The irregularity of a graph G is defined as the sum of weights |d(u) − d(v)| of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. In this paper, some structural properties on trees with maximum (or minimum) irregularity among trees with given degree sequence and trees with given branching number are explored, respectively. Moreover, the correspon...

For polynomial vector fields in R, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a ...

By using a framework where the object of noncommutativity θ represents independent degrees of freedom, we study the symmetry properties of an extended x + θ space-time, given by the group P ’, which has the Poincaré group P as a subgroup. In this process we use the minimal canonical extension of the Doplicher, Fredenhagen and Roberts algebra. It is also proposed a generalized Dirac equation, wh...

A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well known result states that graphs with the same degree sequence may be changed into each other via sequences of 2-switches. We show that if a 2-switch changes the isomorphism class of a graph, then it must take place in one of four configurations. We also pres...

This essay considers the extent to which a concept of emergence can be associated with Effective Field Theories (EFTs). I suggest that such a concept can be characterized by microphysicalism and novelty underwritten by the elimination of degrees of freedom from a high-energy theory, and argue that this makes emergence in EFTs distinct from other concepts of emergence in physics that have appear...

An r.e. set A is called mitotic if there exist a pair of disjoint r.e. sets Al, A2 with A, UA, = A (in this case we write AI UA, =A) such that AI =,.A2 sTA. We refer to such a splitting as a mitotic splitting of A. Lachlan [lo] was the first person to show that not all, r.e. sets are mitotic. More extensive investigations into (non)mitoticity were provided by Ladner [ll, 121 who constructed var...

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.

A given nonincreasing sequence D = (d1, d2, · · · , dn) is said to contain a (nonincreasing) repetition sequence D∗ = (di1 , di2 , · · · , dik ) for some k ≤ n − 2 if all values of D −D∗ are distinct and for any dil ∈ D ∗ there exists some dt ∈ D −D∗ such that dil = dt. For any pair of integers n and k with n ≥ k + 2, we investigate the existence of a graphic sequence which contains a given rep...

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We prove that for points on a plane curve, one of the coordinates has to have high order. We also discuss a conjecture of Poonen for subvarieties of semiabelian varieties for which our result is a weak special case. Finally, we look at some special cases where we...

A sequence S is potentially K4 e graphical if it has a realization containing a K4 e as a subgraph. Let 0'(K4 e, n) denote the smallest degree sum such that every n-term graphical sequence S with O'(S) 2: a(I{4 e, n) is potentially K4 e graphical. Gould, Jacobson, Lehel raised the problem of determining the value of 0'(K4 e, n). In this paper, we prove that 0'(K4 e, n) = 2[(317, 1)/2] for 17, 2...