# Applications of Algebraic Topology: Graphs and Networks The by S. Lefschetz

By S. Lefschetz

This monograph relies, partly, upon lectures given within the Princeton college of Engineering and utilized technological know-how. It presupposes normally an straightforward wisdom of linear algebra and of topology. In topology the restrict is size in general within the latter chapters and questions of topological invariance are rigorously shunned. From the technical perspective graphs is our in basic terms requirement. although, later, questions significantly concerning Kuratowski's classical theorem have demanded an simply supplied therapy of 2-complexes and surfaces. January 1972 Solomon Lefschetz four advent The learn of electric networks rests upon initial thought of graphs. within the literature this thought has consistently been handled by means of certain advert hoc tools. My objective here's to teach that truly this idea is not anything else than the 1st bankruptcy of classical algebraic topology and should be very advantageously handled as such through the well-known tools of that technology. half I of this quantity covers the subsequent flooring: the 1st chapters current, typically in define, the wanted uncomplicated components of linear algebra. during this half duality is handled a bit extra generally. In bankruptcy III the merest parts of basic topology are mentioned. Graph thought right is roofed in Chapters IV and v, first structurally after which as algebra. bankruptcy VI discusses the purposes to networks. In Chapters VII and VIII the weather of the idea of 2-dimensional complexes and surfaces are presented.

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**Example text**

A(-bi) -1 (-bi) = ••• = and we have For Note that b h. (-bi> A 0, We n. Figure 17. Since where Dl (except Zl is the nucleus of 6: Cl + Co we have is the space of the one-chains with non-zero boundary dl Since 0) . Co = ZO' the analogue of Dl for the space Co is DO = O. Let 6D 1 = FO' The space are boundaries of one-chains. FO Since consists of zero-chains which 6 has no nucleus as an 46 V. operation Dl "" F 0' the two spaces Dl and F0 GRAPH ALGEBRA are isomorphic. We also have (1. " 2. HO = CO/FO (recall that according to Chapter II, Section 5 signifies that subspaces have been replaced by isomorphs).

1) any open set containing if2. of the point x A. Then the circular R, while a line, an ellipse R. Cells and spheres. These are two figures of frequent occurrence later and contributing important topological types. A zero-cell is just a point. For n > 0 we have: an open n-cell is the homeomorph of the Euclidean set x'x Replacing < by ~ yields the closed n-cell. The (n-l)-sphere is the homeomorph of the set of sented by x'x = 1. repre- The zero-sphere consists of just two points. A one-cell is called ~, a closed one-cell is called closed arc.

Then ih arrives at or leaves o. The obvious conclusion is that the algebraic sum of the currents arriving at the node nk is VI. 52 Hence Kirchoff's first law means that chain i'b is a cycle. 3) =0 or that the The reformulation of the law is therefore: A current distribution is any vector (1. 2) ELECTRICAL NETWORKS i 'b which is a G. Second Kirchoff law (voltage law). is any cochain vector v'b* A voltage distribution such that the algebraic sum of the voltages along any loop is zero. Reformulation of the second law.