Geodesic
Generalizations of the strong Arnold property and the minimum number of distinct eigenvalues of a graph
Wayne Barrett1 ; Shaun Fallat2 ; H. Tracy Hall1 ; Leslie Hogben3 ; Jephian C.-H. Lin4 ; Bryan L. Shader5
1Brigham Young University
2University of Regina
3Iowa State University and American Institute of Mathematics
4Iowa State University
5University of Wyoming
The electronic journal of combinatorics, Tome 24 (2017) no. 2
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For a given graph $G$ and an associated class of real symmetric matrices whose diagonal entries are governed by the adjacencies in $G$, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdière in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with $G$, denoted by $q(G)$. The graphs for which $q(G)$ is at least the number of vertices of $G$ less one are characterized.
DOI : 10.37236/5725
Classification : 05C50, 15A18, 15A29, 15B57, 58C15
Mots-clés : inverse eigenvalue problem, strong Arnold property, strong spectral property, strong multiplicity property, Colin de Verdière type parameter, maximum multiplicity

Wayne Barrett  1   ; Shaun Fallat  2   ; H. Tracy Hall  1   ; Leslie Hogben  3   ; Jephian C.-H. Lin  4   ; Bryan L. Shader  5

1 Brigham Young University
2 University of Regina
3 Iowa State University and American Institute of Mathematics
4 Iowa State University
5 University of Wyoming 
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     author = {Wayne Barrett and Shaun Fallat and H. Tracy Hall and Leslie Hogben and Jephian C.-H. Lin and Bryan L. Shader},
     title = {Generalizations of the strong {Arnold} property and the minimum number of distinct eigenvalues of a graph},
     journal = {The electronic journal of combinatorics},
     year = {2017},
     volume = {24},
     number = {2},
     doi = {10.37236/5725},
     zbl = {1366.05065},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/5725/}
}
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Wayne Barrett; Shaun Fallat; H. Tracy Hall; Leslie Hogben; Jephian C.-H. Lin; Bryan L. Shader. Generalizations of the strong Arnold property and the minimum number of distinct eigenvalues of a graph. The electronic journal of combinatorics, Tome 24 (2017) no. 2. doi: 10.37236/5725

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