By Richard A. Brualdi

The ebook bargains with the numerous connections among matrices, graphs, diagraphs and bipartite graphs. the fundamental thought of community flows is constructed so one can receive lifestyles theorems for matrices with prescribed combinatorical homes and to procure quite a few matrix decomposition theorems. different chapters conceal the everlasting of a matrix and Latin squares. The e-book ends by means 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)

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1 50, pp. 1 67-1 78. B. Grone and R. Merris [ 1 987] , Algebraic connectivity of trees, Czech. Math. J. , 37, pp. 660-670. R. R. Johnson [ 1985] , Matrix Analysis, Cambridge University Press, Cambridge. B. Mohar [ 1 988] ' The Laplacian spectrum of graphs, Preprint Series Dept. Math. University E. K. Ljubljana, 26 , pp. 353-382. V. Temperiey [ 1964] , On the mutual cancellation of cluster integrals in Mayer ' s fugacity series, Proc. Phys. Soc. , 83, pp. 3-16. 6 M atchings A graph G is called bipartite provided that its vertices may be partitioned into two subsets X and Y such that every edge of G is of the form { a , b} where a is in X and b is in Y.

But the transpose of a permutation matrix is equal to its inverse. Thus A and A' are similar matrices and hence G and G' have the same spectrum. Two nonisomorphic general graphs G and G' with the same spectrum are called cospectral. 1 two pairs of cospectral graphs of orders 5 and 6 with characteristic polynomials f(>.. ) (>.. - 2)(>" + 2)>.. 3 and f(>.. ) (>.. 3 - >.. 2 - 5>" + 1 ) , respectively. 2 The Adjacency Matrix of a Graph 27 + Figure 2 . 1 . Two pairs of cospectral graphs. connected provided that every pair of vertices a and b as endpoints.

277-308. A. Schrijver[1986] , Theory of Linear and Integer Programming, Wiley, New York. D. Seymour[1980] , Decomposition of regular matroids, J. Combin. Theory, Ser. B, 28, pp. 305-359. [1982] ' Applications of the regular matroid decomposition, Colloquia Math. Soc. Janos Bolyai, No. 40 Matroid Theory, Szeged (Hungary) , pp. 345-357. T. Tutte[1958] ' A homotopy theorem for matroids, I and II, Trans. Amer. Math. , 88, pp. 144-174. N. ) [1986] , Theory of Matroids, Encyclopedia of Maths. and Its Ap­ plies.

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