some fixed tree T with t ≥ 3 vertices must be almost regular, i.e., ∑ v∈V (G) |deg(v)− pn| = o(pn ).

Non-regular graphs with minimal total irregularity

The total irregularity of a simple undirected graphG is defined as irrt(G) = 1 2 ∑ u,v∈V (G) |dG(u)− dG(v)|, where dG(u) denotes the degree of a vertex u ∈ V (G). Obviously, irrt(G) = 0 if and only if G is regular. Here, we characterize the non-regular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu, You and Yang [18] about the lower bound on the minimal ...