NDLI: The Spectrum of the Corona of Two Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Sarma, B. K. Barik, S. Pati, S.
Copyright Year	2007
Abstract	We consider only simple graphs. Given two graphs G with vertices $1,\ldots,n$ and H, the corona $G\circ H$ is defined as the graph obtained by taking n copies of H and for each i inserting edges between the ith vertex of G and each vertex of the ith copy of H. For a connected graph G and any rregular graph H we provide complete information about the spectrum of $G\circ H$ using the spectrum of G and spectrum of H. Complete information about the Laplacian spectrum of $G\circ H$ is also provided even when H is not regular. A graph G is said to have the property R if $\frac{1}{\lambda}$ is an eigenvalue of G whenever $\lambda$ is an eigenvalue of G. Further, if $\lambda$ and $\frac{1}{\lambda}$ have the same multiplicity, for each eigenvalue $\lambda$, then it is said to have the property SR. We characterize all trees with property SR and show that such a tree is the corona product of some tree and an isolated vertex. We supply a family of bipartite graphs with property R. As an application we construct infinitely many pairs of nonisomorphic graphs with the same spectrum and the same Laplacian spectrum. We prove some results about the eigenvector related to the second smallest eigenvalue of the Laplacian matrix of $G\circ H$ and give an application.
Starting Page	47
Ending Page	56
Page Count	10
File Format	PDF
ISSN	08954801
DOI	10.1137/050624029
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	21
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2007-01-12
Access Restriction	Subscribed
Subject Keyword	property SR Trees Eigenvalues, singular values, and eigenvectors adjacency matrix Graphs and linear algebra Laplacian matrix property R corona
Content Type	Text
Resource Type	Article
Subject	Mathematics

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