37364536
On regular and strongly-regular self-complementary graphs
Abstract In this paper we solve 3 of the 6 problems of A. Kotzig on regular and strongly-regular self-complementary graphs, mentioned in “Graph Theory and Related Topics” edited by J.A. Bondy and U.S.R. Murty.
205492674
Spectra and energies of iterated line graphs of regular graphs
If G is a graph and L(G)=L1(G) its line graph, then Lk(G), k=2,3,…, defined recursively via Lk(G)=L(Lk−1(G)), are the iterated line graphs of G. If G is a regular graph of degree r, r≥3, then all negative eigenvalues of its iterated line graphs are equal to minus 2. The energy E(G) of a graph G is the sum of absolute values of the eigenvalues of G. If G is a regular graph of order n and of degree r≥3, then for each k≥2, E(Lk(G)) depends solely on n and r. In particular, E(L2(G))=2nr(r−2). This result enables a systematic construction of pairs of non-cospectral connected graphs of the same order, having equal energies.
33854423
Cycles in self-complementary graphs
Abstract Let G be a self-complementary graph of order p ≥ 8. It is shown that for every integer l , 3 ≤ l ≤ p − 2, G has an l -cycle. Further, if G is hamiltonian, then G is pancyclic.
11142529
Segmentation of hybrid motions via hybrid quadratic surface analysis
In this paper, we investigate the mathematical problem underlying segmentation of hybrid motions: given a series of tracked feature correspondences between two (perspective) images, we seek to segment and estimate multiple motions, possibly of different types (e.g., affine, epipolar, and homography). In order to accomplish this task, we cast the problem into a more general mathematical framework of segmenting data samples drawn from a mixture of linear subspaces and quadratic surfaces. The result is a novel algorithm called hybrid quadratic surface analysis (HQSA). HQSA uses both the derivatives and Hessians of fitting polynomials for the data to separate linear data samples from quadratic data samples. These derivatives and Hessians also lead to important necessary conditions, based on the so-called mutual contraction subspace, to separate data samples on different quadratic surfaces. The algebraic solution we derive is non-iterative and numerically stable. It tolerates moderate noise and can be used in conjunction with outlier removal techniques. We show how to solve the hybrid motion segmentation problem using HQSA, and demonstrate its performance on simulated data with noise and on real perspective images.
10307808
A tight bound on approximating arbitrary metrics by tree metrics
In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion Ω(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.
96030
Beyond the flow decomposition barrier
We introduce a new approach to the maximum flow problem. This approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities. Our approach leads to an <italic>O</italic>(min(<italic>n</italic><supscrpt>2/3</supscrpt>, <italic>m</italic><supscrpt>1/2</supscrpt>)<italic>m</italic> log(<italic>n</italic><supscrpt>2</supscrpt>/<italic>m</italic>) log <italic>U</italic>) time bound for a network with <italic>n</italic> vertices, <italic>m</italic> arcs, and integral arc capacities in the range [1, …, <italic>U</italic>]. This is a fundamental improvement over the previous time bounds. We also improve bounds for the Gomory-Hu tree problem, the parametric flow problem, and the approximate <italic>s-t</italic>  cut problem.
504793
A case for end system multicast (keynote address)
The conventional wisdom has been that IP is the natural protocol layer for implementing multicast related functionality. However, ten years after its initial proposal, IP Multicast is still plagued with concerns pertaining to scalability, network management, deployment and support for higher layer functionality such as error, flow and congestion control. In this paper, we explore an alternative architecture for small and sparse groups, where end systems implement all multicast related functionality including membership management and packet replication. We call such a scheme End System Multicast. This shifting of multicast support from routers to end systems has the potential to address most problems associated with IP Multicast. However, the key concern is the performance penalty associated with such a model. In particular, End System Multicast introduces duplicate packets on physical links and incurs larger end-to-end delay than IP Multicast. In this paper, we study this question in the context of the Narada protocol. In Narada, end systems self-organize into an overlay structure using a fully distributed protocol. In addition, Narada attempts to optimize the efficiency of the overlay based on end-to-end measurements. We present details of Narada and evaluate it using both simulation and Internet experiments. Preliminary results are encouraging. In most simulations and Internet experiments, the delay and bandwidth penalty are low. We believe the potential benefits of repartitioning multicast functionality between end systems and routers significantly outweigh the performance penalty incurred.
2039106
Distributed Clustering Algorithm for Finding Virtual Backbone in Ad Hoc Networks
An important objective in designing a protocol is to save scarce resources like energy and bandwidth, and avoid the broadcast storm problem [1]. One way of addressing these problems is by forming a small virtual backbone. In this paper, we present a distributed clustering algorithm for forming a small backbone in ad-hoc network, based on connected dominating set. The time and message complexity of the algorithm is in O(n).