Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph
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The Department of Electrical and Computer Engineering (ECpE) contains two focuses. The focus on Electrical Engineering teaches students in the fields of control systems, electromagnetics and non-destructive evaluation, microelectronics, electric power & energy systems, and the like. The Computer Engineering focus teaches in the fields of software systems, embedded systems, networking, information security, computer architecture, etc.
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The Department of Electrical Engineering was formed in 1909 from the division of the Department of Physics and Electrical Engineering. In 1985 its name changed to Department of Electrical Engineering and Computer Engineering. In 1995 it became the Department of Electrical and Computer Engineering.
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1909-present
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- Department of Electrical Engineering (1909-1985)
- Department of Electrical Engineering and Computer Engineering (1985-1995)
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- College of Engineering (parent college)
- Department of Physics and Electrical Engineering (predecessor)
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Abstract
For a given graph G and an associated class of real symmetric matrices whose off- 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 Verdiere 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.
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This article is published as W. Barrett, S. Fallat, H. T. Hall, L. Hogben, J. C.-H. Lin, B.L. Shader. Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph. Electronic Journal of Combinatorics 24, no. 2 (2017): P2.40.
https://www.emis.de/journals/EJC/ojs/index.php/eljc/article/view/v24i2p40/0