Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic enumeration of labelled 4-regular planar graphs
157
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Building on previous work by the present authors [Proc. London Math. Soc. 119(2):358--378, 2019], we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs. Our estimate is of the form $g_n sim gcdot n^{-7/2} rho^{-n} n!$, where $g>0$ is a constant and $rho approx 0.24377$ is the radius of convergence of the generating function $sum_{nge 0}g_n x^n/n!$, and conforms to the universal pattern obtained previously in the enumeration of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to work with large systems of polynomials equations. In particular, we use evaluation and Lagrange interpolation in order to compute resultants of multivariate polynomials. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.
rate research
Read More
We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.
In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper normal subgroup $N$ with the property that the automorphism group of the digraph fixes each $N$-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.
We provide bivariate asymptotics for the poly-Bernoulli numbers, a combinatorial array that enumerates lonesum matrices, using the methods of Analytic Combinatorics in Several Variables (ACSV). For the diagonal asymptotic (i.e., for the special case of square lonesum matrices) we present an alternative proof based on Parsevals identity. In addition, we provide an application in Algebraic Statistics on the asymptotic ML-degree of the bivariate multinomial missing data problem, and we strengthen an existing result on asymptotic enumeration of permutations having a specified excedance set.
Point-determining graphs are graphs in which no two vertices have the same neighborhoods, co-point-determining graphs are those whose complements are point-determining, and bi-point-determining graphs are those both point-determining and co-point-determining. Bicolored point-determining graphs are point-determining graphs whose vertices are properly colored with white and black. We use the combinatorial theory of species to enumerate these graphs as well as the connected cases.
The energy of a simple graph $G$ arising in chemical physics, denoted by $mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In general for a type of lattice in statistical physics, to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness. In this paper, we show that if ${G_n}$ is a sequence of finite simple graphs with bounded average degree and ${G_n}$ a sequence of spanning subgraphs of ${G_n}$ such that almost all vertices of $G_n$ and $G_n$ have the same degrees, then $G_n$ and $G_n$ have the same asymptotic energy. Thus, for each type of lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions, we have the same asymptotic energy. As applications, we obtain the asymptotic formulae of energies per vertex of the triangular, $3^3.4^2$, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions simultaneously.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
