The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs. These algorithms motivate and rely upon fascinating primitives in graph theory, in...

Linear-time algorithms for testing the planarity of a graph are well known for over 35 years. However, these algorithms are quite involved and recent publications still try to give simpler linear-time tests. We give a conceptually simple reduction from planarity testing to the problem of computing a certain construction of a 3-connected graph. This implies a linear-time planarity test. Our appr...

A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chord...

a communication net is usually represented by a linear graph g whose branches represent channels for information flow and whose vertices represent sources, sinks and switching centers. the purpose of this paper is to show that for modeling digital communication nets, conventional use of the concepts of capacity and terminal capacity is inadequate . we will present an alternative to max-flow min...

a simple and rapid method has been developed for the determination of ascorbic acid in fruit and pharmaceutical products. the method is based on the measurement of the reaction rate between ammonium molybdate and ascorbic acid in acidic media. the reaction was monitored spectrophotometrically at 800 nm by fixed-time and variable-time methods. the calibration graph was linear in the range of 5 0...

in static scheduling problem, where there is no change in situation, the challenge is that the large problems can be solved in a short time. in this paper, the static scheduling problem of automated guided vehicles in container terminal is solved by the network simplex algorithm (nsa). the algorithm is based on graph model and their performances are at least 100 times faster than traditional si...

Register allocation has long been formulated as a graph coloring problem, coloring the conflict graph with physical registers. Such a formulation does not fully capture the goal of the allocation, which is to minimize the traffic between registers and memory. Linear scan has been proposed as an alternative to graph coloring, but in essence, it can be viewed as a greedy algorithm for graph color...

We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case. 1998 ACM Subject Classification G.2.2. Graph Theory

A linear k-forest is a graph whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by lak(G), is the least number of linear k-forests needed to decompose G. In this paper, it is obtained that la4(Kn,n) = d5n/8e for n ≡ 0( mod 5).