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The set of singular values of a digraph with respect to a vertex-degree based topological index is the set of all singular values of its general adjacency matrix. The spectral norm is the largest singular value and the energy the sum of the singular values. In this paper we characterize the digraphs which have exactly one singular value different from zero and the digraphs for which all singular values are equal. As a consequence, we deduce sharp upper and lower bounds for the spectral norm and energy of digraphs. In addition to being a natural generalization, proving the results in the general setting of digraphs allows us to deduce new results on graph energy.
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Abandonment of agricultural lands promotes the global expansion of secondary forests, which are critical for preserving biodiversity and ecosystem functions and services. Such roles largely depend, however, on two essential successional attributes, trajectory and recovery rate, which are expected to depend on landscape-scale forest cover in nonlinear ways. Using a multi-scale approach and a large vegetation dataset (843 plots, 3511 tree species) from 22 secondary forest chronosequences distributed across the Neotropics, we show that successional trajectories of woody plant species richness, stem density and basal area are less predictable in landscapes (4 km radius) with intermediate (40-60%) forest cover than in landscapes with high (greater than 60%) forest cover. This supports theory suggesting that high spatial and environmental heterogeneity in intermediately deforested landscapes can increase the variation of key ecological factors for forest recovery (e.g. seed dispersal and seedling recruitment), increasing the uncertainty of successional trajectories. Regarding the recovery rate, only species richness is positively related to forest cover in relatively small (1 km radius) landscapes. These findings highlight the importance of using a spatially explicit landscape approach in restoration initiatives and suggest that these initiatives can be more effective in more forested landscapes, especially if implemented across spatial extents of 1-4 km radius.
Assuntos
Ecossistema , Florestas , Biodiversidade , Árvores , PlantasRESUMO
We assume that D is a directed graph with vertex set V ( D ) = { v 1 , v n } and arc set E ( D ) . A VDB topological index φ of D is defined as φ ( D ) = 1 2 ∑ u v ∈ E ( D ) φ d u + , d v - , where d u + and d v - denote the outdegree and indegree of vertices u and v, respectively, and φ i , j is a bivariate symmetric function defined on nonnegative real numbers. Let A φ = A φ ( D ) be the n × n general adjacency matrix defined as [ A φ ] i j = φ d v i + , d v j - if v i v j ∈ E ( D ) , and 0 otherwise. The energy of D with respect to a VDB index φ is defined as E φ ( D ) = ∑ i = 1 n σ i ( A φ ) , where σ 1 ( A φ ) ≥ σ 2 ( A φ ) ≥ ⯠≥ σ n ( A φ ) ≥ 0 are the singular values of the matrix A φ . We will show that in case φ = R is the Randic index, the spectral norm of A R is equal to 1, and rank of A R is equal to rank of the adjacency matrix of D. Immediately after, we illustrate by means of examples, that these properties do not hold for most well-known VDB topological indices. Taking advantage of nice properties the Randic matrix has, we derive new upper and lower bounds for the Randic energy E R in digraphs. Some of these generalize known results for the Randic energy of graphs. Also, we deduce a new upper bound for the Randic energy of graphs in terms of rank, concretely, we show that E R ( G ) ≤ r a n k ( G ) for all graphs G, and equality holds if and only if G is a disjoint union of complete bipartite graphs.
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Let D be a digraph with set of arcs A. The Sombor index of D is defined as SO ( D ) = 1 2 ∑ u v ∈ A ( d u + ) 2 + ( d v - ) 2 , where d u + and d v - are the out-degree and in-degree of the vertices u and v of D. When D is a graph, we recover the Sombor index of graphs, a molecular descriptor recently introduced with a good predictive potential and a great research activity this year. In this paper we initiate the study of the Sombor index of digraphs. Specifically, we find sharp upper and lower bounds for SO over the class D n of digraphs with n non-isolated vertices, the classes C n and S n of connected and strongly connected digraphs on n vertices, respectively, the class of oriented trees OT ( n ) with n vertices, and the class O ( G ) of orientations of a fixed graph G.