Partial Characterization of Graphs Having a Single Large Laplacian Eigenvalue
Keyword(s):
The parameter $\sigma(G)$ of a graph $G$ stands for the number of Laplacian eigenvalues greater than or equal to the average degree of $G$. In this work, we address the problem of characterizing those graphs $G$ having $\sigma(G)=1$. Our conjecture is that these graphs are stars plus a (possible empty) set of isolated vertices. We establish a link between $\sigma(G)$ and the number of anticomponents of $G$. As a by-product, we present some results which support the conjecture, by restricting our analysis to cographs, forests, and split graphs.
2012 ◽
Vol 45
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pp. 113-120
2010 ◽
Vol 108
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pp. 323-329
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1971 ◽
Vol 246
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pp. 2584-2593
1966 ◽
Vol 241
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pp. 1530-1536
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1980 ◽
Vol 255
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pp. 5468-5474
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1990 ◽
Vol 265
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pp. 17062-17069
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1976 ◽
Vol 251
(16)
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pp. 4947-4957
1973 ◽
Vol 248
(9)
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pp. 3298-3304