We present upper and lower bounds for the number $i_n$ of interval graphs on $n$ vertices. Answering a question posed by Hanlon, we show that the ordinary generating function $I(x) = \sum_{n\ge 0} i_n\,x^n$ for the number $i_n$ of $n$-vertex interval graphs has radius of convergence zero. We also show that the exponential generating function $J(x) = \sum_{n\ge 0} i_n\,x^n/n!$ has radius of convergence at least $1/2$.
Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks
Dartmouth College（达特茅斯学院）；Concordia University（康考迪亚大学）；The George Washington University（乔治华盛顿大学）；Willamette University（威拉米特大学）；North Dakota State University（北达科他州立大学）；Eastern Illinois University（东伊利诺伊大学）
Sieved Enumeration of Interval Orders and Other Fishburn Structures
根据Eriksen和Sjöstrand（2014）的结果，我们详细介绍了一种从适当的Mahonian结构构建遵循Fishburn分布的结构的技术。这种技术是在Bousquet-Mélou et al.（2010）的bivincular模式上引入的，然后用于引入以前未考虑的一类匹配;明确地，根据右交叉和左嵌套的弧的数量的零对齐匹配。然后，我们定义了Claesson和Linusson（2011）的阶乘偏序集的统计量，计算我们称为错误标记的特征的数量，并证明根据错误标记的数量，阶乘偏序集遵循Fishburn分布。作为我们的方法的结果，我们发现一个身份的菲什伯恩数的Mahonian数。
We introduce the arrow product, a systematic generating function technique for directed graph enumeration. It provides short proofs for previous results of Gessel on the number of directed acyclic graphs and of Liskovets, Robinson and Wright on the number of strongly connected directed graphs. We also recover Robinson's enumerative results on directed graphs where all strongly connected components belong to a given family.
We present new short proofs of known spanning tree enumeration formulae for threshold and Ferrers graphs by showing that the Laplacian matrices of such graphs admit triangular rank-one perturbations. We then characterize the set of graphs whose Laplacian matrices admit triangular rank-one perturbations as the class of special 2-threshold graphs, introduced by Hung, Kloks, and Villaamil. Our work introduces (1) a new characterization of special 2-threshold graphs that generalizes the characterization of threshold graphs in terms of isolated and dominating vertices, and (2) a spanning tree enume…
The Young--Fibonacci graph is the Hasse diagram of one of the two (along with the Young lattice) 1-differential graded modular lattices. This explains the interest to path enumeration problems in this graph. We obtain a formula for the number of paths between two vertices of the Young--Fibonacci graph which is polynomial with respect to the minimum of their ranks.
A Deza graph $Γ$ with parameters $(v,k,b,a)$ is a $k$-regular graph with $v$ vertices such that any two distinct vertices have $b$ or $a$ common neighbours, where $b \ge a$. A Deza graph of diameter 2 which is not a strongly regular graph is called a strictly Deza graph. We find all 139 strictly Deza graphs up to 21 vertices.
We prove that the number of single element extensions of $M(K_{n+1})$ is $2^{{n\choose n/2}(1+o(1))}$. This is done using a characterization of extensions as "linear subclasses".
We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (a cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The present data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant to distinguish graphs in cases where other algebraic invariants, such as those derived from the spectrum, fail. Finally, w…
A permutation graph is a graph that can be derived from a permutation, where the vertices correspond to letters of the permutation, and the edges represent inversions. We provide a construction to show that there are infinitely many connected $r$-regular permutation graphs for $r \geq 3$. We prove that all $3$-regular permutation graphs arise from a similar construction. Finally, we enumerate all $3$-regular permutation graphs on $n$ vertices.
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 bipartite Cayley digraphs and Cayley graphs over abelian groups. Let $A$ be an abelian group and let $ι$ be the automorphism of $A$ defined by $a^ι=a^{-1}$, for every $a\in A$. A Cayley graph $\Cay(A, S)$ is said to have an automorphism group as small as possible if $\Aut(\Cay(A,S)) = \langle A,ι\rangle$. In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley di…
In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of $d$-regular graphs for all $d$, as $n\to\inf…
In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2…
