# 2 — Chordal Graphs

## Abstract

Let P be an undirected graph with vertices V and edges E. Fix an enumeration, {v_{1},v_{2},...,v_{n}}, of V and let M(P) = {A ∈ M_{n} (ℂ)|<Ae_{i},e_{j}> = 0 if (v_{i},v_{j}) ∉ E where e_{i} is the standard orthonormal basis of ℂ^{n}. M_{n} (ℂ)^{+} is the set of positive semi-definite n × n matrices with complex entries. For X ⊂ M_{n} (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 M_{m} (X) denote m x m matrices with entries from X. It is known that a graph P is chordal if and only if ord(M_{m} (M(P))^{+}) = 1 for every positive integer m, where M_{m} (M(P))^{+} = {A ∈ M_{m} (M(P))|A is positive semi-definite}. We characterize, in a graph theoretic way, graphs P for which ord(M_{m} (M(P))^{+}) = ord(M(P)^{+}) ≤ 2 for every positive integer m.

## Keywords

Cholesky Decomposition Minimal Path Chordal Graph Simplicial Vertex Simplicial Pair## Preview

Unable to display preview. Download preview PDF.

## References

- [AHMR]Agier, J., Helton, J.W., McCullough, S., and Rodman, L., Positive Semi-definite Matrices with a Given Sparsity Pattern, to appear in Linear Algebra and its Applications.Google Scholar
- [GJSW]Grone, R., Johnson, C, Sa, E.M., and Wolkowitz, H., Positive Completions of Partial Hermytian Matrices, Linear Algebra and its Applications 58 (1984), 109–124.CrossRefGoogle Scholar
- [PPS]Paulsen, V.l., Power, S.C., and Smith, R.R, Schur Products and Matrix Completions, preprint.Google Scholar
- [PPW]Paulsen, V.l., Power, S.C., and Ward, J.P., Semi-Discreteness and Dilation Theory for Nest Algebras, preprint.Google Scholar
- [R]Rose, D.J., A Graph Theoretic Study of the Numerical Solution of Linear Equations, Graph Theory and Computing, R. Reed Editor, Academic Press, New York, 1973, 183–217.Google Scholar
- [G]Golumbic, M.C., Algorithmic graph theory and perfect graphs, Academic Press, New York, 1980.Google Scholar
- [HPR]Helton, J.W., Pierce, S., and Rodman, L., The ranks of extremal positive semi-definite matrices with a given sparsity pattern, preprint.Google Scholar