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Non-negative Matrices and Digraphs

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Encyclopedia of Complexity and Systems Science

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The interplay between Matrix Theory and Graph Theory isfascinating and fruitful, e.?g. [4,5,8,14,18,21,25,31,32]. Here we only deal with theinterplay between nonnegative matrices and digraphs.

Digraphs are of great importance in Computer Science, SocialSciences and Natural Sciences, see [1,2,55]. Nonnegative matrices haveapplications to a variety of disciplines, such as numericalanalysis, probability, game theory, economics, optimization, dynamicalsystems, and data mining, see [3,6,7,39]. These two important subjects,digraphs and nonnegative matrices, are also stronglyconnected. (Nonnegative) matrices are a natural way to describedigraphs and conversely, digraphs describe the zero-nonzerostructure of a matrix. This survey concentrates on the meetingpoints of the two subjects.

The theory of nonnegative matrices is a hundred years old,and digraphs have accompanied it almost since its beginning. Thiscontinues to be true today, when modern applications stimulate...

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Abbreviations

Nonnegative matrix:

A  nonnegative matrix is a matrix whose entries are realnonnegative numbers. The matrix is positive if all its elements arepositive.

Digraph:

A  digraph (directed graph) G consists ofa finite set V anda set E of ordered pairsof elements of V. Theelements of V are calledvertices and thoseof E are called arcs. It is often represented graphically bypoints as the vertices of G, and arrows between these pointsas the arcs.

Irreducible matrix:

A  reducible matrix is a matrix which is either the \( { 1\times 1 }\) zero matrix, or a squarematrix that has a zero submatrix on complementary sets of rowsand columns. A square matrix which is not reducible is irreducible .

Strongly connected digraph:

A  stronglyconnecteddigraph is a digraph in whichthere is a walk from every vertex to every othervertex.

(Directed) walk:

A directed walk (walk) in a digraph isa sequence of arcs of the digraph such that each arc starts atthe end vertex of itspredecessor.

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Author information

Authors and Affiliations

  1. Technion – Israel Institute ofTechnology, Haifa, Israel

    Abraham Berman

  2. Emek Yezreel College, Emek Yezreel, Israel

    Naomi Shaked-Monderer

Authors
  1. Abraham Berman
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  2. Naomi Shaked-Monderer
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Berman, A., Shaked-Monderer, N. (2009). Non-negative Matrices and Digraphs. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_368

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