On the graph complement conjecture for minimum rank
نویسندگان
چکیده
منابع مشابه
On the graph complement conjecture for minimum rank
The minimum rank of a graph has been an interesting and well studied parameter 6 investigated by many researchers over the past decade or so. One of the many unresolved questions on 7 this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This 8 conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank 9 of its comp...
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The minimum rank of a graph has been an interesting and well studied parameter 6 investigated by many researchers over the past decade or so. One of the many unresolved questions on 7 this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This 8 conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank 9 of its comp...
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Given a graph G, a real orthogonal representation of G is a function from its set of vertices to R such that two vertices are mapped to orthogonal unit vectors if and only if they are not neighbors. The minimum vector rank of a graph is the smallest dimension d for which such a representation exists. This quantity is closely related to the minimum semidefinite rank of G, which has been widely s...
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For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n × n matrices A whose (i, j)th entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound ...
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Orthogonal representations are used to show that complements of certain sparse graphs have (positive semidefinite) minimum rank at most 4. This bound applies to the complement of a 2-tree and to the complement of a unicyclic graph. Hence for such graphs, the sum of the minimum rank of the graph and the minimum rank of its complement is at most two more than the order of the graph. The minimum r...
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ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2012
ISSN: 0024-3795
DOI: 10.1016/j.laa.2010.12.024