Effective Resistance and Generalized Bejaia-Pisa Sequences on Complete Graphs with Circulant Distance Deletions

In this paper, we investigate the effective resistance on the graph $G_N^{(r)}$, which is obtained by deleting all edges corresponding to circular distances $\{\pm1, \pm2, \dots, \pm r\}$ from the complete graph $K_N$. We utilize the cyclic symmetry of the graph to diagonalize the Laplacian matrix via the discrete Fourier basis and derive a finite trigonometric sum representation for the effective resistance between two vertices at distance $\ell$. Specifically, we treat the cases $r=1$ and $r=2$ in detail and provide explicit formulas. For the case of $r=1$, we use Fourier analysis to rederive the closed form in terms of Bejaia and Pisa numbers given by Chair. For the case of $r=2$, we show that the denominator reduces to a quadratic polynomial with complex roots and introduce a generalized Bejaia-Pisa-type complex sequence. Using this sequence, we provide some closed forms for the effective resistance and various related formulas.