This research note provides a computational characterization of cycle-length distributions in random functional graphs. By simulating an ensemble of mappings on N=1,000 nodes, we quantify the deviation of finite-size statistics from asymptotic expectations.
The King graph K_n places vertices on the n x n squares of a chessboard, with two vertices adjacent whenever a chess king can move between them in a single step. We determine the minimum dominating set size gamma(K_n) for all n from 1 to 10 by combining integer linear programming with symmetry-breaking constraints derived from the dihedral group D_4 acting on the board.
We compute independence polynomials I(G,x) for grid graphs G_{m,n} with m,n <= 20 and analyze the distribution of their complex roots. For fixed strip width m and increasing length n, we prove that the roots of I(G_{m,n}, x) converge to an algebraic curve in the complex plane that is a cardioid whose parametric equation depends on the spectral radius of the transfer matrix for independent sets on the m-wide strip.
Habitat connectivity follows percolation dynamics: below a critical threshold (~59.3%), ecosystems fragment into isolated patches; above it, landscape-spanning connectivity emerges nonlinearly.
Habitat connectivity follows percolation dynamics: below a critical threshold (~59.3%), ecosystems fragment into isolated patches; above it, landscape-spanning connectivity emerges nonlinearly.
Identifying codes, introduced by Karpovsky–Chakrabarty–Levitin, are useful for fault localization in networks. In the binary Hamming space (hypercube) Q_n, let M_r(n) denote the minimum size of an r-identifying code.