We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, ...

The eccentric connectivity index ξ is a novel distance–based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as ξ(G) = ∑ v∈V (G) deg(v) · ε(v) , where deg(v) and ε(v) denote the vertex degree and eccentricity of v , respectively. We survey some mathematical properties of this index and furthermore support ...

We present a deterministic, log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log(·) obtained by Armoni, Ta-Shma, Wigderson and Zhou [ATSWZ00]. As undirected st-connectivity is complete for the class of problems solvable by symmetric, non-deterministic, log-space computations (the class SL), this al...

In this appendix we provide mathematical models that can be used to derive connectivity graphs for our three proposed on/off interconnection networks and answer the first and second questions of our 5-step design methodology, i.e. (1) the structure of a connectivity graph determines which links are allowed to turn on/off (designate which links are on/off candidates), and (2) that the proposed r...

This paper considers the concept of restricted edge-connectivity, and relates that to the edgedegree of a connected graph. The author gives some necessary conditions for a graph whose restricted edge-connectivity is smaller than its minimum edge-degree, then uses these conditions to show some large classes of graphs, such as all connected edge-transitive graphs except a star, and all connected ...

We present a deterministic, log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log(·) obtained by Armoni, Ta-Shma, Wigderson and Zhou [ATSWZ00]. As undirected st-connectivity is complete for the class of problems solvable by symmetric, non-deterministic, log-space computations (the class SL), this al...

This work deals with a generalization of the Cartesian product of graphs, the product graph Gm ∗Gp introduced by Bermond et al. [J.C. Bermond, C. Delorme, G. Farhi, Large graphs with givendegree anddiameter II, J. Combin. Theory, Series B 36 (1984), 32–48]. The connectivity of these product graphs is approached by studying the k-restricted edge-connectivity, which is defined as the minimum numb...

How much did a network change since yesterday? How different is the wiring between Bob’s brain (a lefthanded male) and Alice’s brain (a right-handed female)? Graph similarity with known node correspondence, i.e. the detection of changes in the connectivity of graphs, arises in numerous settings. In this work, we formally state the axioms and desired properties of the graph similarity functions,...

Let G be a (multi)graph of order n and let u,v vertices G. The maximum number internally disjoint u–v paths in is denoted by κG(u,v), the edge-disjoint λG(u,v). average connectivity defined κ¯(G)=∑κG(u,v)∕n2, edge-connectivity λ¯(G)=∑λG(u,v)∕n2, where both sums run over all unordered pairs {u,v}⊆V(G). A graph called ideally connected if κG(u,v)=min{deg(u),deg(v)} for {u,v} We prove that every m...

Random walks on graphs have turned out to be a powerful tool in the design of algorithms and other applications. In particular, expander graphs, which are graphs on which random walks have particularly good properties, are extremely useful in complexity and other areas of computer science. In this chapter we study random walks on general regular graphs, leading to a the randomized logspace algo...