A variant on the graph parameters of Colin de Verdiere: Implications to the minimum rank of graphs
By Francesco Barioli, Shaun M. Fallat, Leslie Hogben. 2005
49 citations recorded by the source corpus.
For a given undirected graph G, the minimum rankof G is defined to be the smallest possible rankover all real symmetric matrices A whose (i, j)th entry is nonzero whenever ij and {i, j} is an edge in G. Building upon recent workinvolving maximal corank s (or nu llities) of certain symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corankof a different but related class of symmetric matrices. For this new parameter some properties analogous to the ones possessed by the existing parameters are verified. In addition, an attempt is made to apply these properties associated with ξ to learn more about the minimum rankof graphs - the original motivation.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/7e2b56a7f9fa69742850b241793b03d2f0731ca4
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In Computational Theory and Mathematics
- Leslie Hogben
- Shaun M. Fallat
- Darren D. Row
- My Huynh
- Joshua C. LaGrange
- Christina Edholm
- Samet Oymak
- Necmiye Ozay
In Geometry and Topology
Citations in the corpus
- The minimum rank of symmetric matrices described by a graph: A survey☆ (2007, methodology): “In [8] the parameter ξ(G) was introduced as the Colin de Verdière-type parameter specifically designed for use in computing minimum rank, by removing any unnecessary restrictions (the Strong Arnold Hypothesis seems to be necessary to obtain minor monotonicity).”
- SPECTRAL GRAPH THEORY AND THE INVERSE EIGENVALUE PROBLEM OF A GRAPH (2005, methodology): “The graph G shown in Figure 7 has μ(G) = ν(G) = 2 < 3 = ξ(G) [5].”
- Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph (2012, methodology): “The parameter ξ(G) was introduced in [5] as a Colin de Verdière type parameter intended for use in computing maximum nullity and minimum rank, by removing any unnecessary restrictions while preserving minor monotonicity.”