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LİSANSÜSTÜ ARAŞTIRMALAR SEMPOZYUMU & TANITIM GÜNLERİ 2018

On Normalized Laplacian Spectrum of Some Special Graphs

Nursemin FERATLAR, Roghayeh HAFEZIEH
Matematik ABD

ÖZET
Spectral graph theory looks at the interplay between the structure of a graph and the eigenvalues of a matrix

associated with the graph. Many interesting graphs have rich structure which can help in determining the eigenvalues
associated with some particular matrix of a graph. There are many different matrices that are considered, including
the adjacency matrix A whose entries indicate when two vertices are adjacent, Laplacian matrix L and the normalized
Laplacian matrix. Let G be an undirected graph without an isolated vertex, then the Normalized Laplacian Matrix
*L(G) is de fined as *L(G) = D^-1/2.L(G).D^-1/2 where D is a diagonal matrix whose entries are degrees of vertices of
G. The eigenvalues of *L(G) are called as the normalized Laplacian eigenvalues of G. Furthermore, if G is d-regular,
then *L(G) = I-1/d.A, herefore if we know the spectrum of the normalized Laplacian matrix, we also know the
spectrum of the adjacency matrix as well as the Laplacian. Hence any known results for regular graphs will translate
into the normalized Laplacian matrix. Cavers in [1] focused on those graphs that are co-spectral with respect to the
normalized Laplacian eigenvalues. Furthermore, he discussed properties of graphs with few normalized Laplacian
eigenvalues. Almost at the same time, he and Fallat [2] considered the energy of a simple graph with respect to its
normalized Laplacian eigenvalues, which is called the the *L-energy. Van Dam and Omidi in [4] gave a combinatorial
characterization of those graphs whose normalized Laplacian have three distinct eigenvalues. The goal of this thesis
is to investigate the spectrum of the normalized Laplacian of some special graphs, in particular, we will be looking at
understanding how eigenvalues can be computed for some types of join graphs, as a note on [3].
Anahtar Kelimeler: Normalized Laplacian Matrix, Adjacency, Spectral.

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