For given non-abelian group G, the non-commuting (NC)-graph $\Gamma(G)$ is agraph with the vertex set $G$\ $Z(G)$ and two distinct vertices $x, y\in V(\Gamma)$ areadjacent whenever $xy \neq yx$. The aim of this paper is to compute the spectra ofsome well-known NC-graphs. تفاصيل المقالة

‎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. تفاصيل المقالة