By C. Pozrikidis
An creation to Grids, Graphs, and Networks goals to supply a concise advent to graphs and networks at a degree that's available to scientists, engineers, and scholars. In a pragmatic process, the e-book offers merely the mandatory theoretical innovations from arithmetic and considers numerous actual and conceptual configurations as prototypes or examples. the topic is well timed, because the functionality of networks is well-known as a major subject within the examine of complicated platforms with purposes in power, fabric, and data grid delivery (epitomized through the internet). The booklet is written from the sensible viewpoint of an engineer with a few historical past in numerical computation and utilized arithmetic, and the textual content is observed via a number of schematic illustrations all through.
In the e-book, Constantine Pozrikidis presents an unique synthesis of strategies and phrases from 3 unique fields-mathematics, physics, and engineering-and a proper software of robust conceptual apparatuses, like lattice Green's functionality, to parts the place they've got hardly ever been used. it truly is novel in that its grids, graphs, and networks are attached utilizing techniques from partial differential equations. This unique fabric has profound implications within the examine of networks, and should function a source to readers starting from undergraduates to skilled scientists.
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Extra info for An Introduction to Grids, Graphs, and Networks
Nodes and links are labeled arbitrarily in this illustration. 1(b). 1) k = [ 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4 ], l = [ 2, 3, 4, 1, 6, 7, 8, 5, 6, 7, 8, 1 ]. 1(c). 1 (a) Illustration of a cubic network and its projection on the plane. (b) The adjacency matrix and (c) the oriented incidence matrix. The cubic network consists of N = 8 nodes (vertices) connected by L = 12 links (edges). Nodes and links are labeled arbitrarily in this example. where I is the 8 × 8 identity matrix. 3) L = ⎢ ⎢ 0 0 0 –1 3 –1 0 –1 ⎢ ⎢ –1 0 0 0 –1 3 –1 0 ⎢ ⎣ 0 –1 0 0 0 –1 3 –1 0 0 –1 0 –1 0 –1 3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥.
N contain useful information on the structure of the graph. Although the layout of the adjacency matrix depends on the node labeling, the eigenvalues are independent of node labeling. Since A is symmetric, it has real eigenvalues and a complete set of orthogonal eigenvectors. The sum of the eigenvalues of A is equal to the trace of A, which is zero. A necessary but not sufficient condition for two graphs to be isomorphic is that the spectra of the corresponding adjacency matrices are identical.
9) L = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ N–1 –1 –1 .. –1 N–1 –1 .. –1 –1 N–1 .. ··· ··· ··· .. –1 –1 –1 .. –1 –1 –1 .. –1 –1 –1 .. –1 –1 –1 –1 –1 –1 –1 –1 –1 ··· ··· ··· N–1 1 –1 –1 N–1 –1 –1 –1 N–1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. 10) Lk = N k–1 L, for any positive integer, k. 11) λ1 = 0, λn = N for n = 2, . . , N. 12) ui 1 = √ exp(–i i αn ), N for n = 1, . . 13) αn = n–1 2π N and i is the imaginary unit, i2 = –1. Because of the pronounced multiplicity of the eigenvalues, other sets of eigenvectors can be chosen. , [27, 30]).