2-biplacement without fixed points of (p,q)-bipartite graphs

In this paper we consider $2$-biplacement without fixed points of paths and $(p,q)$-bipartite graphs of small size. We give all $(p,q)$-bipartite graphs $G$ of size $q$ for which the set $\mathcal{S}^{*}(G)$ of all $2$-biplacements of $G$ without fixed points is empty

Suppressors of selection

Inspired by recent works on evolutionary graph theory, an area of growing interest in mathematical and computational biology, we present the first known examples of undirected structures acting as suppressors of selection for any fitness value $r > 1$. This means that the average fixation probability of an advantageous mutant or invader individual placed at some node is strictly less than that of this individual placed in a well-mixed population. This leads the way to study more robust structures less prone to invasion, contrary to what happens with the amplifiers of selection where the fixation probability is increased on average for advantageous invader individuals. A few families of amplifiers are known, although some effort was required to prove it. Here, we use computer aided techniques to find an exact analytical expression of the fixation probability for some graphs of small order (equal to $6$, $8$ and $10$) proving that selection is effectively reduced for $r > 1$. Some numerical experiments using Monte Carlo methods are also performed for larger graphs.Comment: New title, improved presentation, and further examples. Supporting Information is also include

3-biplacement of bipartite graphs

Let $G=(L,R;E)$ be a bipartite graph with color classes $L$ and $R$ and edge set $E$. A set of two bijections $\{\varphi_1 , \varphi_2\}$, $\varphi_1 , \varphi_2 :L \cup R \to L \cup R$, is said to be a $3$-biplacement of $G$ if $\varphi_1(L)= \varphi_2(L) = L$ and $E \cap \varphi_1^*(E)=\emptyset$, $E \cap \varphi_2^*(E)=\emptyset$, $\varphi_1^*(E) \cap \varphi_2^*(E)=\emptyset$, where $\varphi_1^*$, $\varphi_2^*$ are the maps defined on $E$, induced by $\varphi_1$, $\varphi_2$, respectively. We prove that if $|L| = p$, $|R| = q$, $3 \leq p \leq q$, then every graph $G=(L,R;E)$ of size at most $p$ has a $3$-biplacement

Mediatic graphs

Any medium can be represented as an isometric subgraph of the hypercube, with each token of the medium represented by a particular equivalence class of arcs of the subgraph. Such a representation, although useful, is not especially revealing of the structure of a particular medium. We propose an axiomatic definition of the concept of a `mediatic graph'. We prove that the graph of any medium is a mediatic graph. We also show that, for any non-necessarily finite set S, there exists a bijection from the collection M of all the media on a given set S (of states) onto the collection G of all the mediatic graphs on S.Comment: Four axioms replaced by two; two references added; Fig.6 correcte

Resource Allocation for Downlink Multi-Cell OFDMA Cognitive Radio Network Using Hungarian Method

This paper considers the problem of resource allocation for downlink part of an OFDM-based multi-cell cognitive radio network which consists of multiple secondary transmitters and receivers communicating simultaneously in the presence of multiple primary users. We present a new framework to maximize the total data throughput of secondary users by means of subchannel assignment, while ensuring interference leakage to PUs is below a threshold. In this framework, we first formulate the resource allocation problem as a nonlinear and non-convex optimization problem. Then we represent the problem as a maximum weighted matching in a bipartite graph and propose an iterative algorithm based on Hungarian method to solve it. The present contribution develops an efficient subchannel allocation algorithm that assigns subchannels to the secondary users without the perfect knowledge of fading channel gain between cognitive radio transmitter and primary receivers. The performance of the proposed subcarrier allocation algorithm is compared with a blind subchannel allocation as well as another scheme with the perfect knowledge of channel-state information. Simulation results reveal that a significant performance advantage can still be realized, even if the optimization at the secondary network is based on imperfect network information