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2 — Chordal Graphs

  • Scott McCullough
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 35)

Abstract

Let P be an undirected graph with vertices V and edges E. Fix an enumeration, {v1,v2,...,vn}, of V and let M(P) = {A ∈ Mn (ℂ)|<Aei,ej> = 0 if (vi,vj) ∉ E where ei is the standard orthonormal basis of ℂn. Mn (ℂ)+ is the set of positive semi-definite n × n matrices with complex entries. For X Mn (ℂ)+ a cone, define the order of X, denoted ord(X), to be the smallest integer k such that the elements of X of rank at most k generate X as a cone. For any set X, let Mm (X) denote m x m matrices with entries from X. It is known that a graph P is chordal if and only if ord(Mm (M(P))+) = 1 for every positive integer m, where Mm (M(P))+ = {A ∈ Mm (M(P))|A is positive semi-definite}. We characterize, in a graph theoretic way, graphs P for which ord(Mm (M(P))+) = ord(M(P)+) ≤ 2 for every positive integer m.

Keywords

Cholesky Decomposition Minimal Path Chordal Graph Simplicial Vertex Simplicial Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel 1988

Authors and Affiliations

  • Scott McCullough
    • 1
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

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