The cubical dimension of a graph $G$ is the smallest dimension of a hypercube into which $G$ is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with $2^n$ vertices, $n\geq 1$, is $n$. The 2-rooted complete binary tree of depth $n$ is obtained from two copies of the complete binary tree of depth $n$ by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice a 2-rooted complete binary tree and prove that every such balanced tree satisfies the conjecture of Havel.
We compared variation in sun-canopy leaf anatomy, morphology and photosynthetic rates of coexisting woody species (trees and lianas) in an 8-year-old secondary forest (SF) and mature forest (MF) in the wet season in Xishuangbanna, SW China. Variability of leaf traits of 66 species within growth-form groups in each forest was quantified using coefficients of variation (CV). For the mean values, the woody species in the SF had significantly higher leaf thickness and stomatal density, but lower nonmesophyll/mesophyll ratios than those in the MF. The average leaf area and leaf mass area (LMA) in the studied woody species did not change greatly during the successional process, but differed significantly between the growth forms, with trees having higher values than lianas. The light-saturated photosynthetic rate per unit leaf area (Aa) of the woody species in the SF ranged from 11.2 to 34.5 μmol m-2 s-1, similarly to pioneer tree species from literature data in southeast Asia. The Aa and photosynthetic nitrogen-use efficiency (PNUE) were significantly higher than those in the MF; whereas Aa in the MF ranged between 9 to 21 μmol m-2 s-1, with similar values between lianas and trees. For all woody species in both SF and MF, there were no significant differences in the average values of the CV of all measured variables for both lianas and trees. However, considerable variation in leaf anatomy, morphology, and photosynthetic rates within both growth forms and forests existed, as well as substantial variation in leaf size and stomatal density. We concluded that the tropical woody species formed a heterogeneous functional group in terms of leaf morphology and physiology in both secondary and mature forests. and L. Han ... [et al.].
Nástropní freska v oválném štukovém rámu. Figurální scéna v krajině, na terase zámku, v dálce vchod do podsvětí, vpravo Proserpína trhá ze stromu ovoce, uprostřed v popředí výr - Askalafos., Poche 1977#, s. 318., and Autor se inspiroval ilustrovaným vydáním Ovidiových Metamorfóz. Syn boha Acherontu a podsvětní nymfy Orfny byl proměněn ve výra.
The factors that affect the local distribution of the invasive Harmonia axyridis are not yet completely resolved. Hypotheses predicting positive and independent effects of prey abundance and degree of urbanization on the adult abundance of this species in Central Europe were tested. Populations of H. axyridis were sampled in a period when it was most abundant, by sweeping lime trees (Tilia spp.) at 28 sites along a 20 km transect across urban (western Prague) and surrounding rural areas. The sites differed in aphid abundance (number of Eucallipterus tiliae per 100 sweeps) and degree of urbanization (percentage of the surrounding area within a 500 m radius covered by impervious human constructions). Multiple linear regression analysis of log-transformed data revealed that abundance of H. axyridis (number of adults per 100 sweeps) increased significantly with both aphid abundance (P = 0.015) and urbanization (P = 0.045). The positive relationship between degree of urbanization and abundance of H. axyridis was thus not a side effect of variation in aphid abundance, which was also greater in urban than rural areas. The effect of urbanization might constrict the habitat available to H. axyridis and force this species to aggregate in urban green "refugia". These results point to a plurality of factors that determine coccinellid abundance at natural sites.
The maximum nullity over a collection of matrices associated with a graph has been attracting the attention of numerous researchers for at least three decades. Along these lines various zero forcing parameters have been devised and utilized for bounding the maximum nullity. The maximum nullity and zero forcing number, and their positive counterparts, for general families of line graphs associated with graphs possessing a variety of specific properties are analysed. Building upon earlier work, where connections to the minimum rank of line graphs were established, we verify analogous equations in the positive semidefinite cases and coincidences with the corresponding zero forcing numbers. Working beyond the case of trees, we study the zero forcing number of line graphs associated with certain families of unicyclic graphs., Shaun Fallat, Abolghasem Soltani., and Obsahuje seznam literatury
Na východní a jihozápadní fasádě budovy zámeckého zahradnictví reliéfy s antikizujícími motivy. Na jihozápadní straně: Nahý mladík, vlající plášť, se džbánem - Hylas, chce nabrat vodu z potoka, ze kterého vyskakují tři nymfy, které jej zajímají., Samek 1999#, 446-450., and Starší zahradníkův dům byl rekonstruován v roce 1844, 1884-1885. Reliéf je variací na dílo B. Thornvaldsena z roku 1833.
Na východní a jihozápadní fasádě budovy zámeckého zahradnictví reliéfy s antikizujícími motivy. Na jihozápadní straně: Tři Grácie odzbojují Amora - jedna z nich se rukou dotýká hrotu šípu, na zemi toulec se šípy, Amor drží květinový feston, který za druhý konec drží jedna ze tří Gracií., Samek 1999#, 446-450., and Starší zahradníkův dům byl rekonstruován v roce 1844, 1884-1885. Reliéfy jsou kopiemi Thornvaldsenových děl.
The concept of the $k$-pairable graphs was introduced by Zhibo Chen (On $k$-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter $p(G)$, called the pair length of a graph $G$, as the maximum $k$ such that $G$ is $k$-pairable and $p(G)=0$ if $G$ is not $k$-pairable for any positive integer $k$. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved are restricted to be trees. That is, we characterize the trees $G$ with $p(G)=1$ and prove that $p(G \square H)=p(G)+p(H)$ when both $G$ and $H$ are trees.
A proper coloring c : V (G) → {1, 2, . . . , k}, k ≥ 2 of a graph G is called a graceful k-coloring if the induced edge coloring c ′ : E(G) → {1, 2, . . . , k − 1} defined by c ′ (uv) = |c(u) − c(v)| for each edge uv of G is also proper. The minimum integer k for which G has a graceful k-coloring is the graceful chromatic number χg(G). It is known that if T is a tree with maximum degree ∆, then χg(T ) ≤ ⌈ 5⁄3∆⌉ and this bound is best possible. It is shown for each integer ∆ ≥ 2 that there is an infinite class of trees T with maximum degree ∆ such that χg(T ) = ⌈ 5⁄3 ∆⌉. In particular, we investigate for each integer ∆ ≥ 2 a class of rooted trees T∆,h with maximum degree ∆ and height h. The graceful chromatic number of T∆,h is determined for each integer ∆ ≥ 2 when 1 ≤ h ≤ 4. Furthermore, it is shown for each ∆ ≥ 2 that lim h→∞ χg(T∆,h) = ⌈ 5⁄3∆⌉.
In the first part, we assign to each positive integer $n$ a digraph $\Gamma (n,5),$ whose set of vertices consists of elements of the ring $\mathbb {Z}_n=\{0,1,\cdots ,n-1\}$ with the addition and the multiplication operations modulo $n,$ and for which there is a directed edge from $a$ to $b$ if and only if $a^5\equiv b\pmod n$. Associated with $\Gamma (n,5)$ are two disjoint subdigraphs: $\Gamma _1(n,5)$ and $\Gamma _2(n,5)$ whose union is $\Gamma (n,5).$ The vertices of $\Gamma _1(n,5)$ are coprime to $n,$ and the vertices of $\Gamma _2(n,5)$ are not coprime to $n.$ In this part, we study the structure of $\Gamma (n,5)$ in detail. \endgraf In the second part, we investigate the zero-divisor graph $G(\mathbb {Z}_n)$ of the ring $\mathbb {Z}_n.$ Its vertex- and edge-connectivity are discussed.