Normalized laplacian spectrum of two new types of join graphs
محورهای موضوعی : History and biography
M. Hakimi-Nezhaad
1
(Department of Mathematics, Faculty of Science, Shahid Rajaee
Teacher Training University, Tehran, 16785-136, Iran)
M. Ghorbani
2
(Department of Mathematics, Faculty of Science, Shahid Rajaee
Teacher Training University, Tehran, 16785-136, Iran)
کلید واژه: Join of graphs, normalized Laplacian eigenvalue, integral eigenvalue,
چکیده مقاله :
Let $G$ be a graph without an isolated vertex, the normalized Laplacian matrix $\tilde{\mathcal{L}}(G)$is defined as $\tilde{\mathcal{L}}(G)=\mathcal{D}^{-\frac{1}{2}}\mathcal{L}(G)\mathcal{D}^{-\frac{1}{2}}$, where $\mathcal{D}$ is a diagonal matrix whose entries are degree of vertices of $G$. The eigenvalues of$\tilde{\mathcal{L}}(G)$ are called as the normalized Laplacian eigenvalues of $G$. In this paper, we obtain the normalized Laplacian spectrum of two new types of join graphs. In continuing, we determine the integrality of normalized Laplacian eigenvalues of graphs. Finally, the normalized Laplacian energy and degree Kirchhoff index of these new graph products are derived.
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