We define the Wiener-entropy, which is together with the eccentricity-entropy one of the most natural distance-based graph entropies. By deriving the (asymptotic) extremal behaviour, we conclude that the Wiener-entropy of graphs of a given order is more spread than is the case for the eccentricity-entropy. We solve 3 conjectures on the eccentricity-entropy and give a conjecture on the Wiener-entropy related to some surprising behaviour on the graph minimizing it.

Given a knot or link in the form of plat closure of a braid, we describe an algorithm to obtain a braid representing the same knot or link with the standard closure, and vice-versa. We analyze the three cases of knots and links in ℝ3, in handlebodies and in thickened surfaces, and we give for each one a detailed construction of the relative algorithm, with proofs of the results and the calculation of the computational complexity. Indeed, we show that the algorithm has quadratic computational complexity in the number of crossings and loop generators of the braid when passing from plat to standard closure, while it has linear computational complexity in the number of crossings generators of the braid when passing from standard to plat closure. The article includes numerous illustrations to facilitate the understanding of the results, as well as an example of implementation of the algorithm for the general case.

A Cayley graph Cay(G, S) has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay(G, T ), there is a group automorphism α of G such that S α = T . The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group G is a CI-group if every Cayley graph on G has the CI property, and is a DCI-group if every Cayley digraph on G has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group. In 2009, Spiga showed that Z 8 3 is not a DCI-group, by producing a digraph that does not have the DCI property. He also showed that Z 5 3 is a DCI-group (and therefore also a CI-group). Until recently the question of whether there are elementary abelian 3-groups that are not CI-groups remained open. In a recent preprint with Dave Witte Morris, we showed that Z 10 3 is not a CI-group. In this paper we show that with slight modifications, the underlying undirected graph of order 3 8 described by Spiga is does not have the CI property, so Z 8 3 is not a CI-group.

In this article, we determine the complexity function (configurational entropy) of jammed configurations of Rydberg atoms on a one-dimensional lattice. Our method consists of providing asymptotics for the number of jammed configurations determined by direct combinatorial reasoning. In this way we reduce the computation of complexity to solving a constrained optimization problem for the Shannon's entropy function. We show that the complexity can be expressed explicitly in terms of the root of a certain polynomial of degree b, where b is the so-called blockade range of a Rydberg atom. Our results are put in a relation with the model of irreversible deposition of k-mers on a one-dimensional lattice.

Let D be a triple system, and let G be a finite simple group. In this paper we almost determine all possibilities of D admitting G as its flag-transitive automorphism group.

A skew morphism of a finite group B is a permutation φ of B that preserves the identity element of B and has the property that for every a ∈ B there exists a positive integer ia such that φ(ab) = φ(a)φia(b) for all b ∈ B. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism φ of Zn is closely related to a specific skew morphism of Z|⟨φ⟩|, called the quotient of φ. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to 161. During the preparation of this paper we noticed a few flaws in Section 5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68–100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.

A drawing of a graph is k-plane if every edge contains at most k crossings. A k-plane drawing is saturated if we cannot add any edge so that the drawing remains k-plane. It is well-known that saturated 0-plane drawings, that is, maximal plane graphs, of n vertices have exactly 3n−6 edges. For k > 0, the number of edges of saturated n-vertex k-plane graphs can take many different values. In this note, we establish some bounds on the minimum number of edges of saturated 2-plane graphs under various conditions.

We introduce the definition of an Ehrhart limit, that is, a formal power series with integer coefficients that is the limit in the ring of formal power series of a sequence of Ehrhart h*-polynomials. We identify a variety of examples of sequences of polytopes that yield Ehrhart limits, with a focus on reflexive polytopes and simplices.

By mapping the vertices of a graph G to points in ℝ3, and its edges to the corresponding line segments, we obtain a three-dimensional realization of G. A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.

A graph is called a Cayley graph (or bi-Cayley graph, respectively) of a group G if it has a group G of automorphisms acting semiregularly on the vertices with exactly one orbit (or two orbits, respectively). It is known every Cayley graph is vertex-transitive. In this paper, we first present a classification of connected cubic non-Cayley vertex-transitive bi-Cayley graphs of a finite p-group H, where p > 3 is a prime and the derived subgroup of H is either cyclic or isomorphic to Zp × Zp. This is then used to give a classification of connected cubic non-Cayley vertex-transitive graphs of order 2p4 for each prime p.