ABSTRACT.It is possible to determine the optimum time for permanence of vegetative propagules (mini-cuttings) inside a greenhouse for rooting, and this value can be used to optimize the structure of the nursery. The aim of this study was to determine the dynamics of adventitious rooting in mini-cuttings of three clones of Eucalyptus benthamii x Eucalyptus dunnii. Sprouts of H12, H19 and H20 clones were collected from mini-stumps that were planted in gutters containing sand and grown in a semi-hydroponic system. The basal region of the mini-cuttings was immersed in 2,000 mg L -1 indole-3-butyric acid (IBA) solution for 10 seconds. The rooting percentage of the mini-cuttings, the total length of the root system and the rooting rate per mini-cutting were also evaluated at 0 (time of planting), 7, 14, 21, 28, 35, 42, 49 and 56 days. We used logistic and exponential regression to mathematically model the speed of rhizogenesis. The rooting percentage was best represented as a logistic model, and the total length of the root system was best represented as an exponential model. The clones had different speeds of adventitious rooting. The optimum time for permanence of the mini-cuttings inside the greenhouse for rooting was between 35 and 42 days, and varied depending on the genetic material.Keywords: vegetative propagation, clonal forestry, organogenesis, mathematical modeling, rhizogenesis, clonal mini-garden. Dinamica de enraizamento adventício em miniestacas de Eucalyptus benthamii x Eucalyptus dunniiRESUMO. O tempo ideal de permanência de propágulos vegetativos (miniestacas) no interior da casa de vegetação para a rizogênese é possível de ser determinado matematicamente, o que pode otimizar as instalações do viveiro. O objetivo deste estudo foi determinar a dinâmica de enraizamento de miniestacas de três clones de Eucalyptus benthamii x Eucalyptus dunnii. Brotações dos clones H19, H12 e H20 foram coletadas de minicepas plantadas em canaletão com areia e cultivadas sob sistema semi-hidropônico. A região basal da miniestaca foi imersa em solução de 2.000 mg L -1 de ácido indolbutírico (AIB) por 10 segundos. A porcentagem de enraizamento de miniestacas, o comprimento total de raízes e a taxa de enraizamento por miniestaca foram avaliados a cada sete dias (0 -instante da estaquia, 14, 21, 28, 35, 42, 49 e 56 dias). Foram utilizadas as regressões logística e exponencial para a modelagem matemática da velocidade dos processos rizogênicos. O modelo logístico teve o melhor ajuste para o percentual de enraizamento, e o modelo exponencial para o comprimento total do sistema radicular. Os clones apresentaram diferentes velocidades de enraizamento. O tempo ideal de permanência das miniestacas dentro da casa de vegetação para o enraizamento está entre 35 a 42 dias, podendo variar em função do material genético.Palavras-chave: propagação vegetativa, silvicultural clonal, organogênese, modelagem matemática, rizogênese, minijardim clonal.

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There exist pentadiagonal matrices which are diagonally similar to symmetric matrices. In this work we describe explicitly the diagonal matrix that gives this transformation for certain pentadiagonal matrices. We also consider particular classes of pentadiagonal matrices and obtain recursive formulas for the characteristic polynomial and explicit formulas for their eigenvalues.

Let G be a graph on n vertices and G? its complement. In this paper, we prove a Nordhaus-Gaddum type inequality to the second largest eigenvalue of a graph G, ?2(G), ?2(G) + ?2(G?) ? -1 + ? n2/2-n+1, when G is a graph with girth at least 5. Also, we show that the bound above is tight. Besides, we prove that this result holds for some classes of connected graphs such as trees, k-cyclic, regular bipartite and complete multipartite graphs. Based on these facts, we conjecture that our result holds to any graph.

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Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α ∈ [0, 1], Nikiforov [21] and Wang et al. [26] defined the matrices Aα(G) and Lα(G), respectively, as Aα(G) = αD(G)+(1−α)A(G) and Lα(G) = αD(G)+(α − 1)A(G). In this paper, we obtain some relationships between the eigenvalues of these matrices for some families of graphs, a part of the Aα and Lα-spectrum of the spider graphs, and we display the Aα and Lα-characteristic polynomials when their set of vertices can be partitioned into subsets that induce regular subgraphs. Moreover, we determine some subfamilies of spider graphs that are cospectral with respect to these matrices.

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The eigenvalues of a graph are those of its adjacency matrix. Recently, Cioabȃ, Haemers and Vermette characterized all graphs with all but two eigenvalues equal to −2 and 0. In this article, we extend their result by characterizing explicitly all graphs with all but two eigenvalues in the interval [−2, 0]. Also, we determine among them those that are determined by their spectrum.