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

By Richard A. Brualdi

ISBN-10: 0521322650

ISBN-13: 9780521322652

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

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Example text

L) T . Hence each column of adj (F) is a multiple of u. But F = AT A is symmetric and this implies that adj (F) is also symmetric. Hence it follows that adj (A) is a multiple of J. 0 We now obtain a classical formula. 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.

Let A be the incidence matrix of a graph G on and let BL be the adjacency matrix of the line graph L(G) . Then Theorem 2 . 4 . 1 . m edges AAT = 21m + BL. 5) Proof. For i i- j the entry in the (i, j) position of BL is 1 if the edges Q:i and Q:j of G have a vertex in common and 0 otherwise. But the same conclusion holds for the entry in the (i, j) position of AA T . The main diagonal elements are as indicated. 4. 1 implies a severe restriction on the spectrum of a line graph. If>.. is an eigenvalue of the line graph L(G) , then >..

Moreover, 0 is an eigenvalue of F with corresponding eigenvector e n = ( 1 , 1 , . . , 1) T . Let Jl = Jl(G) denote the second smallest eigenvalue of F. In case n = 1 we define Jl to be o. 2 that Jl 2: 0 with equality if and only if G is a disconnected graph. Fiedler[1973] defined Jl to be the algebraic connectivity of the graph G. The algebraic connectivity of the complete graph Kn of order n is n (cf. Exercise 2, Sec. 2) . Let U be the set of all real n-tuples x such that x Tx = 1 and xT en = O.

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Combinatorial Matrix Theory (Encyclopedia of Mathematics and its Applications) by Richard A. Brualdi

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