Geodesic
Minimum semidefinite rank of outerplanar graphs and the tree cover number
The electronic journal of linear algebra, Tome 22 (2011), pp. 10-21.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $\in E(G)$ is a single edge and a ij = 0, i = j, if $ij / \in E(G)$. Let M + (G) denote the maximum multiplicity of zero as an eigenvalue of A $\in S$ + (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).
Classification : 05C50, 15A03, 15A18
Keywords: minimum rank graph, maximum multiplicity, minimum semidefinite rank, outerplanar graphs, tree cover number 
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     author = {Barioli, Francesco and Fallat, Shaun M. and Mitchell, Lon H. and Narayan, Sivaram K.},
     title = {Minimum semidefinite rank of outerplanar graphs and the tree cover number},
     journal = {The electronic journal of linear algebra},
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     year = {2011},
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Barioli, Francesco; Fallat, Shaun M.; Mitchell, Lon H.; Narayan, Sivaram K. Minimum semidefinite rank of outerplanar graphs and the tree cover number. The electronic journal of linear algebra, Tome 22 (2011), pp. 10-21. http://geodesic.mathdoc.fr/item/ELA_2011__22__a75/