Shannon Capacity and the Categorical Product

Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore $C_{\rm OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds for every pair of graphs, where $F\times G$ is the categorical product of graphs $F$ and $G$. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.

Nonlinear elliptic anisotropic problem involving non-local boundary conditions with variable exponent and graph data

We study a nonlinear elliptic anisotropic problem involving non-local conditions. We also consider variable exponent and general maximal monotone graph datum at the boundary. We prove the existence and uniqueness of weak solution to the problem.

On the Existence of Solutions of a Two-Layer Green Roof Mathematical Model

The aim of this article is to fill part of the existing gap between the mathematical modeling of a green roof and its computational treatment, focusing on the mathematical analysis. We first introduce a two-dimensional mathematical model of the thermal behavior of an extensive green roof based on previous models and secondly we analyze such a system of partial differential equations. The model is based on an energy balance for buildings with vegetation cover and it is presented for general shapes of roofs. The model considers a vegetable layer and the substratum and the energy exchange between them. The unknowns of the problem are the temperature of each layer described by a coupled system of two partial differential equations of parabolic type. The equation modeling the evolution of the temperature of the substratum also considers the change of phase of water described by a maximal monotone graph. The main result of the article is the proof of the existence of solutions of the system which is given in detail by using a regularization of the maximal monotone graph. Appropriate estimates are obtained to pass to the limit in a weak formulation of the problem. The result goes one step further from modeling to validate future numerical results.

Estimating parameters associated with monotone properties

There has been substantial interest in estimating the value of a graph parameter, i.e. of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable parameter z is the size of a random sample of a graph G required to ensure that the value of z(G) may be estimated within an error of ε with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}= \text{Forb}\!(\mathcal{F}),$ we study the sample complexity of estimating a bounded graph parameter z that, for an input graph G, counts the number of spanning subgraphs of G that satisfy$\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.

The Extremal Function and Colin de Verdière Graph Parameter

The Colin de Verdière parameter $\mu(G)$ is a minor-monotone graph parameter with connections to differential geometry. We study the conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and $\mu(G) \leq t$, then $|E(G)| \leq t|V(G)|-\binom{t+1}{2}$. We observe a relation to the graph complement conjecture for the Colin de Verdière parameter and prove the conjectured edge upper bound for graphs $G$ such that either $\mu(G) \leq 7$, or $\mu(G) \geq |V(G)|-6$, or the complement of $G$ is chordal, or $G$ is chordal.

Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition

The aim of this paper is to consider the asymptotic behavior of boundary value problems in n-dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which {1<p<n} , {n\geq 3} . In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called “strange term” in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles.

A non-smooth regularization of a forward–backward parabolic equation

In this paper, we introduce a model describing diffusion of species by a suitable regularization of a “forward–backward” parabolic equation. In particular, we prove existence and uniqueness of solutions, as well as continuous dependence on data, for a system of partial differential equations and inclusion, which may be interpreted, e.g. as evolving equation for physical quantities such as concentration and chemical potential. The model deals with a constant mobility and it is recovered from a possibly non-convex free-energy density. In particular, we render a general viscous regularization via a maximal monotone graph acting on the time derivative of the concentration and presenting a strong coerciveness property.

Manipulative Waiters with Probabilistic Intuition

For positive integersnandqand a monotone graph property$\mathcal{A}$, we consider the two-player, perfect information game WC(n,q,$\mathcal{A}$), which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client,q+ 1 edges of the complete graphKnwhich have not been offered previously. Client then chooses one of these edges which he keeps and the remainingqedges go back to Waiter. If, at the end of the game, the graph which consists of the edges chosen by Client satisfies the property$\mathcal{A}$, then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker–Chooser games) for a variety of natural graph-theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameterqis close tonand is reminiscent of phase transition phenomena in random graphs. Namely, we prove that ifq⩾ (1 + ϵ)n, then Client can avoid components of ordercϵ−2lnnfor some absolute constantc> 0, whereas forq⩽ (1 − ϵ)n, Waiter can force a giant, linearly sized component in Client's graph. In the second part of the paper, we prove that Waiter can force Client's graph to be pancyclic for everyq⩽cn, wherec> 0 is an appropriate constant. Note that this behaviour is in stark contrast to the threshold for pancyclicity and Hamiltonicity of random graphs.