2 — Chordal Graphs

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


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.


Cholesky Decomposition Minimal Path Chordal Graph Simplicial Vertex Simplicial Pair 
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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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