Abstract:
The paper describes the wooden structure of a pedestrian bridge of a mixed use building with (residential and working) for disabled people. It has been analyzed from the process of design to the structure itself and questions about materials and treatments. The building was built 13 years ago, when the wooden construction was really reduced in Spain; something to keep in mind in the analysis of this work. El artículo describe la estructura de una pasarela de madera de acceso a un edificio de uso mixto residencial y laboral para discapacitados. Se recorren los aspectos de proyecto, del dise o de la estructura y otros más técnicos relativos al tipo de materiales y tratamientos. La obra se construyó hace unos 13 a os, momento en el que en Espa a la construcción con madera era mínima; algo que sin duda no puede perderse de vista al analizar esta obra.

Abstract:
The even discrete torus is the graph T_{L,d} on vertex set {0,...,L-1}^d (L even) with two vertices adjacent if they differ by 1 (mod L) on one coordinate. The hard-core measure with activity x on T_{L,d} is the distribution pi_x on the independent sets (sets of vertices spanning no edges) of T_{L,d} in which a set I is chosen with probability proportional to x^|I|. This distribution occurs in problems from statistical physics and communication networks. We study Glauber dynamics, a single-site update Markov chain on the set of independent sets of T_{L,d} whose stationary distribution is pi_x. We show that for x > cd^{-1/4}log^{3/4}d (and d large) the convergence to stationarity is exponentially slow in L^{d-1}. This improves a result of Borgs et al., who had shown slow mixing for x > c^d. Our proof, which extends to r-local chains (chains which alter the state of at most a proportion r of the vertices in each step) for suitable r, follows the conductance argument of Borgs et al., adding to it some combinatorial enumeration methods that are modifications of those used by Galvin and Kahn to show that the hard-core model with parameter x on the integer lattice Z^d exhibits phase coexistence for x > cd^{-1/4}log^{3/4}d. The graph T_{L,d} is bipartite, with partition classes E (the vertices the sum of whose coordinates is even) and O. Our result can be expressed combinatorially as the statement that for each sufficiently large x, there is an r(x)>0 such that if I is an independent set chosen according to pi_x, then the probability that ||I \cap E|-|I \cap O|| is at most r(x)L^d is exponentially small in L^{d-1}. In particular, for all eps>0 the probability that a uniformly chosen independent set from T_{L,d} satisfies ||I \cap E|-|I \cap O|| \leq (.25 - eps)L^d is exponentially small in L^{d-1}.

Abstract:
Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If true, this bound would be tight, being achieved by the disjoint union of $N/2d$ copies of $K_{d,d}$. Kahn established the bound for bipartite $G$, and later gave an argument that established $$ i(G)\leq 2^{\frac{N}{2}\left(1+\frac{2}{d}\right)} $$ for $G$ not necessarily bipartite. In this note, we improve this to $$ i(G)\leq 2^{\frac{N}{2}\left(1+\frac{1+o(1)}{d}\right)} $$ where $o(1) \rightarrow 0$ as $d \rightarrow \infty$, which matches the conjectured upper bound in the first two terms of the exponent. We obtain this bound as a corollary of a new upper bound on the independent set polynomial $P(\lambda,G)=\sum_{I \in {\cal I}(G)} \lambda^{|I|}$ of an $N$-vertex, $d$-regular graph $G$, namely $$ P(\gl,G) \leq (1+\gl)^{\frac{N}{2}} 2^{\frac{N(1+o(1))}{2d}} $$ valid for all $\gl > 0$. This also allows us to improve the bounds obtained recently by Carroll, Galvin and Tetali on the number of independent sets of a fixed size in a regular graph.

Abstract:
Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Alavi, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal; Levit and Mandrescu further conjectured that this should hold for all bipartite $G$. We consider the independent set sequence of finite regular bipartite graphs, and graphs obtained from these by percolation (independent deletion of edges). Using bounds on the independent set polynomial $P(G,\lambda):=\sum_{t \geq 0} i_t(G)\lambda^t$ for these graphs, we obtain partial unimodality results in these cases. We then focus on the discrete hypercube $Q_d$, the graph on vertex set $\{0,1\}^d$ with two strings adjacent if they differ on exactly one coordinate. We obtain asymptotically tight estimates for $i_{t(d)}(Q_d)$ in the range $t(d)/2^{d-1} > 1-1/\sqrt{2}$, and nearly matching upper and lower bounds otherwise. We use these estimates to obtain a stronger partial unimodality result for the independent set sequence of $Q_d$.

Abstract:
For graphs $G$ and $H$, a {\em homomorphism} from $G$ to $H$, or {\em $H$-coloring} of $G$, is an adjacency preserving map from the vertex set of $G$ to the vertex set of $H$. Writing ${\rm hom}(G,H)$ for the number of $H$-colorings admitted by $G$, we conjecture that for any simple finite graph $H$ (perhaps with loops) and any simple finite $n$-vertex, $d$-regular, loopless graph $G$ we have $$ {\rm hom}(G,H) \leq \max{{\rm hom}(K_{d,d},H)^{\frac{n}{2d}}, {\rm hom}(K_{d+1},H)^{\frac{n}{d+1}}} $$ where $K_{d,d}$ is the complete bipartite graph with $d$ vertices in each partition class, and $K_{d+1}$ is the complete graph on $d+1$ vertices. Results of Zhao confirm this conjecture for some choices of $H$ for which the maximum is achieved by ${\rm hom}(K_{d,d},H)^{n/2d}$. Here we exhibit infinitely many non-trivial triples $(n,d,H)$ for which the conjecture is true and for which the maximum is achieved by ${\rm hom}(K_{d+1},H)^{n/(d+1)}$. We also give sharp estimates for ${\rm hom}(K_{d,d},H)$ and ${\rm hom}(K_{d+1},H)$ in terms of some structural parameters of $H$. This allows us to characterize those $H$ for which ${\rm hom}(K_{d,d},H)^{1/2d}$ is eventually (for all sufficiently large $d$) larger than ${\rm hom}(K_{d+1},H)^{1/(d+1)}$ and those for which it is eventually smaller, and to show that this dichotomy covers all non-trivial $H$. Our estimates also allow us to obtain asymptotic evidence for the conjecture in the following form. For fixed $H$, for all $d$-regular $G$ we have $$ {\rm hom}(G,H)^{\frac{1}{|V(G)|}} \leq (1+o(1))\max{{\rm hom}(K_{d,d},H)^{\frac{1}{2d}}, {\rm hom}(K_{d+1},H)^{\frac{1}{d+1}}} $$ where $o(1)\rightarrow 0$ as $d \rightarrow \infty$. More precise results are obtained in some special cases.

Abstract:
At most how many (proper) q-colorings does a regular graph admit? Galvin and Tetali conjectured that among all n-vertex, d-regular graphs with 2d|n, none admits more q-colorings than the disjoint union of n/2d copies of the complete bipartite graph K_{d,d}. In this note we give asymptotic evidence for this conjecture, giving an upper bound on the number of proper q-colorings admitted by an n-vertex, d-regular graph of the form a^n b^{n(1+o(1))/d} (where a and b depend on q and where o(1) goes to 0 as d goes to infinity) that agrees up to the o(1) term with the count of q-colorings of n/2d copies of K_{d,d}. An auxiliary result is an upper bound on the number of colorings of a regular graph in terms of its independence number. For example, we show that for all even q and fixed \epsilon > 0 there is \delta=\delta(\epsilon,q) such that the number of proper q-colorings admitted by an n-vertex, d-regular graph with no independent set of size n(1-\epsilon)/2 is at most (a-\delta)^n.

Abstract:
Let $I$ be an independent set drawn from the discrete $d$-dimensional hypercube $Q_d=\{0,1\}^d$ according to the hard-core distribution with parameter $\lambda>0$ (that is, the distribution in which each independent set $I$ is chosen with probability proportional to $\lambda^{|I|}$). We show a sharp transition around $\lambda=1$ in the appearance of $I$: for $\lambda>1$, $\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}=0$ asymptotically almost surely, where ${\cal E}$ and ${\cal O}$ are the bipartition classes of $Q_d$, whereas for $\lambda<1$, $\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}$ is asymptotically almost surely exponential in $d$. The transition occurs in an interval whose length is of order $1/d$. A key step in the proof is an estimation of $Z_\lambda(Q_d)$, the sum over independent sets in $Q_d$ with each set $I$ given weight $\lambda^{|I|}$ (a.k.a. the hard-core partition function). We obtain the asymptotics of $Z_\lambda(Q_d)$ for $\lambda>\sqrt{2}-1$, and nearly matching upper and lower bounds for $\lambda \leq \sqrt{2}-1$, extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution. We also derive a long-range influence result. For all fixed $\lambda>0$, if $I$ is chosen from the independent sets of $Q_d$ according to the hard-core distribution with parameter $\lambda$, conditioned on a particular $v \in {\cal E}$ being in $I$, then the probability that another vertex $w$ is in $I$ is $o(1)$ for $w \in {\cal O}$ but $\Omega(1)$ for $w \in {\cal E}$.

Abstract:
For a simple finite graph G denote by {G \brace k} the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If E_n is the graph on n vertices with no edges then {E_n \brace k} coincides with {n \brace k}, the ordinary Stirling number of the second kind, and so we refer to {G \brace k} as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of E_n, is asymptotically normal --- essentially, as n grows, the histogram of ({E_n \brace k})_{k \geq 0}, suitably normalized, approaches the density function of the standard normal distribution. In light of Harper's result, it is natural to ask for which sequences (G_n)_{n \geq 0} of graphs is there asymptotic normality of ({G_n \brace k})_{k \geq 0}. Do and Galvin conjectured that if for each n, G_n is acylic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that G_n has no more than o(\sqrt{n/\log n}) components. Here we settle Do and Galvin's conjecture in the affirmative, and significantly extend it, replacing "acyclic" in their conjecture with "co-chromatic with a quasi-threshold graph, and with negligible chromatic number". Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in a Weyl algebra, and work of Kahn on the matching polynomial of a graph.

Abstract:
We explain the notion of the {\em entropy} of a discrete random variable, and derive some of its basic properties. We then show through examples how entropy can be useful as a combinatorial enumeration tool. We end with a few open questions.

Abstract:
Let $i_t(G)$ denote the number of independent sets of size $t$ in a graph $G$. Levit and Mandrescu have conjectured that for all bipartite $G$ the sequence $(i_t(G))_{t \geq 0}$ (the {\em independent set sequence} of $G$) is unimodal. We provide evidence for this conjecture by showing that is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph $G(n,n,p)$, and show that for any fixed $p\in(0,1]$ its independent set sequence is almost surely unimodal, and moreover almost surely log-concave except perhaps for a vanishingly small initial segment of the sequence. We obtain similar results for $p=\tilde{\Omega}(n^{-1/2})$. We also consider the problem of estimating $i(G)=\sum_{t \geq 0} i_t(G)$ for $G$ in various families. We give a sharp upper bound on the number of independent sets in an $n$-vertex graph with minimum degree $\delta$, for all fixed $\delta$ and sufficiently large $n$. Specifically, we show that the maximum is achieved uniquely by $K_{\delta, n-\delta}$, the complete bipartite graph with $\delta$ vertices in one partition class and $n-\delta$ in the other. We also present a weighted generalization: for all fixed $x>0$ and $\delta >0$, as long as $n=n(x,\delta)$ is large enough, if $G$ is a graph on $n$ vertices with minimum degree $\delta$ then $\sum_{t \geq 0} i_t(G)x^t \leq \sum_{t \geq 0} i_t(K_{\delta, n-\delta})x^t$ with equality if and only if $G=K_{\delta, n-\delta}$.