Abstract
Let \(G=(V,E)\) be a finite graph on \(n=|V|\) vertices. Numbering the vertices, we write down its adjacency matrix in an explicit form of \(n\times n\) matrix, say A.
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Notes
- 1.
Let \(\varphi _A(x)=|xI-A|\) be the characteristic polynomial of a matrix \(A\in M(n,{\mathbb {C}})\). Then \(\lambda \in {\mathbb {C}}\) is an eigenvalue of A if and only if \(\varphi _A(\lambda )=0\). In that case the algebraic multiplicity of the eigenvalue \(\lambda \) is defined to be the multiplicity of \(\lambda \) as a zero of the polynomial \(\varphi _A(x)\). While, the geometric multiplicity is defined to be the dimension of the eigenspace associated to \(\lambda \). .
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Obata, N. (2017). Spectra of Finite Graphs. In: Spectral Analysis of Growing Graphs. SpringerBriefs in Mathematical Physics, vol 20. Springer, Singapore. https://doi.org/10.1007/978-981-10-3506-7_2
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DOI: https://doi.org/10.1007/978-981-10-3506-7_2
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Publisher Name: Springer, Singapore
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Online ISBN: 978-981-10-3506-7
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