Download Combinatorial Matrix Theory (Encyclopedia of Mathematics and by Richard A. Brualdi PDF

By Richard A. Brualdi

The e-book offers with the various connections among matrices, graphs, diagraphs and bipartite graphs. the fundamental thought of community flows is constructed so one can receive life theorems for matrices with prescribed combinatorical houses and to acquire a number of matrix decomposition theorems. different chapters conceal the everlasting of a matrix and Latin squares. The ebook ends by way of contemplating algebraic characterizations of combinatorical homes and using combinatorial arguments in proving classical algebraic theorems, together with the Cayley-Hamilton Theorem and the Jorda Canonical shape.

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Extra resources for Combinatorial Matrix Theory (Encyclopedia of Mathematics and its Applications)

Example text

The proof is by induction on n. For n = 1 the theorem follows trivially from (i). Suppose that every column of A has two nonzero entries. Then the sum of the rows in B equals the sum of the rows in C and det(A) = 0. This assertion is also valid in case B = 0 or C = 0 . Also if some column of A has all O ' s, then det(A) = 0. Hence we are left with the case in which some column of A has exactly one nonzero entry. We expand det(A) by this column and apply the induction hypothesis. 0 The preceding result implies the following theorem of Poincare[I90I] .

Deutscher Verlag der Wissenschaften, Berlin, Academic Press, New York. D. Cvetkovic, M. Doob, I. Gutman and A. Torgasev [ 1988] , Recent Results in the Theory of Graph Spectra, Annals of Discrete Mathematics No. 36, Elsevier Science Publishers, New York. W. Haemers [ 1979] , Eigenvalue Techniques in Design and Graph Theory, Mathe­ matisch Centrum, Amsterdam. J. J. W. J . Wilson, eds. ) , Academic Press, New York, pp . 307-336. 3 The Incidence Matrix of a Graph 29 The Incidence Mat r ix of a G raph Let G be a general graph of order n with vertices a I , a2 , " " an and edges .

Theorem 2 . 5 . 3 . In the above notation we have adj ( F) = c( G) J. Proof. 2 we need only show that one cofactor of F is equal to c( G) . Let Ao denote the matrix obtained from A by removing the last column of A. It follows that det(Aif Ao) is a cofactor of F. Now let Au denote a submatrix of order n - 1 of Ao whose rows correspond to the edges in an (n - l )-subset U of the edges of G. Then by the Binet-Cauchy theorem we have det(Aif Ao) = L det(AE ) det(Au ) , where the summation is over all possible choices of U .

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