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By Rudolf Kingslake

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Extra resources for Applied optics and optical engineering,Vol.II

Example text

In a variant of this definition the edges E j are associated with independent variables x j, and aij is the sum of the variables associated with the joining edges. Then of course each variable associated with a link appears twice, once in aij and once in aji. In a further modification every such variable Xk is entered as +Xk in one of these places and as -Xk in the other. In that last case loops are ignored, that is, the diagonal elements are made all zero. We then have a skew-symmetric matrix whose determinant, if the matrix is of even order, is the square of a polynomial called a "Pfaffian".

Oystein Ore gave a factor theory for directed graphs, putting restrictions on the numbers of incoming and outgoing edges at each vertex [25]. The theory of I-factors for bipartite graphs derives from this. For a graph with a bipartition {U, V} can conveniently be regarded as directed, each edge going from U to V. Chapter XII is concerned with Petersen's Theorem, published by J. Petersen in 1891 in the guise of a result on factorizing algebraic forms. As usually stated it asserts that every cubic graph without an isthmus has a 1-factor.

He promises more applications of graph theory to set theory in Chapter XIII. More than once I have heard an eminent graph theorist express the suspicion that many theorems about topological spaces could be reduced to graph theory if some researcher would take enough trouble. But to my knowledge not much more has been done along these lines than has already been achieved by Konig. The Unendlichkeitslemma has been a powerful tool for investigating locally finite graphs. In a sense it has been too powerful, causing some of us to lose interest in that kind of graph.

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Applied optics and optical engineering,Vol.II by Rudolf Kingslake

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