By C. Pozrikidis
An advent to Grids, Graphs, and Networks goals to supply a concise creation to graphs and networks at a degree that's available to scientists, engineers, and scholars. In a pragmatic method, the booklet provides purely the required theoretical innovations from arithmetic and considers a number of actual and conceptual configurations as prototypes or examples. the topic is well timed, because the functionality of networks is well-known as an incredible subject within the learn of complicated structures with functions in strength, fabric, and knowledge grid delivery (epitomized through the internet). The booklet is written from the sensible standpoint 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 ebook, Constantine Pozrikidis presents an unique synthesis of techniques and phrases from 3 certain fields-mathematics, physics, and engineering-and a proper software of robust conceptual apparatuses, like lattice Green's functionality, to components the place they've got not often been used. it's 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 may function a source to readers starting from undergraduates to skilled scientists.
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Additional resources for An Introduction to Grids, Graphs, and Networks
Consequently, the eigenvalues of L, denoted by λi , are either zero or positive. The sum of the eigenvalues of L is equal to the trace of L, which is equal to the trace of D, which is equal to the sum of the degrees of all nodes. 7) 0 = λ1 ≤ λ2 ≤ · · · ≤ λN , where the first eigenvalue, λ1 , is always zero. Further or all other eigenvalues may also be zero. The second smallest eigenvalue, λ2 , is of particular interest in spectral graph theory. The value of λ2 is sometimes called the algebraic connectivity of the network.
0 0 0 ··· 0 1 ⎢ ⎣ 0 0 0 ··· 1 0 1 0 0 ··· 0 1 1 0 0 .. ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ 0 ⎥ ⎥ 1 ⎦ 0 22 / / A N I N T R O D U C T I O N T O G R I D S , G R A P H S , A N D N E T W O R K S Two nonzero corner elements appear due to the periodicity condition. The degrees of all nodes are the same, di = 2 for i = 1, . . , N. 8) μn = 2 cos n–1 2π N for n = 1, . . , N. 7) are zero when s is zero or an odd integer and nonzero when s is an even integer. 9) n2 = 26, n4 = 78, n6 = 260, n8 = 910, n10 = 3276. 10) R = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ –1 1 0 ..
N. By convention, the diagonal elements of the adjacency matrix are zero. By construction, the adjacency matrix is symmetric. 1(b). 1) L = N i–1 Aij = i=1 j=i+1 Aij = i=2 j=1 1 2 N N Aij . i=1 j=1 The fraction 1/2 in front of the last double sum accounts for the inherent symmetry of A. 26 G r a p h s a n d N e t w o r k s / / 27 (a) 7 8 11 9 6 10 12 8 5 4 5 6 4 7 2 2 1 3 3 1 (b) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ A=⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥.
An Introduction to Grids, Graphs, and Networks by C. Pozrikidis