## Abstract

Let G = (V,E) be a multigraph with no loops on the vertex set V = {1, 2,..., n}. Define S_{+}(G) as the set of symmetric positive semidefinite matrices A = [a_{ij}] with a_{ij} ≠ 0, i ≠ j, if ij ∈ E(G) is a single edge and a_{ij} = 0, i ≠ j, if ij /∉ E(G). Let M_{+}(G) denote the maximum multiplicity of zero as an eigenvalue of A ∈ S_{+}(G) and mr_{+}(G) = |G|-M_{+}(G) denote the minimum semidefinite rank of G. The tree cover number of a multigraph G, denoted T(G), is the minimum number of vertex disjoint simple trees occurring as induced subgraphs of G that cover all of the vertices of G. The authors present some results on this new graph parameter T(G). In particular, they show that for any outerplanar multigraph G, M_{+}(G) = T(G).

Original language | English |
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Pages (from-to) | 10-21 |

Number of pages | 12 |

Journal | Electronic Journal of Linear Algebra |

Volume | 22 |

DOIs | |

State | Published - 2011 |

## Keywords

- Maximum multiplicity
- Minimum rank graph
- Minimum semidefinite rank
- Outer- planar graphs
- Tree cover number