Graph Theory

On Construction and Identification of Graphs by B. Weisfeiler

By B. Weisfeiler

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What is the resistance of the whole system? The ratio of the horizontal side of the original big rectangle to the vertical side, that is Since the process above is reversible, that is every squared rectangle can be obtained from some network we have an effective tool to help us in our search for squared squares. Take a connected planar graph G and turn it into an electrical network by giving each edge resistance 1. Calculate the total resistance from a vertex s to a vertex t. If this is also 1, the network may correspond to a suitably squared square.

Prove that if a rectangular parallelepiped can be decomposed into cubes then the ratios of its sides are rational. 15. Show that a cube cannot be dissected into finitely many incongruent cubes. 16. Show that in a plane graph the boundaries of bounded faces form a. cycle basis. 17. Show that the cycle space is the kernel of the map C I (G) incidence matrix B. 18. -+ Co( G) defined by the Let B' be the transpose of B. Show that the cut space is the image of the map -+ C1(G) defined by B'. Co(G) 19.

3 Vector Spaces and Matrices Associated with Graphs 35 found with the help of computers, but the first examples were found without computers by Sprague in 1939 and by Brooks, Smith, Stone and Tutte in 1940. 8 shows this tiling, due to Duijvestijn. In fact, this is the only tiling of order 21. The connection between squaring a rectangle and electrical networks gives us immediately a beautiful result first proved by Dehn in 1903. Corollary 3 tells us that if each edge has resistance 1 and a current of size 1 flows through the system then in each edge the value of the current is rational.

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