Abstract For a connected graph, the first Zagreb eccentricity index $\xi _{1}$ ? 1 is defined as sum of squares eccentricities all vertices, and second _{2}$ 2 products pairs adjacent vertices. In this paper, we mainly present different universal approa...

Many applications like pointer analysis and incremental compilation require maintaining a topological ordering of the nodes of a directed acyclic graph (DAG) under dynamic updates. All known algorithms for this problem are either only analyzed for worst-case insertion sequences or only evaluated experimentally on random DAGs. We present the first averagecase analysis of incremental topological ...

We present two online algorithms for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm takes O(m) amortized time per arc and our second algorithm takes O(n/m) amortized time per arc, where n is the number of vertices and m is the total number of arcs. For sparse graphs, our O(m) bound improves the best p...

Topological relativity is a concept of interest in geographic information theory. One way of assessing the importance of topology in spatial reasoning is to analyze commonplace terms from natural language relative to conceptual neighborhood graphs, the alignment structures of choice for topological relations. Sixteen English-language spatial prepositions for regionregion relations were analyzed...

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (τ -covers). We deal with two types of combinatorial questions which arise from this study. (1) Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. (2) We study the additiv...

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (τ -covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivit...

We begin with trying to find the shortest path in a directed acyclic graph (DAG). Recall that a DAG has directed edges and contains no cycles. Recall the definition of a topological sort: Definition 1. Let G = (V,E) be a graph. Let v1, . . . ,vn be an ordering of the vertices in V . v1, . . . ,vn are in topologically sorted order if for all edges (vi,v j) ∈ E, i < j. If G is a DAG, then it is a...

We present an on-line algorithm for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created. Our algorithm takes O(m) amortized time per arc, where m is the total number of arcs. For sparse graphs, this bound improves the best previous bound by a logarithmic factor and is tight to within a constant factor for a natural class of al...

Many applications like pointer analysis and incremental compilation require maintaining a topological ordering of the nodes of a directed acyclic graph (DAG) under dynamic updates. All known algorithms for this problem are either only analyzed for worst-case insertion sequences or only evaluated experimentally on random DAGs. We present the first average-case analysis of online topological orde...

Active systems, from bacterial suspensions to cellular monolayers, are continuously driven out of equilibrium by local injection of energy from their constituent elements and exhibit turbulent-like and chaotic patterns. Here we demonstrate both theoretically and through numerical simulations, that the crossover between wet active systems, whose behaviour is dominated by hydrodynamics, and dry a...