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Graphs for which the least eigenvalue is minimal, I |
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| Authors: | Francis K Bell Drago&#x; Cvetkovi&#x; Peter Rowlinson Slobodan K Simi&#x; |
| Institution: | aDepartment of Computing Science and Mathematics, University of Stirling, Stirling FK9 4LA, Scotland, United Kingdom bDepartment of Mathematics, Faculty of Electrical Engineering, University of Belgrade, P.O. Box 35-54, 11120 Belgrade, Serbia cMathematical Institute SANU, Kneza Mihaila 36, 11001 Belgrade, Serbia |
| Abstract: | Let G be a connected graph whose least eigenvalue λ(G) is minimal among the connected graphs of prescribed order and size. We show first that either G is complete or λ(G) is a simple eigenvalue. In the latter case, the sign pattern of a corresponding eigenvector determines a partition of the vertex set, and we study the structure of G in terms of this partition. We find that G is either bipartite or the join of two graphs of a simple form. |
| Keywords: | Graph spectrum Largest eigenvalue Least eigenvalue Nested split graph |
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