We establish rigorous estimates for the Hausdorff dimension of spectra Laplacians associated with Sierpiński lattices and infinite gaskets other post-critically finite self-similar sets.

Here, we study the Fučik spectrum for $ p, q $-Laplacian$ -\Delta_p u-\Delta_q u = \alpha \left((u^+)^{p-1}+(u^+)^{q-1}\right)-\beta \left((u^-)^{p-1}+(u^-)^{q-1}\right) $in a bounded domain \Omega in \mathbb{R}^N ($ N\geq1 $), with Dirichlet boundary condition and 1<q<p<\infty $.

The use of spectral methods in graph theory has allowed for some amazing results where an arithmetic invariant (i.e., diameter, chromatic number, and so on) has been bounded and analyzed using analytic tools. The key has been to examine the spectrum of various matrices associated with graphs and to try to “hear the shape” of the graph from the spectrum. The three most widely used spectrums are ...