On the extremal properties of the average eccentricity

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc (G)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease $ecc (G)$. Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number, the Randi\' c index and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure

Estimating the weight of metric minimum spanning trees in sublinear time

In this paper we present a sublinear-time $(1+\varepsilon)$-approximation randomized algorithm to estimate the weight of the minimum spanning tree of an $n$-point metric space. The running time of the algorithm is $\widetilde{\mathcal{O}}(n/\varepsilon^{\mathcal{O}(1)})$. Since the full description of an $n$-point metric space is of size $\Theta(n^2)$, the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in $o(n)$ time the weight of the minimum spanning tree to within any factor. We also show that no deterministic algorithm can achieve a $B$-approximation in $o(n^2/B^3)$ time. Furthermore, it has been previously shown that no $o(n^2)$ algorithm exists that returns a spanning tree whose weight is within a constant times the optimum

CliqueStream: an efficient and fault-resilient live streaming network on a clustered peer-to-peer overlay

Several overlay-based live multimedia streaming platforms have been proposed in the recent peer-to-peer streaming literature. In most of the cases, the overlay neighbors are chosen randomly for robustness of the overlay. However, this causes nodes that are distant in terms of proximity in the underlying physical network to become neighbors, and thus data travels unnecessary distances before reaching the destination. For efficiency of bulk data transmission like multimedia streaming, the overlay neighborhood should resemble the proximity in the underlying network. In this paper, we exploit the proximity and redundancy properties of a recently proposed clique-based clustered overlay network, named eQuus, to build efficient as well as robust overlays for multimedia stream dissemination. To combine the efficiency of content pushing over tree structured overlays and the robustness of data-driven mesh overlays, higher capacity stable nodes are organized in tree structure to carry the long haul traffic and less stable nodes with intermittent presence are organized in localized meshes. The overlay construction and fault-recovery procedures are explained in details. Simulation study demonstrates the good locality properties of the platform. The outage time and control overhead induced by the failure recovery mechanism are minimal as demonstrated by the analysis.Comment: 10 page