Document Type: Research Paper


Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, Iran


‎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.


Main Subjects

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