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Meindl, Compact Distributed RLC Interconnect Molels Part II: Coupled Line Transient Expressions and Peak Crosstalk in Multivel Networks, IEEE Trans Electron Devices, vol 47, no 11, pp 2078 2087, Nov 2000 21 R Venkatesaran, J Davis, and J D Meindl, Compact Distributed RLC Interconnect Molels Part III: Transients in Single and Coupled Lines with Capacitive Load Termination, IEEE Trans Electron Devices, vol 50, no 4, pp 1081 1093, Apr 2003 22 R Venkatesan, J A Davis, and J D Meindl, Compact Distributed RLC Interconnect Models Part IV: Uni ed Models for Time Delay, Crosstalk and Repeater Insertion, IEEE Trans Electron Devices, vol 50, pp 1094 1102, Apr 2003 23 F Y Chang, Transient Simulation of Nonuniform Coupled Lossy Transmission Lines Characterized with Frequency-Dependent Parameters Part I: Waveform Relaxation Analysis, IEEE Trans Circuits Syst, vol 39, pp 585 603, Aug 1992 24 F Y.

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Researchers at the University of California at Berkley and at the Dana Farber Cancer Institute at Harvard Medical School have used aberration multigraphs to model chromosome aberrations [65] A multigraph is a graph that allows multiple edges between two vertices Aberration multigraphs characterize and interrelate three basic aberration elements: (1) the initial con guration of a chromosome; (2) the exchange process whose cycle structure helps to describe aberration complexity; and (3) the nal con guration of rearranged chromosomes An aberration multigraph refers in principle to the actual biophysical process of aberration formation We nd that graphical invariants provide information about the processes involved in chromosome aberrations High diameter for the multigraph corresponds to many different cycles in the exchange process, linked by the fact that they have some chromosomes in common Girth 2 in a multigraph usually corresponds to a ring formation and girth 3 to inversions.

Chang, Transient Simulation of Nonuniform Coupled Lossy Transmission Lines Characterized with Frequency-Dependent Parameters Part II: Discrete-Time Analysis, IEEE Trans Circuits Syst I, vol 39, pp 907 927, Nov 1992 25 J E Bracken, V Raghavan, and R A Rohrer, Interconnect Simulation with Asymptotic Waveform Evaluation (AWE), IEEE Trans Circuits Syst I, vol 39, pp 869 878, Nov 1992 26 L M Silveira, I M Elfadel, J K White, M Chilukuri, and K S Kundert, Ef cient Frequency-Domain Modeling and Circuit Simulation of Transmission Lines, IEEE Trans Comp Packag Manufact Technol B, vol 17, pp 505 513, Nov 1994 27 D B Kuznetsov and J E Schutt-Aine, Optimal Transient Simulation of Transmission Lines, IEEE Trans Circuits Syst I, vol 43, pp 110 121, Feb 1996 28 J S Roychowdhury, A R Newton, and D O Pederson, Algorithms for the Transient Simulation of Lossy Interconnect, IEEE Trans Computer Aided Design, vol.

Aberration multigraphs are closely related to cubic multigraphs An enormous amount is known about cubic multigraphs, mainly because they are related to work on the four-color theorem Results on cubic multigraphs suggest a mathematical classi cation of aberration multigraphs The aberration multigraph models the entire process of DNA damage, beginning with an undamaged chromosome and ending with a damaged one A relation is symmetric if a is related to b" implies b is related to a" Clearly, not all relations are symmetric If a graph models a relation that is not symmetric, then directions are assigned to the edges Such graphs are known as digraphs and networks are usually modeled by digraphs Some network applications exist in chemical graph theory [66] Since a reaction network in chemistry is a generalization of a graph, the decomposition of the associated graph re ects the submechanisms by closed directed cycles.

13, pp 96 104, Jan 1994 29 Y I Ismail and E G Friedman, Effects of Inductance on the Propagation Delay and Repeater Insertion in VLSI Circuits, IEEE Trans VLSI Syst, vol 8, pp 195 206, Apr 2000 30 Y Cao, X Huang, D Sylvester, N Chang, and C Hu, A New Analytical Delay and Noise Model for On-Chip RLC Interconnect, Proc IEDM, San Francisco, CA, 2000, pp 823 826..

If this ratio is much larger or smaller than 1, then that implies that electromigration may take place [151]. 5.1.5 General Guidelines for Testing Electromigration

A reaction mechanism is direct if no distinct mechanisms for the same reaction can be formed from a subset of the steps Although the decomposition is not unique, the set of all direct mechanisms for a reaction is a unique attribute of a directed graph Vingron and Waterman [67] utilized the techniques and concepts from electrical networks to explore applications in molecular biology A variety of novel modeling methods that exploit various areas of mathematical graph theory such as random graph theory are emerging with exciting results For more examples applications of graphs in molecular biology, see the work by Boncher et al [68]..

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