Extremal problems for the p-spectral radius of graphs

The $p$-spectral radius of a graph $G\$of order $n$ is defined for any real number $p\geq1$ as $$\lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{ and }\left\vert x_{1}\right\vert ^{p}+\cdots+\left\vert x_{n}\right\vert ^{p}=1\right\} .$$ The most remarkable feature of $\lambda^{\left( p\right) }$ is that it seamlessly joins several other graph parameters, e.g., $\lambda^{\left( 1\right) }$ is the Lagrangian, $\lambda^{\left( 2\right) }$ is the spectral radius and $\lambda^{\left( \infty\right) }/2$ is the number of edges. This paper presents solutions to some extremal problems about $\lambda^{\left( p\right) }$, which are common generalizations of corresponding edge and spectral extremal problems. Let $T_{r}\left( n\right)$ be the $r$-partite Tur\'{a}n graph of order $n.$ Two of the main results in the paper are: (I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n,$ then $$\lambda^{\left( p\right) }\left( G\right) <\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) ,$$ unless $G=T_{r}\left( n\right) .$ (II) Let $r\geq2$ and $p>1.$ If $G\$is a graph of order $n,$ with $$\lambda^{\left( p\right) }\left( G\right) >\lambda^{\left( p\right) }\left( T_{r}\left( n\right) \right) ,$$ then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1,$ where $c$ is a positive number depending only on $p$ and $r.$Comment: 21 pages. Some minor corrections in v

From sea to land and beyond : new insights into the evolution of euthyneuran Gastropoda (Mollusca)

Background The Euthyneura are considered to be the most successful and diverse group of Gastropoda. Phylogenetically, they are riven with controversy. Previous morphology-based phylogenetic studies have been greatly hampered by rampant parallelism in morphological characters or by incomplete taxon sampling. Based on sequences of nuclear 18S rRNA and 28S rRNA as well as mitochondrial 16S rRNA and COI DNA from 56 taxa, we reconstructed the phylogeny of Euthyneura utilising Maximum Likelihood and Bayesian inference methods. The evolution of colonization of freshwater and terrestrial habitats by pulmonate Euthyneura, considered crucial in the evolution of this group of Gastropoda, is reconstructed with Bayesian approaches. Results We found several well supported clades within Euthyneura, however, we could not confirm the traditional classification, since Pulmonata are paraphyletic and Opistobranchia are either polyphyletic or paraphyletic with several clades clearly distinguishable. Sacoglossa appear separately from the rest of the Opisthobranchia as sister taxon to basal Pulmonata. Within Pulmonata, Basommatophora are paraphyletic and Hygrophila and Eupulmonata form monophyletic clades. Pyramidelloidea are placed within Euthyneura rendering the Euthyneura paraphyletic. Conclusion Based on the current phylogeny, it can be proposed for the first time that invasion of freshwater by Pulmonata is a unique evolutionary event and has taken place directly from the marine environment via an aquatic pathway. The origin of colonisation of terrestrial habitats is seeded in marginal zones and has probably occurred via estuaries or semi-terrestrial habitats such as mangroves

A hybrid strategy for real-time traffic signal control of urban road networks

The recently developed traffic signal control strategy known as traffic-responsive urban control (TUC) requires availability of a fixed signal plan that is sufficiently efficient under undersaturated traffic conditions. To drop this requirement, the well-known Webster procedure for fixed-signal control derivation at isolated junctions is appropriately employed for real-time operation based on measured flows. It is demonstrated via simulation experiments and field application that the following hold: 1) The developed real-time demand-based approach is a viable real-time signal control strategy for undersaturated traffic conditions. 2) It can indeed be used within TUC to drop the requirement for a prespecified fixed signal plan. 3) It may, under certain conditions, contribute to more efficient results, compared with the original TUC method