The signed edge domination number of a graph is an edge variant of the signed domination number. The closed neighbourhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having a common end vertex with e. Let f be a mapping of the edge set E(G) of G into the set {−1, 1}. If ∑ x∈N[e] f(x) ≥ 1 for each e ∈ E(G), then f is called a signed edge dominating function on G. The minimum of the values ∑ x∈E(G) f(x), taken over all signed edge dominating function f on G, is called the signed edge domination number of G and is denoted by γ s(G). If instead of the closed neighbourhood NG[e] we use the open neighbourhood NG(e) = NG[e] − {e}, we obtain the definition of the signed edge total domination number γ st(G) of G. In this paper these concepts are studied for trees. The number γ s(T) is determined for T being a star of a path or a caterpillar. Moreover, also γ s(Cn) for a circuit of length n is determined. For a tree satisfying a certain condition the inequality γ s(T) ≥ γ (T) is stated. An existence theorem for a tree T with a given number of edges and given signed edge domination number is proved. At the end similar results are obtained for γ st(T).
These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded operators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounded operators, and the set of Banach spaces whose set of Banach-Saks operators is complemented in its bounded operators, are all non Borel in ${\rm SB}$. At last, we give several applications of those results to non-universality results.
In this paper we investigate the effect on the multiplicity of Laplacian eigenvalues of two disjoint connected graphs when adding an edge between them. As an application of the result, the multiplicity of 1 as a Laplacian eigenvalue of trees is also considered.
For any two positive integers n and k\geqslant 2, let G(n, k) be a digraph whose set of vertices is {0, 1, ..., n − 1} and such that there is a directed edge from a vertex a to a vertex b if ak ≡ b (mod n). Let n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} be the prime factorization of n. Let P be the set of all primes dividing n and let P_{1},P_{2} \subseteq P be such that P_{1\cup P_{2}}= P and P_{1\cup P_{2}}=\emptyset . A fundamental constituent of G(n, k), denoted by G_{{P_2}}^*(n,k), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \prod\nolimits_{{p_i} \in {P_2}} {{p_i}} and are relatively prime to all primes q\in P_{1}. L. Somer and M. Křižek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic., Amplify Sawkmie, Madan Mohan Singh., and Obsahuje seznam literatury
Štukový reliéf: nalevo polonahá Koronis, drží se koruny stromu a otáčí se dozadu, na Apollóna (plášť, luk, toulec, boty), který k ní přichází., Vlček 1996#, 185-188., and Pojetí mýtu se liší od kanonického typu v tom, že se Dafné nemění ve strom, ale pouze se drží jeho větví.
Pískovcový reliéf se dvěma výjevy, nalevo Faun (kozí nohy) pronásleduje nahou ženu se syrinx v ruce (Pan a Syrinx ?), na druhé straně je Marsyás (kozí nohy) pověšený za nohy na strom, vedle stojí bezvousý muž (Apollón ?) a stahuje mu kůži., Vlček 1996#, 185-188., and Reliéf zjevně úmyslně nezobrazuje žádnou kanonickou scénu. Syrinx v ruce nahé ženy napravo je zcela neobvyklý motiv, protože tento nástroj byl vyhrazen mužům, nejčastěji je má v ruce Pan nebo Marsyás. Faun ženu evidentně pronásleduje, takže lze uvažovat o tom, zda výjev není shrnutím báje o Panovi pronásledujícím Nymfu Syrinx známé především z Ovidiových Proměn (1, 690-706). Bohové panenskou Nymfu zachránili tím, že ji proměnili v rákos, z něhož si Pan udělal syrinx. Podle jiné verze mýtu, kterou zaznamenává také Ovidius (fast. 06, 697-708) píšťalu vynalezla Minervaa a když ji odhodila, zmocnil se jí Faun, který potom vyzval Apollóna na soutěž a byl mu stažena kůže z těla. Žena nalevo však nemůže být Minerva, protože je nahá. Výjev napravo se bezpochyby vztahuje k mýtu o Faunovi, který vyzval Apollóna na hudební soutěž, srov. Exemplum: Apollón a Marsyás (Pán).
For any ordinal $\lambda $ of uncountable cofinality, a $\lambda $-tree is a tree $T$ of height $\lambda $ such that $|T_{\alpha }|<{\rm cf}(\lambda )$ for each $\alpha <\lambda $, where $T_{\alpha }=\{x\in T\colon {\rm ht}(x)=\alpha \}$. In this note we get a Pressing Down Lemma for $\lambda $-trees and discuss some of its applications. We show that if $\eta $ is an uncountable ordinal and $T$ is a Hausdorff tree of height $\eta $ such that $|T_{\alpha }|\leq \omega $ for each $\alpha <\eta $, then the tree $T$ is collectionwise Hausdorff if and only if for each antichain $C\subset T$ and for each limit ordinal $\alpha \leq \eta $ with ${\rm cf}(\alpha )>\omega $, $\{{\rm ht}(c)\colon c\in C\} \cap \alpha $ is not stationary in $\alpha $. In the last part of this note, we investigate some properties of $\kappa $-trees, $\kappa $-Suslin trees and almost $\kappa $-Suslin trees, where $\kappa $ is an uncountable regular cardinal.
Symmetrical temperature difference also known as the sap flow index (SFI) forms the basis of the Heat Field Deformation sap flow measurement and is simultaneously collected whilst measuring the sap flow. SFI can also be measured by any sap flow method applying internal continuous heating through the additional installation of an axial differential thermocouple equidistantly around a heater. In earlier research on apple trees SFI was found to be an informative parameter for tree physiological studies, namely for assessing the contribution of stem water storage to daily transpiration. The studies presented in this work are based on the comparative monitoring of SFI and diameter in stems of different species (Pseudotsuga menziesii, Picea omorika, Pinus sylvestris) and tree sizes. The ability of SFI to follow the patterns of daily stem water storage use was empirically confirmed by our data. Additionally, as the HFD multipointsensors can measure sap flow at several stem sapwood depths, their use allowed to analyze the use of stored water in different xylem layers through SFI records. Radial and circumferential monitoring of SFI on large cork oak trees provided insight into the relative magnitude and timing of the contribution of water stored in different sapwood layers or stem sectors to transpiration.
By a ternary system we mean an ordered pair $(W, R)$, where $W$ is a finite nonempty set and $R \subseteq W \times W \times W$. By a signpost system we mean a ternary system $(W, R)$ satisfying the following conditions for all $x, y, z \in W$: if $(x, y, z) \in R$, then $(y, x, x) \in R$ and $(y, x, z) \notin R$; if $x \ne y$, then there exists $t \in W$ such that $(x, t, y) \in R$. In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G, T)$, where $G$ is a connected graph and $T$ is a spanning tree of $G$. If $(G, T)$ is a ct-pair, then by the guide to $(G,T)$ we mean the ternary system $(W, R)$, where $W = V(G)$ and the following condition holds for all $u, v, w \in W$: $(u, v, w) \in R$ if and only if $uv \in E(G)$ and $v$ belongs to the $u-w$ path in $T$. By Proposition 1, the guide to a ct-pair is a signpost system. We say that a signpost system is tree-controlled if it satisfies a certain set of four axioms (these axioms could be formulated in a language of the first-order logic). Consider the mapping $\phi $ from the class of all ct-pairs into the class of all signpost systems such that $\phi ((G, T))$ is the guide to $(G, T)$ for every ct-pair $(G, T)$. It is proved in this paper that $\phi $ is a bijective mapping from the class of all ct-pairs onto the class of all tree-controlled signpost systems.
In this paper we show that in a tree with vertex weights the vertices with the second smallest status and those with the second smallest branch-weight are the same.