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This paper investigates the asymptotic nature of graph spectra when some edges of a graph are subdivided sufficiently many times. In the special case where all edges of a graph are subdivided, we find the exact limits of the ▫$k$▫-th largest and ▫$k$▫-th smallest eigenvalues for any fixed ▫$k$▫. Giv ...

For a Hermitian matrix ▫$A$▫ of order ▫$n$▫ with eigenvalues ▫$\lambda_1(A)\ge \cdots\ge \lambda_n(A)$▫, define ▫$\mathcal{E}_p^+(A)=\sum_{\lambda_i > 0} \lambda_i^p(A), \quad \mathcal{E}_p^-(A)=\sum_{\lambda_i<0} |\lambda_i(A)|^p$▫, to be the positive and the negative ▫$p$▫-energy of ▫$A$▫, respect ...