Interpolation theorem for a continuous function on orientations of a simple graph

summary:Let $G$ be a simple graph. A function $f$ from the set of orientations of $G$ to the set of non-negative integers is called a continuous function on orientations of $G$ if, for any two orientations $O_1$ and $O_2$ of $G$, $|f(O_1)-f(O_2)|\le 1$ whenever $O_1$ and $O_2$ differ in the orientation of exactly one edge of $G$. We show that any continuous function on orientations of a simple graph $G$ has the interpolation property as follows: If there are two orientations $O_1$ and $O_2$ of $G$ with $f(O_1)=p$ and $f(O_2)=q$, where $p<q$, then for any integer $k$ such that $p<k<q$, there are at least $m$ orientations $O$ of $G$ satisfying $f(O) = k$, where $m$ equals the number of edges of $G$. It follows that some useful invariants of digraphs including the connectivity, the arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of $G$

Interpolation theorem for a continuous function on orientations of a simple graph

Let G be a simple graph. A function f from the set of orientations of G to the set of Iron-negative integers is called a continuous function on orientations of G if, for any two orientations O-1 and O-2 of G, \f(O-1) - f(O-2)\ less than or equal to 1 whenever O-1 and O-2 differ in the orientation of exactly one edge of G. We show that any continuous function on orientations of a simple graph G has the interpolation property as follows: If there are two orientations O-1 and O-2 of G with f(O-1) = p and f(O-2) = q, where p < q, then for any integer k such that p < k < q, there are at least m orientations O of G satisfying f(O) = k, where m equals the number of edges of G. It follows that some useful invariants of digraphs including the connectivity, the arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Weighted dependency graphs

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality of sums of random variables. In this article, we introduce a more general notion of weighted dependency graphs and give normality criteria in this context. We also provide generic tools to prove that some weighted graph is a weighted dependency graph for a given family of random variables. To illustrate the power of the theory, we give applications to the following objects: uniform random pair partitions, the random graph model $G(n,M)$, uniform random permutations, the symmetric simple exclusion process and multilinear statistics on Markov chains. The application to random permutations gives a bivariate extension of a functional central limit theorem of Janson and Barbour. On Markov chains, we answer positively an open question of Bourdon and Vall\'ee on the asymptotic normality of subword counts in random texts generated by a Markovian source.Comment: 57 pages. Third version: minor modifications, after review proces

Graphs isomorphisms under edge-replacements and the family of amoebas

Let $G$ be a graph of order $n$ and let $e\in E(G)$ and $e'\in E(\overline{G}) \cup \{e\}$. If the graph $G-e+e'$ is isomorphic to $G$, we say that $e\to e'$ is a \emph{feasible edge-replacement}. We call $G$ a \emph{local amoeba} if, for any two copies $G_1$, $G_2$ of $G$ that are embedded in $K_n$, $G_1$ can be transformed into $G_2$ by a chain of feasible edge-replacements. On the other hand, $G$ is called \emph{global amoeba} if there is an integer $T \ge 1$ such that $G \cup tK_1$ is a local amoeba for all $t \ge T$. We study global and local amoebas under an underlying algebraic theoretical setting. In this way, a deeper understanding of their structure and their intrinsic properties as well as how these two families relate with each other comes into light. Moreover, it is shown that any connected graph can be a connected component of an amoeba, and a construction of a family of amoeba trees with a Fibonacci-like structure and with arbitrary large maximum degree is presented.Comment: 21 pages, 9 figure

Combinatorics

[no abstract available

Mod-phi convergence I: Normality zones and precise deviations

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$. Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of "weakly dependent" random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new section on mod-Gaussian convergence coming from the factorization of the generating function ; the multi-dimensional results have been moved to a forthcoming paper ; and the introduction has been reworke