On the power graph and the reduced power graph of a finite group

In this paper, for a finite group, we investigate to what extent its directed (resp. undirected) reduced power graph determines its directed power graph (resp. reduced power graph). Moreover, we investigate the determination of the orders of the elements of a finite group from its directed (resp. undirected) reduced power graph. Consequently, we show that some classes of finite groups are recognizable from their undirected reduced power graphs. Also, we study the relationship between the isomorphism classes of groups corresponding to the equivalence relations induced by the isomorphism of each of these graphs on the set of all finite groups.     

A dynamic free-entry oligopoly with sluggish entry and exit adjustments

We study a dynamic free-entry oligopoly with sluggish entry and exit of firms under general demand and cost functions. We show that the number of firms in a steady-state open-loop solution for a dynamic free-entry oligopoly is smaller than that at static equilibrium and that the number of firms in a steady-state memoryless closed-loop solution is larger than that in an open-loop solution.     

Existential monadic second order logic of undirected graphs: The Le Bars conjecture is false

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ? such that $\mathsf{P}(G_n??)$ does not converge as $n\to \infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1,\dots ,n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.     

Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation

This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example, a new mixed finite element method (FEM) is established and the $τ$ -independent superclose results of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{\rightarrow}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is an error splitting technique, with which the time-discrete and the spatial-discrete systems are constructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or require certain time step conditions. Finally, some numerical results are provided to confirm the theoretical analysis, and show the efficiency of the proposed method.     

Meyniel's conjecture holds for random d‐regular graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d‐regular graphs when d = d(n) ≥ 3.     

On the Differential Polynomial of a Graph

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial B(G; x) :=∑?(G)k=-nB_k(G) x~(n+k), where B_k(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G;x) and its coefficients.In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.     

Weight enumerators of reducible cyclic codes and their dual codes

Let ${\mathbb{F}}_q$ be a finite field and $n$ a positive integer. In this paper, we find a new combinatorial method to determine weight enumerators of reducible cyclic codes and their dual codes of length $n$ over ${\mathbb{F}}_q$, which just generalize results of Zhu et al. (2015); especially, we also give the weight enumerator of a cyclic code, which is viewed as a partial Melas code. Furthermore, weight enumerators obtained in this paper are all in the form of power of a polynomial.     

Optimal packings of two to four equal circles on any flat torus

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.     

Synthesis and Structures of Arene Ruthenium (II)–NHC Complexes: Efficient Catalytic α‐alkylation of ketones via Hydrogen Auto Transfer Reaction

A panel of six new arene Ru (II)‐NHC complexes 2a‐f , (NHC = 1,3‐diethyl‐(5,6‐dimethyl)benzimidazolin‐2‐ylidene 1a , 1,3‐dicyclohexylmethyl‐(5,6‐dimethyl)benzimidazolin‐2‐ylidene 1b and 1,3‐dibenzyl‐(5,6‐dimethyl)benzimidazolin‐2‐ylidene 1c ) were synthesized from the transmetallation reaction of Ag‐NHC with [(η⁶‐arene)RuCl₂]₂ and characterized. The ruthenium (II)‐NHC complexes 2a‐f were developed as effective catalysts for α‐alkylation of ketones and synthesis of bioactive quinoline using primary/amino alcohols as coupling partners respectively. The reactions were performed with 0.5 mol% catalyst load in 8 h under aerobic condition and the maximum yield was up to 96%. Besides, the different alkyl wingtips on NHC and arene moieties were studied to differentiate the catalytic robustness of the complexes in the transformations.